DYNAMIC portfolio optimization is one of the important


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1 , July 24, 2014, London, U.K. A Simulaionbased Porfolio Opimizaion Approach wih Leas Squares Learning Chenming Bao, Geoffrey Lee, and Zili Zhu Absrac This paper inroduces a simulaionbased numerical mehod for solving dynamic porfolio opimizaion problem. We describe a recursive numerical approach ha is based on he Leas Squares Mone Carlo mehod o calculae he condiional value funcions of invesors for a sequence of discree decision daes. The mehod is daa driven raher han resriced o specific asse model, also imporanly inermediae ransacion coss associaed wih porfolio rebalancing is considered in he dynamic opimisaion mehod, and invesors risk preferences and risk managemen consrains are also aken ino accoun in he curren implemenaion. In his paper, he presened mehod is used for a case sudy on a global equiy porfolio invesed in five equiy markes, and foreign exchange risks are also included. We examine he porfolio performance wih hree opimizers in a ouofsample simulaion sudy ogeher wih a benchmark porfolio which is passively managed wih equal weighed posiion. Index Terms Porfolio Opimizaion, Leassquares Mone Carlo, Approximae Sochasic Dynamic Programming, Opimal Asse Allocaion. I. INTRODUCTION DYNAMIC porfolio opimizaion is one of he imporan applicaions of decisionmaking under uncerainy in asse managemen. Invesors wih longerm views, paricularly major insiuional invesors such as pension funds, uni russ and muual funds, normally hold a diversified porfolio across a range of asse classes wih differen financial markes including equiies, bonds, properies, infrasrucure, hedge funds, ec. For invesors, heir objecives of invesmen may vary grealy, from seeking excess reurns over benchmark indices o hedging agains specific risk facors. For mos asse classes, asse reurns are uncerain and sochasic. There is a cerain degree of correlaion across differen asse classes and across differen geographical markes. Addiionally, he coss of ransacion can make a meaningful impac on asse reurns, so in porfolio rebalancing and risk managemen, ransacion coss should be included. Muliperiod dynamic porfolio opimizaion has increasingly become a popular approach, mainly due o he fac ha such muliperiod opimisaion schemes can now achieve soluions of manageable accuracy wihin accepable compuing ime. In [5], [6], he auhors used a Mone Carlo mehod and a Sample Approximaion Algorihm o consruc scenario rees, hen applied a sochasic programming algorihm o compue he opimal porfolio posiions for each branch of he scenario ree. A Taylor expansion of he value funcion up o second order a rebalancing ime was used in [7]. The auhors hen showed ha he opimal posiion can be calculaed recursively by a dynamic programming scheme wih a leassquare learning. Manuscrip received March 21, Chenming Bao, Geoffrey Lee and Zili Zhu are from CCI CSIRO, Ausralia. In his paper we inroduce a new compuaional mehod o solve he dynamic porfolio opimizaion problem numerically. The Mone Carlo mehod is used for simulaing a large number of hypoheical sample pahs of asse reurns and sae variables. We call hese sample pahs he raining se. The key idea is ha hese sample pahs should incorporae he invesors belief abou he sochasic properies of he fuure asse dynamics and sae variable. For example, he sample pahs may have an arbirarily complex marginal join disribuion, correlaion srucure, pahdepency, and nonsaionariy. Given he simulaed raining se, we can solve he opimal porfolio allocaion problem by using an approximae sochasic dynamic programming framework in he form of he Leas Squares Mone Carlo (LSM) mehod. LSM was inroduced iniially by [3] as a numerical mehodology o value American or Bermudian opions by a leassquares regression. In his paper we ex he LSM approach o solve a muliple swiching opions problem which also incorporaes he complex feaures of he inermediae ransacion cos and nonlinear uiliy funcions. In he paper, he porfolio weigh of each asse is resriced o discree incremen of equal inerval from 0% o 100%. Addiionally, we denoe a sraegy as one possible combinaion of he discree porfolio weighs of all asses subjec o porfolio operaion consrains. All he possible sraegies ha he invesor can adop as he opimal arge porfolio posiion a any rebalancing dae form he sraegy se. A similar approach for discreizing porfolios is used in [9] where a finie discreizaion mehod is used o ransform he coninuous mining operaional raes o a se of combinaions of discree operaional raes. In he Leas Squares Mone Carlo (LSM) implemenaion, each condiional value funcion for every sraegy in he whole sraegy se is approximaed as a linear combinaion of basis funcions, and is sored for opimisaion on each rebalancing dae. The opimal exercise boundaries for each sraegy form he opimal decision rules as a funcion of underlying risk facors and sae variables. Up o his poin, he LSM model has been fully calibraed. The calibraed LSM model of his paper can hen be used as a decision suppor ool for invesors in achieving opimal porfolio rebalancing for various scenarios. A any decision dae, he inpus of he calibraed model are values of realized or hypoheical underlying risk facors and pas porfolio sraegies. The arge opimal porfolio weigh can be chosen by comparing he coninuaion funcions for all he possible sraegies in he sraegy se. One of he key feaures of his porfolio opimizing algorihm is ha i can serve as an informaion ranslaor. An invesor may have a differen belief or forecas for fuure marke performances. The invesor s forecas abiliy may
2 , July 24, 2014, London, U.K. dep on all he qualiaive or quaniaive research or even luck. The opimal porfolio model implemened in his paper provides an efficien ranslaion of he marke forecas and risk managemen requiremen ino he corresponding porfolio. In he nex secion, we describe in deail he framework of he dynamic porfolio opimizaion problem and he numerical implemenaion of he algorihm for he approximae sochasic dynamic programming. In Secion III, we apply our mehod on a case sudy of a global equiy porfolio. Some compuaional resuls and implemenaion issues are discussed in Secion IV. Secion V concludes. A. The invesor s problem II. THE FRAMEWORK We consider he dynamic porfolio opimizaion problem a ime of an invesor. There are N asses ha he invesor can inves in, he uni price of buying or selling he ih asse a ime is given by S. i For each asse, he price S i deps on some underlying sochasic processes called risk facors on he probabiliy space (Ω, F, P). We consider he planning ime horizon wih mauriy dae T, and he porfolio can be rebalanced a a sequence of discree rebalancing daes, + 1, + 2,..., T. The invesor s problem is V (ω ) = max x {E [f (x, ω ) + V +1 (ω +1 ) F ], (1) ω F where x is a vecor of porfolio posiion on he N asses a ime, V is he value funcion a ime wih boundary condiion a he mauriy dae T : V T (ω T ) = 0 a.s.. The uiliy funcion f ( ) represens he invesor s preference of he porfolio performance which will be discussed furher in Secion III. The value funcion V can be seen as he expeced oal fuure uiliy of he invesor a ime wih condiion ha all he porfolio posiion weighs x, x +1,..., x T are opimally chosen wih respec o o all random evens ω s F s, s =, + 1,..., T. We assume ha he value funcion in Equaion (1) is he objecive funcion of he invesor. This is a ypical decision under uncerainy problem where he decision maker has o make a decision based only on he realizaions of hisorical performance of he porfolio and aking ino consideraion he dynamics of fuure scenario wih all he possible fuure decisions which would no be unveiled unil he fuure decision daes. The vecor of dynamic porfolio posiion x (ω) is a F  adaped random variable. The value of x is decided by all he informaion available a ime. This may include curren value of all risk facors and he porfolio hisory x 0, x 1,...x 1. B. Consrucing Sraegy Se The posiion vecor x = {w 1, w 2,..., w N a ime represens he weigh in percenage of he oal book size value of he porfolio invesed in asse S i, i = 1, 2,.., N. We discreize he posiion weigh w i of he asse S, i i = 1, 2,..., N in he following way: an msep discree grid is TABLE I STRATEGY SET FOR 5 ASSETS, 5STEP DISCRETIZE CASE. x (1) : (0, 0, 0, 0, 1) x (2) : (0, 0, 0, 0.2, 0.8) x (3) : (0, 0, 0, 1, 0) x (4) : (0, 0, 0.2, 0.8, 0) x (5) : (0, 0, 1, 0, 0) x (6) : (0, 0.2, 0.8, 0, 0) x (7) : (0, 1, 0, 0, 0) x (8) : (0.2, 0.8, 0, 0, 0) x (9) : (1, 0, 0, 0, 0) x (10) : (0.2, 0, 0.8, 0, 0) x (11) : (0, 0.4, 0.4, 0, 0) x (12) : (0.4, 0.4, 0.2, 0, 0) used o represen he porfolio weigh as (0, 1 m, 2 m,..., 1). The discreized porfolio weigh value vecor x are hen defined as all possible combinaions of he discree porfolio weigh values for all individual asses. Of course, we also have he condiion ha saisfy Σ N i=1 w i = 1. The invesor hus has a se of possible porfolio weighing posiions in vecor form, and each of he possible porfolio weigh composiion represens a socalled sraegy. The full se of sraegies for possible adopion can be lised as Θ = {x (1), x (2), x (3),... As an example, for a porfolio wih 5 asses o inves, each asse weigh is discreized by a 5sep grid (i.e. 0%, 20%, 40%, 60%, 80% and 100%), here are in oal 126 possible sraegies for poenial adopion. We lis some of he sraegies in Table I for his example case. C. Consrains Sraegies of he porfolio are subjec o some consrains. The firs consrain is on he posiion limis, his is he individual upper and lowerbound of he posiion (porfolio) weigh of each asse: w i [a i, b i ], i = 1, 2,..., N, a i < b i where a i, b i R. The limis for individual asse porfolio weigh or posiion may be defined by legislaions or from invesor s risk managemen requiremens. Invesors may be required o operae he porfolio under some hresholds of risk exposures in form of risk measures such as VaR. Oher consrains may apply on he urnover and resricions caused by he rading abiliy of he invesor. Also, large orders in he marke may adversely affec marke price movemen iself. Liquidiy consrains can apply for all rebalancing daes. Deping on assumpions on liquidiy, a minimum absolue liquidiy can be specified for he porfolio. For insance, he firs 20% of he local marke raded equiy is highly liquid whereas any amoun larger han 20% of he local marke equiy is assumed o ake more ime and slippage o liquidae. For simpliciy, we make an assumpion on liquidiy consrain by seing a maximum oal urnover so as o resric any possible large rebalancing rades wihin a shor period. D. Leas Squares Mone Carlo Mehod In his secion we describe he approximaion algorihm we use o esimae he expeced value funcion in Equaion (1). The inpu of he LSM model akes:
3 , July 24, 2014, London, U.K. n + s Mone Carlo simulaed sample pahs of he M underlying risk facors a ime = 0, 1, 2,..., T, {X (i,k), where i is he risk facor index, k is he sample pahs number and is he ime index. The firs n sample pahs is he raining se and we use he res for ouofsample ess. The realizaion of ω is a vecor of simulaed risk facor up o ime and all he decision hisory of he porfolio posiion x 0, x 1,..., x 1. The srucure of basis funcion L( ) and he runcaed order parameer K; The invesor s uiliy funcion f( ); A ransacion cos funcion T C(x 1, x, ω ); The N asses S 1, S 2,..., S N and he sep size 1 m for he discree weigh values in he porfolio; and, Some consrain condiions, we can wrie i as an indicaor funcion 1 (ω, x) which reurns a value of eiher 1 if he posiion x is wihin all he consrains oherwise value 0 is reurned a ime given all he sae variable informaion ω. {X (i,k) The firs sep is o consruc he Θ = {x (1), x (2), x (3),.... For any x = {w 1, w 2,..., w N, w i = 0, 1/m, 2/m,.., 1, if 1 (ω, x ) = 1 hen add x ino se Θ. The nex sep is o approximae he condiional value funcion: V (l) (ω ) = E [ f (x (l), ω ) + V +1 (ω +1 x = x (l) ) F ],(2) ω F. The condiional value funcion is he expeced value of he oal uiliy a ime given he sraegy a ime is x (l). Here, we approximae Equaion (2) by using a crosssecional leassquare regression scheme. Daa: {X (i,k), i = 1, 2,...,n Resul: ĉ,k iniializaion; Se ĉ (k) T,j := 0; for = T 1 o 1 do for j = 1 o size(θ) do Q j = n i=1 + max x (l) Θ K + ( f(x (j), ω +1 ) { f( T C(x (j) ĉ (k) +1,l L(k) (X (1,i) +1 k=1 K ĉ (k),j L(k) (X (1,i) k=1 ĉ (k),j = arg {ĉ min Q j ;, x (l) +1, ω +1)),..., X(M,i) +1 ) ) 2,..., X (M,i) ) Algorihm 1: Algorihm o esimae ĉ Algorihm 1 describes he algorihm in pseudo code form. The coefficien parameers c are esimaed by minimizing he sum of sample squared difference (funcion Q) beween he sample uiliy a ime wih he expeced condiional value funcion a ime + 1 wih sraegy l, given he sraegy l provides maximum value of he oal uiliy of ransacion cos and he condiional value funcion a + 1. Afer all he coefficien parameers for he basis funcions are esimaed, we hen calculae he opimal sraegy a ime = 0, he algorihm is described in psuedo code form in Algorihm 2. The algorihm compues he mean of he condiional value funcion a = 0 for all sraegies; he one ha gives he highes value is chosen as he iniial opimal sraegy. Daa: {X (i,k), i = 1, 2,...,n Resul: x 0 iniializaion; for j = 1 o size(θ) do for i = n + 1 o n + s do l = arg x (l) 0 EV (j) 0 + = f(x (j) 0, ω 1) + max x (l) 1 Θ1 { f( T C(x (j) 0, x(l) 1, ω 1)) +Σ K k=1ĉ (k) 1,l L(k) (X (1,i) 1,..., X (M,i) 1 ) { max EV (l) 0 Algorihm 2: Algorihm o esimae x 0 III. A CASE STUDY OF GLOBAL EQUITY PORTFOLIO We consider an invesor who manages an equiy porfolio invesed in five major equiy markes globally he Ausralia (AU), he Unied Saes of America (US), he Unied Kingdom (UK), he Japan (JP) and he emerging equiies markes (EM). The invesor operaes he porfolio from he viewpoin of he home currency. In his paper we assume he Ausralian Dollar (AUD) as he home currency; all valuaions will need o be convered o he home currency. Excep for he AU marke, he reurns from he four foreign equiy markes are subjec o foreign exchange risks. In his case we have o consider he dynamics of he foreign exchange raes for he currency pairs of he local currencies of he equiy markes and he home currency: EAUD USD US dollar o Ausralian dollar; E GBP Briish serling o Ausralian dollar; AUD EAUD JP Y Japanese Yen o Ausralian dollar; and A baske of Emerging marke currencies o Ausralian dollar. A. Daa Our daases will have an eighdimensional srucure: five equiy indices and hree exchange raes. Table II ses ou he daa used as proxies for he variables of he marke indices and heir local currency. We use he adjused closing price of equiy marke indices a he las rading day of he monh. We don have access
4 , July 24, 2014, London, U.K. TABLE II Asse Corresponding proxy Local currency AU ASX200 AUD US S&P 500 USD UK FTSE 100 GBP JP NIKKEI 225 JPY EM ishares MSCI Emerging Markes ETF USD o MCSI Emerging marke daa which is a common index represening he invesmen performance of he EM markes. So insead, we use he price of ishares MSCI Emerging Markes ETF raded in NYSE Arca (Ticker: EMM). In order o model he global equiy porfolio in home currency we need o include he dynamics of some foreign currency exchange raes. The ishares MSCI Emerging Markes ETF is raded in NYSE Arca which is accouned in USD. Therefore, he foreign exchange exposures from emerging marke currency o US dollar has been incorporaed ino he ETF rading price. For he currency risk, we herefore add 3 ime series daa of exchange raes of respecively he EAUD USD, EGBP AUD and. The daa series are he middlerae prices of bid and E JP N AUD ask for each exchange rae a he las rading day for each monh. The full range of daa runs from he las rading day of April 2003 o he las rading day of March 2014, represening 132 observaions for each index. All he marke daases described in his secion are from Yahoo Finance. B. Sochasic Models X c denoes a risk facor c valued a ime. c is a componen of he curren risk facor se: Ψ = {AU, US, UK, JP, EM, EAUD USD, EGBP AUD, EJP AUD N, he risk facors of equiy indices are valued in local currency uni of he global equiy markes. R is he eigh dimensional logreurn rae of risk facors X c, c Ψ: [ ( )] X c R = ln, X c 1 We assume he invesor uses an 8dimensional meanrevering model as he canonical model for R. R = R 1 + Ξ[R 1 µ] + Σ W (3) where Ξ and Σ are 8x8 real value marices, W is an eighdimensional Brownian moion wih zero mean vecor and a 8x8 uni covariance marix. In addiion o he canonical model in Equaion (3), he invesor could have his/her own forecas/view of he fuure asse reurns. These forecas/view can be readily incorporaed in calibraing he sochasic models. Addionally, more risk facors can be readily added ino he canonical model sysem. For example, addiional risk facors based on Arbirage Pricing Theory (APT) syle models [2] can be included. There can be hundreds of risk facors ha can be derived from echnical analysis, paern recogniion, saisical analysis, or behaviour finance, as documened in he lieraure for being significan in hisorical backess. The well known models are he FamaFrench hree facor models inroduced in [1]. TABLE III PORTFOLIO POSITION LIMITS.( % OF THE BOOK SIZE OF THE PORTFOLIO) equiy index upperposiion limi lowerposiion limi AU US 60 0 UK 60 0 JP 60 0 EM 50 0 Some of hese risk facors are insignifican in ouofsample saisical es, eiher due o speculaive rading agains hese facors (consequenly removing marke inefficiency) or due o he socalled error of he second kind (daa scooping) in research. In his sudy, he parameers µ, Ξ and Σ of he canonical model in Equaion (3) are esimaed hrough a sandard maximum likelihood algorihm on hisorical marke daase. However, in general, an invesor can choose o use heir own fuure forecas for hese risk facors o calibrae hese parameers. For discouning, bond yields in ermsrucure form can also be readily used in he curren implemenaion. For he example of his paper, we assume a consan bond yield, and use he Ausralia 10 Year bond yield which is 4.10% a he ime of his sudy (Febuary 2014). C. Transacion Cos and Consrains We assume a 50 basis poins proporional ransacion cos rae charged on absolue urnover of he porfolio a every porfolio rebalancing ime, given by T C = 0.5% Σ N i=1 Φ i (+) Φ i (), where N is he number of asses in he porfolio, Φ i () is he posiion of he asse i a ime before rebalancing and Φ i (+) is he posiion of he asse i a ime afer rebalancing. We ignore any slippage for he rebalancing and assume no shor selling and borrowing for all he asse classes. For simpliciy, in his example, we have prese posiion limi consrains for all he rebalancing ransacions. These consrains are lised in Table III. We also assume he oal urnover mus be less han 80 percen of he oal book size for each rebalancing. D. Invesor s uiliy We consider hree cases of differen risk preference of an invesor by defining differen uiliy funcions as he value funcion in Equaion (1). The firs uiliy funcion we consider is a linear uiliy funcion: u(w) = kw + c, (4) where k and c are consans. The linear uiliy represens an invesor wih no risk preference and only aims for he highes expeced oal reurn for he porfolio. The second uiliy funcion is a Power uiliy funcion which is given by: u(w) = w1 α, α > 0. (5) 1 α
5 , July 24, 2014, London, U.K. The Power uiliy funcion is also known as he consan relaive risk aversion (CRRA) uiliy funcion. I s a relaive measure of risk aversion, defined by wu (w)/u (w), and α is a consan. IV. RESULT AND DISCUSSIONS The algorihm presened here is implemened in sofware package RiskLab, and is used o compue he numerical resuls in his secion 1. Once he model parameers of hese risk facors are calibraed, we can proceed o calculae all he expeced value funcions a any fuure asse prices for all he porfolio sraegies a ime. A ime, if previous sraegy a ime 1 is known, we can readily selec he curren opimal arge porfolio sraegy. In oher words, he algorihm can be used as whaif forecasing ool for porfolio sraegies. For example, for any given scenario of he risk facors or a mahemaically generaed random even ω, he curren mehodology produces opimal arge porfolio posiions a every rebalance dae for given scenario evens. For his example, we firs generae 5000 Mone Carlo sample scenarios from he sochasic models of he 8 risk facors as he raining se, hen we generae anoher se of 5000 simulaion scenarios as he ouofsample daa o analyse he decision oupu for opimal porfolios. We also se up four invesmen syles for managing he porfolio: P0: A consan posiion sraegy wih equal weighed posiion on each of he five asses: 20% of he oal book size of he porfolio is allocaed o each of he five asses a he every rebalancing ime. P1: The invesor chooses o manage he porfolio hrough a linear uiliy funcion of Equaion (4). P2: The invesor selecs he risk aversion uiliy funcion of Equaion (5) wih parameer α = 5. P3: The invesor selecs he risk aversion uiliy funcion of Equaion (5) wih parameer α = 7. Table IV shows he porfolio performance: in values of CRRA uiliy funcions, achieved expeced excess reurns and volailiies over he 10 years invesmen horizon wih 5sep sraegy se (porfolio weigh seps). Firs we looked a he expeced oal reurns and volailiies over he enire invesmen horizon. The P0 invesmen syle gives 1.77% excess reurns agains he 10year bond yield, wih a 43.20% volailiy which is an average 13.66% for each year. The P0 porfolio follows a passively managemen syle and does no require decision making or marke forecasing by he invesor. We choose P0 syle invesmen as he benchmark porfolio. The P1 invesmen syle produces he highes expeced reurn bu also he highes volailiy among he four invesmen syles. Recall in his case sudy we use he same calibraed sochasic asse models o generae he Mone Carlo sample se for model raining and he ouofsample es. The overall superior performance in reurn from his P1 invesmen syle indicaes ha he algorihm has successfully capured he properies of he fuure asse dynamics hrough he raining daa se, and he superior performance in porfolio reurns for he ouofsample daa indicaes he implemened dynamic 1 RiskLab is a sofware package developed by CSIRO for asse modelling, simulaion, decision supporing, real opion pricing and porfolio opimizaion. TABLE IV STATISTICS OF THE PORTFOLIO RETURNS AND CRRA UTILITY FUNCTION VALUES FOR PORTFOLIO OPTIMIZER P0,P1,P2,P3 OF OUTTHESAMPLE TESTS. excess reurns opimizer CRRA5 CRRA7 mean volailiy P % 43.20% P % 56.99% P % 49.60% P % 45.85% porfolio opimizaion scheme achieved he objecive as se ou in he P1 invesmen syle. For he P1 syle invesmen, he high volailiy of he porfolio reurn brings down he calculaed corresponding CRRA uiliy value. An invesor wih risk preferences expressed in he CRRA uiliy funcion will find he P1 invesmen syle as no achieving he objecive maximizing he CRRA uiliy funcion. For he invesmen syle of maximizing he CRRA uiliy funcion, we have chosen o evaluae wo invesmen syles: P2 and P3 respecively for α = 5 and α = 7. The expeced value of excess reurns for he P2 syle is 6.66% which is lower han he P1 syle, whereas he reurn volailiy is reduced by 7.39%. This suggess he invesor following he decision rule given by P2 syle would up wih a lower risk in he form of reduced volailiy and smaller reurn han he P1 invesmen syle. The difference in expeced excess reurns can be seen as he risk premium paid o reduce he volailiy of he porfolio. A similar behaviour is observed for he P3 invesmen syle for which he CRRA uiliy funcion is adoped wih he parameer α = 7. The P3 syle invesmen obviously achieved he highes CRRA7 value. Ineresingly, we observe ha he calculaed CRRA7 value of he P0 syle invesmen is very close o he CRRA 7 value of he P3 syle invesmen. Correspondingly, he P2 invesmen syle maximizes he value of he CRRA5, or he uiliy funcion value wih parameer α = 5. A. Visualizaion of dynamic decisions One inuiive way o show he real ime dynamics of porfolio posiion changing wih respec o differen scenarios is by using a moion plo. Figure 1 shows a snapsho of he moion plo creaed by he visualizaion ool buil in he RiskLab sofware package. We have also generaed moion plos for he invesmen syles P1, P2 and P3, which can be accessed hrough he web link: hps://dl.dropboxuserconen. com/u/788580/presenaion/iaeng/googleembedded.hml. B. Basis funcions One imporan issue when applying he LSM algorihm is he choice of basis funcions. The selecion of basis funcions deps on he applicaion in hand. The work of [3] suggess Laguerre (weigh) should be seleced as he basis orhogonal funcion for single asse American pu opions. The robusness and convergence of LSM algorihms have also been an issue when selecing basis funcions. For example, [4] shows ha he LSM mehod is more efficien han eiher a finie difference or a binomial mehod when valuing opions
6 , July 24, 2014, London, U.K. parallel. The calibraion for each sraegy in he sraegy se a dae only relies on he resuls calculaed from he previous ime sep and can be done simulaneously in a parallel fashion. Thus he LSM algorihm has he poenial for speeding up he calibraion process significanly by adoping mulihreads programming such as on he GPU raher han he raw sequenial calculaion we currenly use. Fig. 1. A moion plo of 30 seleced sample pahs opimal weigh calculaed by P2 TABLE V PERFORMANCE OF RISKLAB PORTFOLIO OPTIMIZER ON SELECTED TEST CASES number of scenarios calculaion ime in minues calibraion opimizaion calibraion opimizaion P P P on muliple asses, and Monomials are suggesed as possible basis funcions. For he case sudy of his paper, we have esed LSM wih a se of differen basis polynomial funcions including Laguerre, Nominal, Hermie, Hyperbolic, Legre. We use he oal sandard deviaion of he leas square residuals as a measure of goodnessoffi. We observe ha essenially all he he esed orhogonal funcions provide comparable resuls. For his paricular example, Laguerre polynomials wih order greaer han 3 provide he lowes error among he 5 basis polynomial funcions. As a sandard approach for selecing a numerical basis approximaion funcion, we sugges esing muliple possible orhogonal funcions for each new applicaion before choosing he appropriae basis funcions. C. Compuaional ime Table V shows he calibraion and opimizaion ime running on a PC wih Inel Core i GHz, 4 GB ram, compiled using Visual C bi version. One can see from he figures of Table V, he calibraion phase for 5000 simulaed sample pahs ook more han four hours o finish. The compuaion ime also deps on he size of he sraegy se and he number of he risk facors. We have also esed for using smaller number of sample pahs, differen number of sraegies in he sraegy se and basis funcions. (Resuls no lised.) The compuaional ime for calibraion is asympoically linear wih respec o he number of sraegies and he number of simulaion sample pahs. Once he LSM model is calibraed, i normally akes less han a few seconds for he LSM algorihm o calculae he opimal porfolio posiions for he 5000 ouofsample scenarios. I is worh noing ha he calibraing process for he LSM model, as described in Algorihm 1, can be performed in V. CONCLUSION We have presened a simulaionbased numerical mehod for solving dynamic porfolio opimizaion problems. There is no resricion on he choice of asse models, invesor preferences, ransacion cos, and liquidiy posiion consrains. We have applied he mehod o managing an equiy porfolio invesed across five global equiy markes. For he case sudy shown in his paper, he views of an invesor on fuure marke reurns is modelled and calibraed by a mulifacor meanrevering process wih eigh risk facors, and auo and cross asse correlaion srucures are also considered. Four invesmen syles are chosen in he es case, and a Leas Square Mone Carlo approximaion mehod has been developed o calibrae he dynamic porfolio model. Through he es case, we have shown ha he hree dynamic invesmen syles ouperform he benchmark porfolio for ouofsample ess. Viewed on a meanvariance plane, he performance of he dynamic porfolios are locaed on a new efficien fronier whereas he benchmark saic porfolio is less efficien wih a higher risk premium. Some compuaional issues wih he LSM model have also been discussed. REFERENCES [1] E.F. Fama and K.R. French The CrossSecion of Expeced Sock Reurns, The Journal of Finance, number. 2, volume XLVII, June 1992 pp [2] R. Grinold and R. Kahn Acive Porfolio Managemen: A Quaniaive Approach for Producing Superior Reurns and Conrolling Risk,(2nd ed.) McGrawHill, [3] F.A. Longsaff and E.S. Schwarz Valuing American opions by simulaion: A simple leassquares approach, Review of Financial sudies, number. I, volume 14, 2001 pp [4] L. Senof Assessing he Leas Squares MoneCarlo Approach, Review of Derivaives Research, number. 7, 2004 pp [5] G. Consigli, M.A.H. Dempser, Dynamic sochasic programming for asseliabiliy managemen, in Annals of Operaions Research , Volume 81, Issue 0,, pp [6] M.A.H. Dempser and M. Germano and E.A. Medova and M. Villaverde Global Asse Liabiliy Managemen, in Briish Acuarial Journal, 9, pp doi: /s [7] M.W. Brand and A. Goyal and P. SanaClara and J.R. Sroud A Simulaion Approach o Dynamic Porfolio Choice wih an Applicaion o Learning Abou Reurn Predicabiliy, in Review of Financial Sudies 18, 2005, pp [8] C. Bao and Z. Zhu Land use decisions under uncerainy: opimal sraegies o swich beween agriculure and afforesaion, in MOD SIM2013, 20h Inernaional Congress on Modelling and Simulaion. Modelling and Simulaion Sociey of Ausralia and New Zealand, December 2013,, pp ISBN: [9] C. Bao and M. MorazaviNaeini and S. Norhey and T. Tarnopolskaya and A. Monch and Z. Zhu Valuing flexible operaing sraegies in nickel producion under uncerainy, in MODSIM2013, 20h Inernaional Congress on Modelling and Simulaion. Modelling and Simulaion Sociey of Ausralia and New Zealand, December 2013,, pp ISBN:
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