PORTFOLIO OPTIMISATION

Transcription

1 OptRsk Systems: Whte Paper Seres Doma: Face Referece Number: OPT 002 PORTFOLIO OPTIMISATION Last Update 24 Aprl 2008

2 Portfolo Optmsato AStatusRevew November 200 Ths whte paper comprses two parts: Portfolo Optmsato (Part I) The rse of Markowtz Mea-Varace Models ad Beyod Prepared by: M. Prbha Portfolo Optmsato (Part II) The Markowtz Mea-Varace Model: A Techcal Perspectve Prepared by: M. Guertler, M.S. Med The three authors are PhD research studets the Departmet of Mathematcal Sceces, Bruel Uversty ad ackowledge Professor C. Ioads, Dr. C.A. Lucas ad Professor G. Mtra. Cotets Scope of whte paper Itroducto A Itroducto to H. Markowtz work o Moder Portfolo theory Asset Prcg Models What portfolo optmsato software are avalable Cocludg Remarks Itroducto ad Backgroud Mea-Varace Models Real-World Extesos Factor Models Idex Trackg Rollg Markowtz Backtestg Cotact Detals: Departmet of Mathematcal Sceces Bruel Uversty Uxbrdge UB8 3PH Mddlesex Eglad

3 Scope Ths whte paper troduces Markowtz mea-varace model wth a geeral overvew ad sets out to expla why ad how the face dustry has fully embraced ths as method of choce for portfolo plag. The ma focuse of the whte paper s to brg out may aspects of the portfolo plag problem whch are addressed by ehaced mea-varace models that meet the growg requremets of the face dustry. Portfolo aalyss s a leadg ssue wth fud maagers who apply such models may stuatos such as dex trackg, performace evaluato ad hstorcal data/backtestg. The techcal uderpg of these methods are descrbed Part II. A umber of curretly avalable software systems are also revewed together wth a broad overvew of usage of these systems Part I. Ths whte paper wll be of terest to: Fud maagers Tradg desk staff Back offce staff Quattatve aalysts who wsh to kow the geeral developmet the market place.

4 Portfolo Optmsato (Part I) I the last few years, the vestmet commuty has expereced revolutoary chages o may frots. It has bee felt more so the mdst of the portfolo fud maagers, hedge fuds, ad wealthy tycoosthough t s ow spreadg eve to the commo prvate vestor. I ths artcle, we dscuss the chagg methodologes ad techologes fdg ther way to may aspects of the fud maagemet dustry of today. So what are the uderpg reasos that are brgg a about-face the day-to-day lfe of the dustry? I a world where competto s the buzzword ad volatlty om-preset the facal dustry, a reasoable level of returs s a task dffcult to be acheved wthout pror plag ad evaluato of vestmet alteratves. The emergece of progressvely sophstcated techologes has hasteed the modersato of the facal dustry. So how has the market reacted to the ew wave of techology avalable to t? If the umber of operators marketg ther servces s a bechmark to measure the popularty of optmsato software, the we could safely say that they have become a weapo of choce the fud-maagers arseal. I the ext sectos, we frst portray Professor Harry Markowtz s poeerg work o moder portfolo theory. We follow ths up by a road-map of the roads made by the theory of Markowtz. The we brefly gve a overvew o the several models used the facal marketplace such as CAPM, APT ad QP models. Fally we sum up wth the ma optmsato software curretly avalable the facal software marketplace. A overvew of H. Markowtz work o Portfolo Theory Moder Portfolo Theory (MPT) was troduced by Harry Markowtz wth hs paper "Portfolo Selecto" whch appeared the 952 Joural of Face. Thrty-eght years later, he shared a Nobel Prze wth Merto Mller ad Wllam Sharpe for what has become a broad theory for portfolo selecto ad corporate face. It has to be stressed that Harry s theory was scatly used utl the late 80 s whe ts popularty grew. MPT explores how rsk-averse vestors ca costruct optmal portfolos takg to cosderato the tradeoff betwee market rsk ad expected returs. The theory quatfes the beefts of dversfcato. Out of a uverse of rsky assets, a effcet froter of optmal portfolos ca be costructed. Each portfolo o the effcet froter offers the maxmum possble expected retur for a gve level of rsk. Ivestors should hold oe of the optmal portfolos o the effcet froter ad adjust ther total market rsk by leveragg or deleveragg that portfolo wth postos the rsk-free asset such as govermet bods. MPT provdes a broad cotext for uderstadg the teractos of systematc rsk ad reward. It has profoudly shaped how sttutoal portfolos are maaged, ad motvated the use of passve vestmet maagemet strateges. The mathematcs of MPT s also used extesvely facal rsk maagemet. Also, ote that the Markowtz model s a sgleperod approach. It s assumed that a vestor has a gve tal edowmet to vest. The vestmet wll be held for a specfc legth of tme referred to as the vestor s holdg perod. At the ed of that perod, the vestor wll lqudate hs holdgs ad wll ether re-vest t or use t for hs ow cosumpto eeds (or a combato of both). That s otherwse kow as a fxed mx or a buy-ad-hold strategy. Hece, we ca wrte the equato of the holdg-perod rate of retur o a securty as: Retur = (ed-of-perod accumulated wealth - startg-perod wealth)/ startg-perod wealth Asset-Prcg Models Markowtz work sparked off further research the area of asset prcg models face. We cosder two well establshed models; the Captal Asset Prcg Model (CAPM), ad the Arbtrage Prcg Theory (APT). The Captal Asset Prcg Model (CAPM) was frst troduced by Wllam Sharpe 964. It exteded Moder Portfolo Theory to clude the otos of Portfolo Optmsato Page

5 systematc ad specfc rsk. CAPM cosders a smplfed world where: There are o taxes or trasacto costs. All vestors have detcal vestmet horzos. All vestors have detcal perceptos regardg the expected returs, volatltes ad correlatos of avalable rsky vestmets. I such a smple world, Tob's super-effcet portfolo (see Captal Market Le) must be the market portfolo. All vestors wll hold the market portfolo, leveragg or deleveragg t wth postos the rsk-free asset. CAPM dvdes the rsk of holdg rsky assets to systematc ad specfc rsk. Systematc rsk s the rsk of holdg the market portfolo. As the market moves, each dvdual asset s more or less affected. To the extet that ay asset s affected by such geeral market moves, that asset etals systematc rsk. Specfc rsk s the rsk, whch s uque to a dvdual asset. It represets the compoet of a asset's volatlty, whch s ucorrelated wth geeral market moves. Total Rsk= Systematc Rsk + Specfc Rsk Accordg to CAPM, the marketplace compesates vestors for takg systematc rsk, but ot for takg specfc rsk. Ths s because specfc rsk ca be tackled through dversfcato. Whe a vestor holds the market portfolo, each dvdual asset that portfolo etals specfc rsk, but through dversfcato, the vestor's et exposure s just the systematc rsk of the market portfolo. Systematc rsk ca be measured usg a parameter beta- whch measures the rsk of a specfc stock relatve to the market as a whole. The expected excess returofaportfoloabovethersk-freeratesjustthe portfolo's beta multpled by the expected excess retur of the market portfolo. The Arbtrage Prcg Theory (APT) s a model of facal strumets ad portfolo behavour based o the proposto that f the returs of a portfolo of assets ca be descrbed by a factor structure or model, the expected retur of each asset the portfolo ca be descrbed by a lear combato of the factors wth the returs of the asset. The factors ca be statstcal artfacts; they ca be market or dustry related; or they ca be macroecoomc varables such as terest rates, flato, dustral producto, etc. The resultg factor model ca be used to create portfolos that track a market dex, to estmate ad motor the rsk of a asset allocato strategy, or to estmate the lkely respose of a portfolo to ecoomc developmets. Startg from a tal model proposed by Stephe Ross, APT models have bee created for applcatos most cash ad dervatves markets. What portfolo optmsato software are avalable A umber of portfolo optmsato software are avalable for a quattatve aalyst tet o usg the latest techologes as a ad for decso-makg. The followg s a represetatve lst, though by o meas exhaustve. We wll start wth Lattce Facal. The compay tres to reder ts optmsato software more realstc. It cludes some busess realtes to the software--- such as taxes, legal requremets ad varable trasacto costs. Lattce also clams ts software s advaced as t allows someoe to eter the realm of o-covex problems oce all mportat factors have bee modelled. There s the avalablty of Dyamc Optmsato Tools, whch the compay clams to be a sophstcated optmsato techology that utlses a combato of optmsato algorthms to teract wth a smulato model. Other key features of the Dyamc Optmzato Tools are: They ca be tegrated to other systems, Ther hgh-speed ad accuracy, They cover a wde rage of applcatos, ad, They solve o-lear, ocovex problems, as well as teger problems. The secod software o offer s the Wager Math Face software. WAoptmze s the persoal verso of WMF portfolo optmsato software. It provdes the dvdual vestor wth sophstcated Portfolo Optmsato Page 2

6 aalytcs for rebalacg ther portfolo. There are two key quattes every vestor seeks to cotrolretur ad rsk, ad we kow that the key to cotrollg rsk s dversfcato. But t s dffcult to reduce your rsk whle matag the same expected retur. WAoptmze was desged specfcally to address ths asset allocato problem. The user terface s a Mcrosoft Excel workbook, whle a customzed dyamc lk lbrary (DLL) based o WMF versatle MVO Lbrary performs the portfolo optmsato. Advaced features clude addtoal lear costrats ad hedgg strateges. The most commo use of lear costrats s for placg upper ad lower bouds o holdgs for a group of assets; for example, you wat o more tha 30% of your portfolo eergy stocks. Wager offers Waoptmze-C to meet ths eed. Ivestors/employees who wsh to hedge ther postos-for e.g as a result of owg share optos from ther compaes by usg optos, could make use of Wager s Waoptmze-H to address ths specalzed problem. Below s a lst of stadard features for Waoptmze. Some of the ehaced features lsted here are stadard for M-V Optmzer, Wager's other Mcrosoft Excel-based optmzer. WAoptmze ca be stalled o the Wdows system. The features are: Hadles up to 50 assets/asset classes User specfed expected returs ad volatltes User specfed correlato matrx Dsplays etre effcet froter Hadles log ad short postos assets User specfed asset-specfc mmum ad maxmum holdg costrats User specfed bechmark portfolo for comparso purposes Computes optmal portfolo from a user specfed target retur Computes optmal portfolo from a user specfed target volatlty Computes optmal Sharpe Rato portfolo. There are also some advaced features such as: Up to 25 geeral lear costrats (WAoptmze-C) More assets/asset classes (up to 250) Proportoal trasacto costs Trackg portfolos Calculate correlato structure from tme seres of returs Hedgg strateges (WAoptmze-H) The ext oe we cosder s Fportfolo.com. It provdes gudace through every step of the vestmet decso-makg process wth a hghly persoalzed ad teractve sute of facal plag ad vestmet aalyss tools: The attrbutes are: FPortfolo's Strategc Asset Allocato helps you determe optmal mx of assets (such as Large Cap stocks, Small Cap stocks, Iteratoal stocks, U.S. Bods ad Cash) that wll comprse your vestmet. FPortfolo's results page allows you to teractvely chage your Ital Ivestmet, Tme Horzo, Yearly Cotrbuto, Tax Impact (Taxable, Tax Deferred, or No Tax), ad the amout of rsk you are wllg to take. Your projected wealth ad recommeded asset allocato wll reflect real-tme ay chages you make to these assumptos. The software s Premum Servce allows you to place costrats o asset classes, eablg you to esure a mmum or maxmum weghtg of a gve asset class wth portfolo. Moreover, ts Portfolo Asset Selecto module gves you tremedous power ad flexblty screeg prospectve portfolo assets, allowg you to select agast a sgle crtera, a bechmark, or a combato of crtera. Portfolo Optmsato Page 3

7 The resultg assets from your asset selecto query ca be added drectly to ay of your exstg portfolos. Quattatve asset selecto tools lke these caot be foud aywhere else o the Web. Portfolo Optmzato s hghly teractve results page eables you to see, real-tme, the asset allocatos ecessary to acheve a rage of optmal portfolos alog the effcet froter. I addto, ts Premum Servce allows you to place costrats o portfolo assets, gvg you a added dmeso of cotrol over your portfolo. Ths s especally mportat f, for reasos of polcy or preferece, you wat to lmt how much or how lttle of your portfolo a gve asset represets. FPortfolo's Premum Servce allows you to set alerts. Our automated system wll otfy you whe your portfolo's degree of optmzato drops below a certa percetage. Also, ts Portfolo Rsk Aalyss module calculates the hstorcal average aual retur o your portfolo ad the varablty or ucertaty of that retur, ad provdes you wth a rsk-adjusted retur rato (the Sharpe rato) to compare your portfolo to others ad to the market. FPortfolo's Retur-based Style Aalyss module forms you to what extet your portfolo s behavg lke each of the major asset classes. I.e., t tells you what combato of holdgs across varous style dexes would have most closely replcated the actual performace of your portfolo. Whle Retur-based Style Aalyss s typcally appled to dvdual mutual fuds, FPortfolo takes the techque oe step further, ad apples t to your etre portfolo of mutual fuds ad stocks. FPortfolo's ovatve Asset Aalyss module eables you to coduct a quck aalyss of assets that you may be cosderg for cluso a portfolo. All securtes FPortfolo's database are updated daly so that you have the most up-todate formato o the rsk ad retur characterstcs of each securty. Cotug our survey, we preset the APT System produced by APTLtd whch s essetally a mplemetato of the Arbtrage Prcg Theory. Whle the APT theorem (Ross, 976) focuses o expected asset prces a arbtraged market, oe of ts most powerful mplcatos cocers prce fluctuatos across these assets. APTLtd provdes software ad rsk models for Rsk Measuremet Portfolo optmzato Rsk Attrbuto Performace Aalyss I estmatg rsk ( the form of covarace matrx) APT uses a algorthm whch helps to avod problems lke cocetrato rato" leadg to based estmate of the covarace matrx. These are problems that plaque the stadard prcpal compoet aalyss. They are able to detfy rsk attrbuto: shared ad specfc compoets. Products volve: APTPro 5 whch provdes a rage of portfolo maagemet fuctos from rsk measuremet, through portfolo optmzato to rsk attrbuto ad performace aalyss. It offers a theory-drve way to attrbute rsk to multple types of famlar real world factors all wth the accurate framework of a statstcal factor model. By mappg real world varables oto the APT Factors, APTPro s capable of performg the same sort of famlar rsk attrbuto foud fudametal models, but matag the accuracy of the APT Factors. Portfolo Optmsato Page 4

8 APTLte s a coveet portfolo maagemet tool for optmzato, ad rsk ad performace measuremet. Attrbutes are: Rsk Measuremet wth the hghly accurate APT statstcal factors Portfolo Optmzato usg APT Factors ad the APT Quadratc optmser Performace Aalyss assesses how your vestmet decsos affect your portfolo performace. APTStore Is a fully programmable, mult-user database maager for storg ad rapdly queryg hstorcal facal data.itseaslytegratedtoexstgdatasystems because of ts user-defable mport/export stadard. We proceed wth the key features of Northfeld Iformato Servces Ic. They are as follows: It s possble to use ay factor of rsk wth a ulmted umber of factors, It eables the user to perform may optmsatos at a tme, There s the possblty to corporate chages dustres, sectors ad coutres of ay securty the data fle, Ca apply arbtrary quadratc pealtes ad hard costrats o ay user-defed descrptors, Coforms the optmsato to pre-specfed trackg error targets, The objectve fucto takes lot-by-lot captal gas to accout, mmzg taxes whle allowg actve strateges: Ca make use of log/short strateges, The corporato of trasacto costs ad taxes. We wrap up wth the software from UNICOM. There s also a bref descrpto ad tercoecto of part of the work from the paper Computatoal aspects of alteratve portfolo selecto models the presece of dscrete asset choce costrats (NJ Jobst, MD Horma, CA Lucas ad G Mtra) the Quattatve Face Volume I (200). The FORTMP/QMIP s desged to use both the teror pot method ad the sparse smplex (SSX) soluto capabltes. Moreover, the system has udergoe extesve tests by usg QLIB test data (Maros ad Meszaros 997) ad models from the face dustry. To get a realstc vew of the portfolo plag problem, some dscrete restrctos have bee placedsuch as: () A buy- threshold whch bascally s the mmum level below whch a asset s ot purchased, (2) The placg of cardalty costrats where the umber of assets to be purchased s specfed, (3) Makg roudlots trasactos. There s a vestgato of the shape of the Dscrete Costrats Effcet Froter (DCEF) whe corporatg the above costrats. The models are mplemeted usg MPL, a mathematcal programmg laguage. The above authors try to exted the cardalty model to address the portfolo re-balacg problem. Ther am s to detfy the trades requred to adjust the tal asset holdgs such that the optmsed portfolo ( terms of varace) tracks a target portfolo or dex. What the authors of the paper above foud out s that the use of quadratc mxed-teger problem eabled the use of dscrete costrats ad thus captured mportat characterstcs of real-world problems. They also hghlghted the dscotutes- for DCEFthat arose as a cosequece of the mposto of dscrete costrats. Moreover, by teger restartg the QMIP wth the prevous soluto, they are able to geerate a reasoable umber of optmal ad ear optmal pots wth a restrcted brach-ad-boud Portfolo Optmsato Page 5

9 search. They also troduced a reoptmzato heurstc, whch proved to be computatoally very effcet costructg parts of the DCEF. We ow tur o to the may merts of the soluto systems: The soluto of covex quadratc programs by SSX (Mtra 976) ad Joes ad Mtra (997) are ow deeply-rooted. A brach-ad-boud solver, whch uses SSX, s by far the most appealg aveue. The other very good features are: Speed. I fact, the QMIP problems are solved to the secod, mprovg, feasble teger soluto subject to a lmt of 500 odes the brach-ad-boud algorthm. Accuracy level. The results are better; for each data set, the mea error s below 0.02% wth the meda error below 0.05%. Scalablty. There s the possblty to use a szeable umber of assets the systemprovgawaytsrobustess. Below s a dagram depctg the Data-modellgsolver archtecture: Cocludg Remarks So are we gog to see a frezy towards these techologcally advaced optmsato software? Well, t wll take a bt more tme for people to realse the help ad mportace of such software. Furthermore, the success of these optmsato tools depeds o the extet to whch t replcates real-world stuatos. Gve that vestmet strateges may be very proftable before takg to accout trasacto costs ad taxato ssues, t s very mportat that these factors are fully corporated to reflect the stuato o the groud. Oe thg s however assured- the gee s deftely out of the box ad rapd developmets the feld wll crop up; wth the evtablty that dfferg software varyg magtudes wll be avalable for the fud-maagers to acheve ther vestmet strateges ad goals. Nobody may realse ths, but may wll be spolt for choce- ther selecto for the software they wat to acqure. Excel/VBA -datastorage -drvg applcato Adjusts MPL/AMPL Model fle reads data calls MPL/ AMPL reads soluto seds to Soluto fle results FortMP/ QMIP Portfolo Optmsato Page 6

10 Portfolo Optmsato II Itroducto ad Backgroud Portfolo Optmsato s the process of aalysg a portfolo ad maagg the assets wth t, to obta the hghest retur gve a level of rsk. Ths paper ams at portfolo maagers, fud maagers, or dvduals wshg to assess the mpact of dfferet vestmet strateges or to actually trace out a vestmet strategy for a pre-specfed vestmet tme horzo. The basc portfolo optmsato theory hges o the dscrete tme, cotuous outcome paradgm otherwse kow as the mea-varace or Markowtz paradgm. I 952, Harry Markowtz troduced ths approach, whch s wdely used applcatos volvg vestmet portfolos. Mea-varace theory assumes that amog portfolos wth the same stadard devato, the oe wth the greatest expected value s the most effcet. Effcet the sese that for a specfed level of expected retur, the correspodg rsk s mmzed; alteratvely, for a gve level of rsk, t yelds the hghest expected retur. He showed how ratoal vestors could buld optmal portfolos uder codtos of ucertaty by usg statstcal measures for expectato ad varace of retur. Ths set of portfolos s kow as the effcet set ad ca be detfed by solvg a parametrc quadratc program. I the rsk-retur space, the effcet set forms the so-called effcet froter. However, the real world vestors are terested extedg the basc mea-varace approach wth restrctos such as lmtg the umber of trades, defg a mmum level of trade for a asset, reducg taxato costs, etc. Descrbed portfolo optmsato s statc ad useful to bult a tal portfolo. Over tme, the portfolo s rebalaced to justfy the actualtes of the stuato the. As a techque for evaluatg the qualty of the portfolo strategy, backtestg s used. The essece of the techque s to compare actual tradg results wth model-geerated measures ad help to refe portfolo optmsato techques. Mea-Varace Models Two mea-varace (MV) models are descrbed below. The frst MV model mmzes the rsk subject to a desred level of retur whereas the secod mmzes ad maxmzes the rsk ad retur respectvely.. The followg MV model determes the effcet portfolo set out of the gve vestmet opportuty set gve a specfed level of retur QP: Mmze Subject to = j= x x jσ j r x = d = x = x 0 () = (2) =,..., (3) where r defes the expected rate of retur of asset, d the level of retur for the portfolo, σ j the covarace betwee asset ad asset j ad x the fracto of the portfolo value vested asset. The objectve fucto mmzes the covarace term, whch tur mmzes the rsk of the portfolo. Costrat () specfes the retur expected from the portfolo ad costrat (2) assures that the whole budget s vested. By specfyg a level of expected retur o the portfolo, the above quadratc model computes the correspodg mmum rsk of the portfolo. The framework s demostrated the fgure below. Hstorcal data s aalysed ad reorgazed to a collecto of aalytcal data otherwse kow as a datamart. The mea-varace model s the bult for a Portfolo Optmsato Page

11 certa desred retur level ad passed to the optmser, whch works as the plag ege. Hstorcal Data Datamart Mea Varace Model Rsk Averso Optmsato Uless otherwse stated, we use a dataset cosstg of 60 mothly returs for stocks from the FTSE00 to llustrate effcet froter for the model QP descrbed. Retur Rsk Fgure 2: Cotuous effcet froter Hedged Decsos Fgure : Data flow chart Solvg the model for alteratve values of retur level d ad plottg the mmum varace agast the retur gves a curve cotag a set of effcet portfolos. Ths curve s called the effcet froter. These portfolos have the lowest rsk for a gve level of retur. The secod MV model approach may be stated as follows mmzg the rsk ad maxmzg the retur: QP2: Real World Extesos () Addtoal restrctos ca be troduced to the geeral structure to capture some of the real problems faced durg portfolo plag. Buy- Threshold Costrat l δ x u δ =,..., The costrat defes the mmum level at whch a asset ca be purchased. It elmates the problem where urealstcally small trades ca be cluded a optmum portfolo. Addg ths costrat chages the effcet froter as follows Mmze Subject to λ = j= x = x 0 x x σ ( λ) j j = r T = () =,..., (2) x Retur Rsk where λ s a parameter 0<λ<. By varyg the parameter λ betwee zero ad oe, the effcet froter s computed. Fgure 3: Buy- effcet froter Portfolo Optmsato Page 2

12 Cardalty Costrat = δ = k =,.., Such a costrat restrcts the umber of stocks allowed the portfolo ad forms the followg effcet froter. Retur Roudlots Rsk Fgure 4: Cardalty effcet froter Roudlots are restrctos used to defe the basc ut of vestmet. Ivestors are allowed oly to make trasactos multples of these roudlots. Whe modellg trasacto roudlots, the portfolo weghts ca be represeted as x = y f =,..., where f s a fracto of portfolo wealth ad y teger umber of roudlots. s a I order to represet threshold ad cardalty costrats, zero-oe varables are troduced ad to realze roudlots, teger varables are requred. As a result the QP model becomes a Quadratc Mxed Iteger Programmg (QMIP) model. Ths creases the sze ad complexty of the problem. Addg these costrats to the QP model, gves the followg QMIP formulato: = y f + lδ y f uδ =,..., = y δ x + ε ε = (2) (3) δ = k (4) teger bary, ε, ε + 0 =,..., (5) =,..., (6) =,..., (7) The effcet froter s plotted as follows Retur Rsk Fgure 5: Roudlots effcet froter The estmato of the uderlyg parameters.e. the expected returs, varaces ad covaraces, s very mportat. Already small chages these puts data ca produce a large effect o the optmal portfolo weghts (see Chopra ad Zemba 993). The expected mothly rate of retur ca be estmated from the mea seres of past mothly returs. The covarace betwee two assets ca be estmated o the bass of sample covarace over the same seres of returs. The estmato of expected returs as sample estmates based o hstorcal tme seres data ca be urelable estmates of the future. Mmze = j= y f y j f σ j j + γε + γε Subject to r y f = d = () Portfolo Optmsato Page 3

13 Real World Extesos(2) Sector/Idustry (jot) Costrat A further costrat the portfolo model s a sector, dustry restrcto. The set of assets s grouped varous sectors respectvely dustres whch have a mmum ad maxmum exposure. I I I I j Sector costrat : uverse of assets : set of stocks whch belog to sector j ' I ' j ' I ' j x L ' x U ' Where L j deotes the mmum exposure sector j ad U j the maxmum exposure. Immuzato Durato Matchg Immuzato s a portfolo strategy that matches terest rate rsk of a asset portfolo agast the projected stream of labltes to acheve zero et market exposure. Cashflow Matchg costrat m P x = = ' ' P Wth P the portfolo cashflow ad P L the lablty cashflow stream. Uder ths codto, ad assumg revestmet rates do ot chage from the assumed values, the portfolo cashflows wll match the lablty cashflows. There s o guaratee that the ecessary codto wll hold whe the rates chage. A frst order estmate of the chage of the preset value of a stream of cashflows, ether a asset or lablty, wth chages the term structure s measured by durato. Hece, a frst-order suffcet codto for portfolo mmuzato s that the durato of assets ad labltes match. L Durato Matchg costrat m = D FW F x = D FW L Where D DW s the Fsher-Wel durato of the labltes. Taxato Taxable portfolos represet a specal challege to the vestmet maager. The mposto of taxes such as captal gas ca make may strateges that wll be proftable o a pre-tax bass to certa losses o a after-tax bass. Ivestmet maagers must smultaeously ot oly cosder the retur ad rsk aspects of each portfolo but also the tax crcumstaces of the vestor. Strateges to address tax effcecy must have the followg propertes: Reduce the cost of taxes, whch s spread betwee pre-tax ad after-tax returs to vestors. Provde mmum hbto (.e. ot lmtg turover) to the portfolo maagemet process, so as to allow actve maagemet strateges to be effectvely pursued. Tax effcet strateges must such that they ca be mplemetedaautomatedfashoorderto lesse the burde of hadlg (potetally) hudreds or thousads of accouts. A trcky aspect of the taxable accout problem s that realty, t s a mult-perod problem, whereas the tradtoal mea-varace Optmsato looks at the futureasasgleperod.thelogcofthetaxmodels s usually captured usg dscrete 0- varables. A trasacto amortzato factor ca be troduced to deal wth the mult-perod aspect of ths process. The choce of the factor wll deped o the tax stuato as well as the market outlook of the partes volved. F L Portfolo Optmsato Page 4

14 Factor Models To compute the optmal weghts for a mea-varace problem, estmates of the expected retur ad the covaraces betwee the securtes the avalable populato s eeded as put. Oe way of makg these estmates s by samplg from past returs. Though ths s the most straghtforward approach to obtag these estmates, t has some problems. Sample meas of stock returs are urelable estmates of the true expected retur. I order to reduce samplg error, sample estmates requre also a legthy hstory of past returs. Ths has the ufortuate cosequece that the farther oe goes back tme, the more lkely the seres of the stock returs does ot reflect the cotemporary character from ts past. The frm today may have sgfcatly dfferet character from ts past. Accordgly, sample estmatesmayprovetobeurelableestmatesofthe future. Better estmates of covaraces ad expected returs ca be obtaed by employg factor models. Sgle factor models are such that the covarace betwee the returs o stocks ca be attrbuted to a sgle factor, usually a market dex. I multfactor models, the covaraces betwee the stocks are attrbuted two or more factors. The factors ca be detfed as ay varable that flueces the retursothestocks.examplesoffactorscludethe market dex, growth dustral producto, chage the rate of employmet, terest rates or flato rates. The varace of returs s made up of two compoets, the systematc rsk ad resdual varace. Systematc rsk s a rsk that caot be dversfed ad resdual varace represets rsk that s uque to a orgazato ad ca be elmated through dversfcato. Captal Asset Prcg Model (CAPM) The Captal Asset Prcg Model (CAPM) s a prme example of a sgle factor model, whch represets the relatoshp betwee a o-dversfable rsk of vestmets ad expected retur. The paradgm uderlyg CAPM s as follows: It looks at the rsk ad rates of retur ad compares them to the overall stock market. The model assumes that most vestors are rsk adverse ad expect to be rewarded. Arbtrage Prcg Theory (APT) The arbtrage prcg theory (APT) was frst troduced by Ross 976. It focuses o the expected asset returs a arbtraged market. APT ca be more geeral tha the CAPM because t allows for multple rsk factors. Ulke CAPM, APT does ot requre the detfcato of the market portfolo. The Factors used are assumed to be both ucorrelated wth the specfc returs, whch are tur ucorrelated. I other words, we cosder several depedet rsk factors. The theorem shows that a asset s expected retur beyod the rsk-free rate wll smply be the sum of ts exposure to some shared sources of rsk, weghted by the prces the market assgs to these rsks the rsk prema. Uder the Arbtrage Prcg theory, market rsk s see as the effect of whch s measured by the covetoal beta (for stocks) or durato (for bods). Idex Trackg Factor models assume that vestors use meavarace models to buld a portfolo such that rsk s reduced gve a desred level of retur. However, o sgfcat fuds are actually vested to formally maage ths relatoshp. Some sttutos, however, trade expected retur agast trackg error (relatve to a bechmark of stock dex portfolo). Trackg error s a method of followg such a dex ad the error ca be defed as the stadard devato of the dfferece betwee the performace of a bechmark gve by the dex ad the plaed portfolo. The mea-varace model ca be used to track targets. It ca fd uque portfolos such that they ether mmse volatlty resdual retur gve ther target beta, maxmse correlato wth the target, or mmse volatlty the dfferece betwee ther returs ad the target returs. Portfolo Optmsato Page 5

15 Rollg Markowtz t=t 0 t=t,t 2, t k I the real world, portfolos are bult such that they ca be traded at ay tme. Ths creates a stuato that s mult-tme perod. Ufortuately all the models metoed the prevous secto, (CAPM ad APT) are sgle perod models. However, they are appled to mult-perod stuatos. Whe a MV model s solved, the optmal portfolo weghts are determed. Accordg to the CAPM theory, these weghts should be detcal to the market portfolo. If we assume that the statstcal propertes of the returs from the ext tme perod are equvalet to those from the prevous tme perod, ad that both returs are ucorrelated, the the ew weghtgs of the portfolo should equal the prevous portfolo weghtgs. As a result the optmal weghts ca be carred through to the ext tme perod. However, as tme passes o, the asset prces chage, whch tur meas the market value chages, sce market weghts are captalzed weghts. The portfolo weghts carred forward o loger correspod to the market portfolo weghts. Ths s oe of the cotradctos of the MV models. A mult-perod approach ca be developed. However ths approach cotradcts some assumptos of the sgle-perod model. For example, the mult-perod theory prefers prce volatlty whereas the sgle-perod model does ot. I order to address a mult-perod stuato, a large umber of orgazatos follow the Rollg Markowtz approach over the plag perod. Ths method volves buldg a portfolo the tal tme perod for a gve budget. To respod to market chages, the portfolo s re-balaced at regular tme tervals. Fgure below shows ths approach more clearly. At t =t 0 =0, we compute a tal portfolo. Subsequetly for t, (=,, k) ths portfolo s rebalaced. Fud maager bulds a portfolo for a gve budget Fud maager re-balaces the portfolo from the prevous tme perod Fgure 6: Geeral procedure for re-balacg portfolos It ca be see from the above dagram that oce the fud maager bulds a portfolo the he/she wll re-balace t at regular tme tervals t=t, t 2,, t k. The factors may be observable (e.g. chages flato) or may be derved through factor aalyss (Prcpal Compoet Aalyss). Implemetg factor models ca produce better estmates of stock returs ad covarace betwee assets. Aother advatage of factor models s that the computatoal complexty, whe estmatg portfolo volatlty, s reduced. A algebrac formulato of the factor model developed for trackg a target dex s stated below. I c defes the degree to whch a dex s affected by factor c. A model ca be stated whch mmzes the dfferece betwee the target portfolo ad the vestor s portfolo terms of ther rsk factors. Such a model ca be defed as RebalaceQP: C Mmz 2 2 e yp cσ f + c Subject to x σ 2 2, ε c= = y P, c = xβc = I c c =,...,C ( ) r x d = (2 ) x = = (3 ) Portfolo Optmsato Page 6

16 l δ x = x δ δ = k u δ =,..., (4 ) 0 =,..., (6 ) bary (5 ) =,..., (7 ) Where y P,c s the dfferece betwee the portfolo beta ad the beta value of the dex I c. The portfolo beta s the weghted average of the betas (of the assets) the portfolo wth respect to factor c. By defg ths varable, the dfferece betwee the factor beta of the portfolo ad the beta value of the target portfolo s mmzed. I the objectve fucto, the frst term represets the rsk assocated wth each factor ad the secod models the resdual varace for the portfolo ad summg them together accouts for the total rsk of the portfolo. For the tme tervals t=t, t 2,, t k the portfolo model has the same format as the model RebalaceQP, exceptthatthasaextracostrat. Ths costrat s resposble for the re-balacg actvty of the portfolo. It defes the optmal portfolo weghts terms of the tal holdgs the amouts bought b ad the amouts sold s. Hece we have Backtestg Backtestg s a techque whereby a set of tradg rules (ths could be the form of a valuato or a forecastg model) s appled to hstorcal stock market data to test the ecoomc mpact of followg those tradg rules. I practce, may vestors do ot have the tme, tools or programmg experece to do ths.therdecsosarebasedohghlyemotoal crtera. However, professoal traders study tradg patters over hstorcal tme perods ad as a result, they ca estmate the rsk of usg a ew tradg strategy. Varous factors ca be examed usg back testg, such as eargs, equtable aalyss, dex trackg, rsk/retur trade off etc. Back testg does ot, ay maer, mply that hstory wll repeat tself. However people's reactos to the market do ot chage dramatcally over tme. If strog eargs reports have made people buy stocks the past, the they may certaly do so the future. Whe we use Markowtz M-V model, backtestg meas reapplyg the RebalacgQP model repeatedly over the hstorcal data steppg through gve tme tervals. The crcle of the backtestg strategy s show below. Step tme dex t:=t+ x = + b s =,..., I addto, the cardalty ad the buy- costrats are modfed accordgly δ LB b b b δ UB b b =,..., (4. ) Ital Portfolo t:=0 Optmse Portfolo at tme perod t usg RebalacgQP model Optmsed Portfolo at tme perod t δ LB δ s s b + s δ = b s b s ( δ + δ ) k δ, δ bary s δ UB s s =,..., (4.2 ) =,..., (6 ) =,..., (7 ) =,..., (9 ) Iput of data varables to the optmzato model: (From database/ spreadsheet) Fgure 7 : Backtestg cycle The represetatve example performaces of the portfolo plag models (see fgure below) are compared wth a equvalet savgs vestmet ad the movemet of a market dex (S&P 500 stock dex), betwee February 996 ad February 999. Portfolo Optmsato Page 7

17 The results therefore do ot fully represet the beefts of re-balacg sce t s rather rratoal to retra the same portfolo despte chagg market codtos. The fgure also shows that performace of portfolo plag models surpasses the performace of a equvalet savgs accout vestmet. The low retur o a savgs accout vestmet s compesated by eglgble rsk. Ths suggests that t may be advatageous to combe a rsk-free vestmet wth a (rsker) stock market vestmet. Ths uderps the CAPM theory, whch ecourages vestors to seek to atta the market portfolo ad some rsk free assets. The cotuous re-balacg model slghtly outperforms the dex for the majorty of the vestmet perod. However the dex recovers dramatcally to beat the portfolo plag models for the last sx moths. Ths aga s coformty wth the CAPM theory, whch matas that the market portfolo caot be out-performed o a systematc bass over a perod of tme. 25,000,000 20,000,000 5,000,000 0,000,000 Portfolo Growth over Tme Selected Refereces [] N.J. Jobst, M.D. Horma, C.A. Lucas ad G. Mtra, Computatoal aspects of alteratve portfolo selecto models the presece of dscrete asset choce costrats, Quattatve Face, (200), p [2] Joh, C. Hull, Optos, Futures, & Other Dervatves, (2000), Fourth Edto, Pretce Hall Iteratoal, Ic. [3] Lter, J., The Valuato of Rsk Assets ad the Selecto of Rsky Ivestmet Stock Portfolos ad Captal Budgets, Revew of Ecoomcs ad Statstcs, (965), p [4] Markowtz, H. M., Portfolo Selecto, (952), Joural of Face, 7, p [5] Markowtz, H. M., Portfolo Selecto: Effcet Dversfcato of Ivestmets, (959), Wley, New York, NY. [6] Markowtz, H. M., Portfolo Selecto: Effcet Dversfcato of Ivestmets, (99), Secod Edto, Basl Blackwell, Cambrdge, MA. [7] Ross, S. A., The Arbtrage Theory of Captal Asset Prcg, Joural of Ecoomc Theory, (976), 3, p ,000, Tme Perods S&P500 SavgsAccout Cotuous Re-balacg Model [8] Sharpe, W. F., Captal Asset Prces: A Theory of Market Equlbrum Uder Codtos of Rsk, (964), Joural of Face, 9, p Fgure 8: Portfolo growth comparso Portfolo Optmsato Page 8