Computational methods for selecting optimal financial investment strategies

Thesis

Reference

Computational methods for selecting optimal financial investment strategies

LULA, Jonela

Abstract

This thesis focuses on empirical asset allocations problems. The nonconvex optimization problem arising from our models specification is solved by means of heuristic optimization methods. Three empirical applications of a particular heuristic, the Threshold Accepting method, are proposed. The first problem that we consider is the replication of the Credit Suisse/Tremont (CST) Hedge Fund Index using liquid instruments such as equities, commodities and bonds. Our specification yields portfolios appearing to be an attractive substitute to the tracked index. In the second application we explore whether an asset allocation approach to Foreign Exchange Market is profitable. Our approach dominates the benchmark portfolio and technical trading model, as well as being less volatile. In the last application, we incorporate Asset-Liability Management in the asset allocation decision. The model used is the multistage programming, which outperforms the other approaches such as mean-variance or minimum downside risk.

LULA, Jonela. Computational methods for selecting optimal financial investment strategies . Thèse de doctorat : Univ. Genève, 2013, no. SES 796

URN : urn:nbn:ch:unige-266898

DOI : 10.13097/archive-ouverte/unige:26689

Available at:

http://archive-ouverte.unige.ch/unige:26689

Disclaimer: layout of this document may differ from the published version.

Computational methods for

seleting optimal nanial

investment strategies

Thèse

présentée à laFaulté des sienes éonomiques etsoiales de

l'Université de Genève

par

Jonela Lula Chamay

pour l'obtention du grade de

Doteur ès sienes éonomiques et soiales

mention éonométrie

Membres du jury de thèse :

Prof. Manfred Gilli, direteur de thèse,Université de Genève

Prof. JayaKrishnakumar,présidentedu jury, Université deGenève

Prof. Dietmar Maringer, Université de Bâle

Prof. DanielRoyer, Université de Genève

Thèse n o

796

Genève,le26février2013

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surlespropositionsquis'ytrouventénonéesetquin'engagentquelaresponsabilité

deleurauteur.

Genève,le26février2013

Ledoyen

BernardMORARD

Impressiond'aprèslemanusritdel'auteur

Résumé iii

Abstrat v

Aknowledgements vii

Introdution 1

1 Repliatinghedgefund indieswith optimizationheuristis 7

1.1 Hedgefunds . . . 7

1.2 Themodels: Indextraking . . . 8

1.3 Appliation: Trakinghedgefundindies . . . 11

1.4 Conludingremarks . . . 18

2 FXtrading: An empirial study 21 2.1 Introdutiontoforeignexhange . . . 21

2.2 Tehnialanalysis inurrenymarkets . . . 22

2.3 Themodels . . . 24

2.3.1 Assetalloationapproah . . . 24

2.3.2 Tehnialtradingapproah . . . 26

2.4 Empirialresults . . . 27

2.4.1 Dataandbaktestingsheme . . . 27

2.4.2 Simulationresultforassetalloationapproah . . . 29

2.4.3 Simulationresultsfortehnialtrading. . . 33

2.5 Robustnesshek . . . 34

2.5.1 Resultsfor2010 . . . 34

2.5.2 Comparingresultstothebenhmark . . . 35

2.6 Conludingremarks . . . 36

3 Asset-Liability Management for Individual Investors in a Multi- Senario and Multi-PeriodSetting 39 3.1 Introdution. . . 39

3.2 MultistageModel . . . 43

3.3 TheMultivariateSwithingRegimeModel. . . 44

3.4 OptimizationModel . . . 47

3.5 Comparisontootherinvestmentstrategies . . . 49

3.6 Empirialresults . . . 50

3.6.1 Dataandbaktestingsheme . . . 50

3.6.2 Example: Firstportfolioat31-De-2004 . . . 51

3.6.3 Results . . . 53

3.7 Conludingremarks . . . 57

Conlusion 59

Cettethèsetraitedesproblèmesempiriques d'alloationdesatifs.

Nous utilisons des modèlessans ontraintes et, par onséquent et étant donnéla

non-onvexitédesproblèmesprésentés,lesméthodesd'optimisationquis'imposent

sontlesméthodesditesheuristiques.

Sontproposéesainsitroisappliationsd'uneméthodeheuristiquepartiulière,soit

laméthodeThresholdAepting,

Lepremiersujettraité,dansleadredeetravail,esttel: répliquerlaperformane

de l'indie CreditSuisse/Tremont(CST) HedgeFunden utilisantdesinstruments

liquides,tels queles ations,lesmatièrespremièreset lesobligations. Aussi nous

onstatonsquenotremodèleproduitdesportefeuillesquisuiventdeprèsl'évolution

de l'indie. Cesportefeuilles sontplusliquides, plustransparentset les fraissont

moins élevés.

Dansladeuxièmeappliation,noustentonsdesavoirsiuneapprohed'alloation

d'atifspourlemarhédesdevises(FOREX)peutêtrerentable. Etils'avèreque

notreapprohesemblepluseaequeleportefeuillederéférene,meilleureaussi

que eluiqu'uneapprohed'analyse tehniquepeutproduire. Le portefeuille que

nousobtenonsparl'approhed'alloationd'atifsest égalementmoinsvolatile.

Dansladernièreappliation,nousonsidéronslagestionatif-passif(Asset-Liability

Management,ALM)dans ladéisiond'alloationd'atifs. Lemodèlepourlequel

nousavonsoptéestlaprogrammationmultistage. Etlàenore,lesrésultatsnous

amènent à la onlusion que que notre modèle semble plus judiieux que elui

produit par d'autres approhes, telles notamment les models mean-variane ou

minimumdownsiderisk.

This thesis fouses on empirial asset alloations problems. We speify models

without any restritions, as a onsequene the nononvex optimization problem

arisingfrom ourmodelsspeiationissolvedbymeansof heuristioptimization

methods.

Three empirial appliations of a partiular heuristi, the Threshold Aepting

method, areproposed.

TherstproblemthatweonsideristherepliationoftheCreditSuisse/Tremont

(CST) Hedge Fund Indexusing liquid instruments suh as equities,ommodities

and bonds. Our speiation yieldsportfolios appearing tobean attrative sub-

stitute to thetrakedindex. These portfoliosare moreliquid, theomposition is

transparentand theyhavelowerfees.

IntheseondappliationweexplorewhetheranassetalloationapproahtoFor-

eignExhangeMarketisprotable. Ourapproahdominatesthebenhmarkport-

folioandtehnialtradingmodel, aswellas beinglessvolatile.

In the last appliation, we inorporate Asset-Liability Management in the asset

alloationdeision. The model used isthe multistageprogramming. This model

outperforms the other approahes suh as mean-variane or minimum downside

risk.

First of all I would liketo thank my supervisor, Professor Manfred Gilli, for his

supportandguidane during alltheseyears. It was anopportunityworkingwith

him,from whomIwas taughttoworkwithpreisionandexatitude.

I am grateful to Professor Daniel Royer, with whom I had the hane to teah

Mathematis in thisFaulty. Throughouttheyears,he hasbeenarolemodelfor

his generosityandpassioninteahing.

I am also partiularly thankful to Professor Jaya Krishnakumar and Professor

DietmarMaringerfor aeptingtotakepartin mythesisommitteeandfortheir

onstrutivesuggestions.

I would also like to thank Gerda Cabej. It was a pleasure working and sharing

withhertheseintense moments.

I would also liketo express my gratitude to IlirRoko andEnrio Shumman for

theirollaborationandtheirvaluablesuggestions.

ThisworkwouldnotbethesamewithoutthehelpofEvisKëllezi,withoutwhom

I wouldhavenotstartedthisprojet.

IamsinerelythankfultoallprofessorsoftheformerDepartmentofEonometris,

for theirpassionofteahing andpartiularly, ProfessorGabrielleAntilleGaillard

forherhelpandsupport throughoutmystudiesinthisFaulty.

ThanksmustgotoMaroRigoforkindlyreadingthisthesisandprovidingvaluable

suggestions.

Andagain,manythankstoallmyolleaguesinthisdepartment,formero-workers

ofAvendis Capitalandtoallmyfriendsfortheirenouragement.

Andlastbutnotleast,themostspeialgratitudegoestomyfamily,fortheontin-

uoussupportandmotivation, in partiularthanksto InaFurerajforallher help,

Charles-Antoineforhislove,andtomyfatherwhohasneverstoppedghting.

Theoptimizationmodelsthatwillbepresentedinthisthesis,aresuhthatannot

be solved by means of lassial optimization methods. As a matter of fat, the

lassialoptimization methods (suh as linear andquadrati programming)relay

onananalytialmodeloftheobjetivefuntion. Thesetehniques areeientfor

problemswithsmoothobjetivefuntion.

Unfortunately,inrealitywehavetofaeomplexobjetivefuntionsthatannotbe

solvedbyusinglassialoptimizationmethods. Thisiswhereheuristitehniques

taketheleadastheyoeranaturalwaytooveromethedeieniesofthelassial

methods. Heuristisdonotrequirepartiularpropertiesfortheobjetivefuntion,

in partiular derivatives are not needed (only funtion evaluations). The main

purpose of this thesis is to illustrate how, these omplex optimization problems

an beeientlysolvedbyresortingto heuristioptimization tehniques.

Heuristis have been put into pratie relatively reently on a larger sale, as a

result of the impressive development of omputing power at a low ost. These

methods adapt easily to any optimization, are robust to hanges, give 'optimal

enough'solution(GilliandShumann,2011a)totheproblemathandandareeasy

to implement.

Heuristis may bedivided in two ategories: trajetory methods and population

based methods. Trajetory methods work with a single solution, whereas the

population-based methods onsider a population of solution simultaneously. In

thisresearh,onlyatrajetorymethodbasedonaloalsearhisused.

AmongthetrajetorymethodswehaveSimulatedAnnealing,ThresholdAepting

andTabuSearh. Thesemethodsdierfromeahotherinthewaytheaeptane

riterionishosen. ApartfromTabuSearh,allthesemethodsaeptuphillmoves

whihallowthemtoesapeloalminima.

Inourresearh,weonlyuseThresholdAepting(TA).Thismethodwasproposed

simultaneouslyby Duek andSheuer(1990)and Mosatoand Fontanari(1990).

Throughitsaeptaneriterionwhihinludes adereasingthreshold(henethe

name), the algorithm aepts not only improvements, but also impairments of

the objetive funtion in order to esapeloal minima. It begins with an initial

solutionandperformsanumberofiterations. Ateahiteration,anewandidateis

generated in theneighbourhoodoftheurrentsolution. Basedontheaeptane

riterion, thenewsolutioniskept. If thenewsolutionisrejeted, TA retainsthe

urrentsolutionandthealgorithmsearhesforanewsolutionintheneighbourhood

ofthisurrentsolution.

Inwhat followstheThresholdAeptingisusedfortheseletionofoptimalasset

alloationstrategies.

There already is a signiant body of researh that deals with the problem of

portfolio optimization. Dierent diretions have been pursued. Rokafellar and

Uryasev(2000,2002)werethersttolinearizetheportfoliooptimizationproblem

in ordertoeientlysolveitwithstandardlinearprogrammingtehniques. Mor-

ton,Popova,Popova,andZhong(2006)showthat whenreturnsarenotnormally

distributed,theportfoliooptimizationproblemannotbesolvedviastandardnon-

linearprogrammingtools. Theyuseanapproximationapproahrooted inMonte

Carlosimulation,usingabranh-and-boundsolutionproedureforsolvingadi-

ultmixed-integerprogram.

As mentionedabove,wetakeadierentresearh diretion basedonanumberof

ontributions where heuristis, Threshold Aepting in partiular, is applied for

theportfoliooptimization problem. DuekandWinker(1992)are therst touse

heuristisoptimization,speiallytheThresholdAepting,inaportfolioseletion

problem. LaterGilliandKëllezi(2002a,b)useThresholdAeptingforminimizing

Value-at-Risk and expeted shortfall. Maringer (2005)applies heuristi methods

tosolveavarietyofportfoliomanagementproblems,dealingamongotherwithin-

tegervariablesand ardinalityonstraints. Chang,Meade, Beasley,and Sharaiha

(2000)disussardinalityonstrainedportfoliosandMansiniandSperanza(1999)

usealinearprogrammingapproah tosolvetheportfolioproblem withminimum

tradingsizeonstraints.

Meanwhile aninreasing numberof papershave been written providing evidene

of thegoodperformane ofheuristioptimizationtehniques appliedto problems

in nane 1

.

Ourstudyfollowsthesameapproah(inthatweuseheuristisinnane),applied

to aseriesofempirialappliations. Theseonsistin:

•

•

•

The traking problem 2

for Hedge Fund Index is presented in Chapter 1. The

problem isapplied toCreditSuisse/TremontHedgeFundIndex(CST).

Consideringbenhmarkreturnsintheobjetivefuntionreetsinamorerealisti

way the motivation of today's investmentmanagers. In pratie, besides aiming

atthebestpossiblerisk-adjustedreturnontheirinvestments,amajoronernfor

portfoliomanagersisndingawayto performbetterthanabenhmarkspeied

in themandateguidelines andquite oftenhowtoinrease thehanes ofoutper-

1

fori.e. Gilli,Maringer,andShumann(2011a)andthereferenestherein.

2

ExploredinGilliandKëllezi(2002b).

formingtheirpeers. Browne(2000)is,toourknowledge,thersttoformalizethe

problemofportfoliooptimizationrelativetoabenhmarkinontinuoustime. The

benhmarkperformaneanbedeterministi,i.e.: axedrateofreturnperyear,

in asewhen an absolute return is the investmentobjetive,or random, suh as

thereturn onapredened market index (likeS&P 500). Thelateraseonerns

themajorityoflong-onlymandates inbondorequityinvestments.

There arevarious waysofintroduing benhmark relatedreturnsinthe objetive

funtion. One an try to maximize the probability of ahieving the benhmark

return, ignoring the magnitude by whih the target is missed or exeeded. An-

other possibility would be to minimize the downside or themagnitude by whih

thebenhmark ismissed,dened by Demboand King(1992)as theexpeted re-

gret. Mortonet al.(2006)proposeafamily ofobjetivefuntions onstruted as

a weighted sumof theprobabilityof ahievingabenhmark and expeted regret

relativetoanother(lessaggressive)benhmark,whihanbeinterpretedasaom-

binationofmeasuresofrewardandrisk. Thesemeasuresarerelatedtothealready

widelyknownriskmeasures,Value-at-RiskandExpetedShortfall.

In ourstudy, we rstminimize themaximumdrawdownand then wetest anew

objetive funtion inluding also a measure of orrelation between the portfolio

and theindex. These objetivefuntions aresubjetto aset ofonstraints,more

speiallybudget onstraints,minimum andmaximumholding sizesfor thesets

ofassets,totaltransationostsandardinality. Theoptimizationproblemweare

faingisanon-onvexone,giventheobjetivefuntionandsomeoftheonstraints.

Inordertosolveit,theThresholdAeptingmethodis used.

Heuristi optimization tehniques are also used for the appliation presented in

Chapter 2. Weapply anassetalloationapproah toaset oftik-by-tikdataof

veurrenypairsandoptimizeamulti-objetivefuntion (maximumdrawdown,

Omega,momentum andombinationof them). Wealsoapplyatehnialtrading

rulebasedmodel. Inpartiular,weexploreappropriatelevelsoftimeaggregation

andrebalaningfrequenies,andweretainhourlyaggregationanddailyrebalane-

ment. Results are afterwards ompared to a benhmark portfolio (buy-and-hold

strategy).

In the last hapter, we introdue long term liabilities in the asset management

problem of an investor. This involves long planning horizons and multi-period

realloationsinordertoadapttotheevolutionofthemarket.Thisorretsforone

ofthelimitationsoftheMeanVarianeapproahwhihisitssingleperiodhorizon

(not favorable forlongterminvestors). TheAsset-LiabilityManagementaims to

ontroltheriskassoiatedtofutureliabilities,aswellastondtheoptimalportfolio

as a trade-o betweenthehighest returnof theportfolio and thelowest possible

assoiatedrisk. Dynamis ofthe assetsin thesuessive periods are represented

in a senario tree and a multistage programming approah is used to ompute

realloations. The optimization problem is one again solved by means of the

ThresholdAeptingmethod.

It wasourintentioninthisthesistomakeeahhapter astand-alonereading. As

aonsequene,Chapters13mayontainsomeredundantinformation.

Repliating hedge fund indies

with optimization heuristis

∗†

1.1 Hedge funds

Startingfromtheearly90s,hedgefundsisonsideredtobeoneofthemostpopular

setorof thenanialindustryandthis ismainly due to theperformanesofthe

hedge funds. Amongthedesirablefeatures ofhedge funds wehavealpha genera-

tion,loworrelationwiththemarket,attrativerisk-returnproleandalternative

riskpremia 1

. Themaindrawbaksarehighmanagementfees,lakoftranspareny

and illiquidity. This leadsto theinentivetorepliatetheattrativefeaturesofa

hedge fund with liquid assets, thus providing afully transparent, liquid and low

ost alternativeto a hedge fund. There exists a largeliterature giving morede-

tailedaboutthemotivationforhedgefundrepliation(egTill(2004)Shneeweiss,

Kazemi,andMartin (2002),Shneeweiss,Kazemi,andMartin(2003)).

∗

ThishapterisbasedonGilli,Shumann,Cabej,andLula(2010).

†

AllauthorsgratefullyaknowledgenanialsupportfromtheeuCommissionthroughmrtn-

t-2006-034270omisef. EvisKëlleziprovidedhelpfulomments.

1

seeLhabitant(2006),Lo(2008)andCoggan(2011)foroverviews

Several tehniques are used to repliate the properties of hedge fund returns.

Among themain approaheswehavefatormodels (Sharpe(1992), Hasanhodzi

and Lo (2007)), trading rules (Kat and Palaro (2005)) and traking tehniques

(Roll (1992), Rudolf, Wolter, and Zimmermann (1999), Alexanderand Dimitriu

(2005)). Kat(2007)proposetradingstrategiestogeneratereturnswithstatistial

properties,similartothoseofhedgefunds.

Our approah diers in that we proeed in speifying the models without any

restritionswithrespettotheirtratabilitywithlassialoptimizationtehniques.

Thisexibilityispossiblebeauseweuseheuristioptimizationmethods allowing

forthesolutionofvirtuallyanyoptimizationproblemgivenofourseitissound.

We test a varietyof models of inreasing omplexity going from a pure traking

objetive to drawdown. In between, we explore ombinations of traking error

withexessreturn,orrelationoftheportfoliowiththeindexandthemarket. The

omputed portfolios satisfyrealistionstraints,suh asminimum andmaximum

holding size, maximum ardinality and also take into onsideration transation

osts. Thenononvexoptimization problemarisingfrom thesespeiationsan-

not besolvedwith lassial gradientbased methods. Our heuristi optimization

tehniqueprovesveryeientforthiskindofproblems(Maringer(2008),Maringer

andOyewumi (2007),Maringeranddi Tollo(2009),Krink,Mittnik,andPaterlini

(2009)).

The hapter is organizedas follows: in Setion 2 dierent models are presented

andtheoptimizationproblemisformulated. Setion3givesinformationaboutthe

analyzeddataanddetailstheresultsforthedierentmodels. Setion4onludes.

1.2 The models : Index traking

A straightforwardapproah is to onstrut a trakingportfolio whih minimizes

the distane between historial portfolio and index returns. We denote

r P

r I

r E = r P − r I

deviationsforaportfoliowealsoonsiderthemeanexessreturn

r E = ¹ _n P

(r P − r I )

leadingtothefollowingobjetivefuntion

λ k r E k ^α − (1 − λ) r E

where

α

λ ∈ [0, 1]

betweentrakingerrorandexessreturn. ThisapproahhasbeenexploredinGilli

andKëllezi(2002b)usingartiialdata.

One of the main goalsis to onstrut a portfolio following the index as lose as

possiblebut beinglittlesensitivetoadversemarketmovements. This suggeststo

inlude the orrelation between traking portfolio and market into the objetive

funtion

λ k r E k ^α − (1 − λ) r E + ρ r P ,r M

where

ρ r P ,r M

ingtheobjetivefuntion minimizesthisorrelation.

Model(1.1)fousingonthetrakingerror,alreadyleadstoaportfoliohighlyor-

relatedwiththeindex. Neverthelessoneanthinktoontrolthisorrelationmore

speiallybyintroduingitintotheobjetivefuntion. Denoting

ρ r P ,r I

lationbetweenportfolioandtheindex,itanbemaximizedwiththenewobjetive

funtion

λ k r E k ^α − (1 − λ) r E + ρ r P ,r M − ρ r P ,r I .

Afurtherdesirablefeatureofthetrakingportfoliowouldbetohavelowdrawdown.

Foraseriesofportfoliovalues

v t , t = 0, 1, 2 . . . T

D t = v _t ^max − v t

where

v ^max _t

v _t ^max = max { v s | s ∈ [0, t] }

denition

D

or the maximum element

D ^max = max( D )

objetivefuntion. Inotherwords

D ^max

value overthe time horizon. Inarst stepweonsider onlythe minimization of

themaximumdrawdownforouranalysis. Inaseondstepweombinemaximum

drawdownminimizationwiththeobjetivefuntion denedin (1.3)yielding

λ k r E k ^α − (1 − λ) r E + ρ r P ,r M − ρ r P ,r I + D max .

The optimization problem

Thegeneraloptimization problemanbestatedas follows

min x Φ(x)

X

j ∈J

x j p 0j = v 0

′

)

x ^inf _j ≤ x j ≤ x ^sup _j j ∈ J

^′′

K inf ≤ # {J } ≤ K sup

′′′

)

.

.

.

where

Φ

x

avetorwith

x j

j

′

)with

v 0

theinvestablewealthand

p 0j

j

period. Constraint(1.5

′′

)speiesminimumandmaximumholdingsizefortheset

ofasset(

J

^′′′

allowsfortratabilityoftheresultingportfolios. Furtheronstraintsmightinlude

totaltransationost,setoronstraintsandotherliquidityissues.

2

Theoptimization problems(1.5)arenononvexdueto theformulationoftheob-

jetivefuntionandpartoftheonstraints. Theyannotbesolvedwithalassial

approah relyingon informationfrom gradientvalues. We thereforeuse Thresh-

oldAeptingwhihisaheuristioptimizationmethodfromthelassoftrajetory

methods. DetaileddesriptionofthetehniqueanbefoundinWinker(2001),Gilli

and Winker(2009) and for appliations see Gilli and Shumann(2010). Matlab

odeisavailableathttp://omisef.eu.

Heuristimethodsdonotproeedrandomlybutexplorethesearhspaeaording

to ertainrules. Neverthelesstheresultingnumberof funtionevaluationsisrela-

tivelyimportant. Intheaseoftheoptimizationofasingleportfoliotheomputing

timestaysintherangeofafewseonds. Howeverthebaktestingweondutedfor

ourappliationneededtheevaluationof approximativelyhalfamillionportfolios.

InordertokeepouromputationsmanageabletheyweredistributedwithMatlab's

ParallelComputing Toolboxonthe MyrinetClusterof theUniversity ofGeneva.

Myrinet is a Linux Cluster with 32 nodes, eah on a Sun V60x dual IntelXeon

2.8GHzwith2GB ofRAM.For moredetailsseehttp://sp.unige.h/.

1.3 Appliation : Traking hedge fund indies

Data and baktesting sheme

The index to repliate is the Credit Suisse/Tremont Hedge Fund Index (CST)

availableatwww.hedgeindex.om. Aordingtotheinformationonsite thisindex

isasset-weighted,inludesmorethan5000fundswithaminimumofUS$50million

under management. The observations are monthly and over the period from

January1999toOtober2009.

The instruments used for the repliation omprise equity, ommodity and bond

indies. Inthesetofequityindieswehaveabout54seriesinludingbroadmarket,

bluehips,setoraswellassizeandstyleindies. Thereare12ommodityindies

and 12 bond indies from government,orporate andemerging markets. The set

ofdatahasbeenolletedfromBloomberg.

In order to analyze the performane of the suggested portfolios we baktest the

strategiesovertenyearswithrebalaningwhereweaountfor10basispointsof

2

OptimizationwithonstraintsontotaltransationosthavebeenusedinGilliand Këllezi

(2002b),setorandliquidityonstraintsinGilli,Shumann,diTollo,andCabej(2011b).

transation osts. Therolling window hasahistoriallengthof

H

periodof

F

H

F

to trajetoriesofportfoliosvalueswheretheportfolioshavebeenrebalanedforty

times. Theshemebelowsummarizesthetehnique.

Period

1

t 1 − ^H t 1 t 1 + ^F

F

H

invest

Period

2

t 2 − ^H t 2 t 2 + ^F

rebalane

Forallportfoliostheminimumholdingandmaximumholdingforanassetis

x ^inf _j = 1%

x ^sup _j = 20%

K sup = 10

Togaininsight into thestohastis of the simulatedportfolioswejakknife from

the historial observations so as to ompute a set of results 3

for whih we then

onsider empirialdistributions fordierentfeaturesoftheportfolio.

Results

In the following we present for eah objetive funtion the median path of the

portfolio value for varying parameter settings, aplot of the kernel estimation of

the density the empirial distribution of the mean yearly return of the traking

portfolioandaplotoftheempirialdistributionoftheorrelationoftheoptimized

portfolios withthe market. Themedian path isdened withrespetto the nal

wealthoftheportfoliosgeneratedwiththejakkning.Forallmodelsweset

α = 2

forthenormmeasuringthetrakingerror.

Performanewillalsobesummarizedandomparedwiththeindexintablesshow-

ingthetrakingerror

TE

r E

(

S

r

vol

ρ r P ,r M

the market of the portfolios minimizing the objetive funtion, realized overthe

simulationperiod.

3

Inthisaseweomputed100trajetoriesforeahspeiationoftheobjetivefuntion.

Optimization oftraking error and exess return

Weomputedresultsfortheobjetivefuntiondesribedin(1.1)forvaryingvalues

of the parameter

λ

Figure1.1showsthemedianpathsfor100simulationsfor

λ = 1

orrespondstominimizingonlytrakingerror. Weantradetrakingerroragainst

nalwealthbydereasing

λ

λ = 0.75

portfolios loseto theindex whereashigher weights for thereward (lowervalues

for

λ

Jan00 May01 Oct02 Feb04 Jul05 Nov06 Apr08 Aug09

0.5 1 1.5 2 2.5 3

λ=1.00 λ=0.75 CST S&P500

Figure 1.1: Medianpaths for portfolios minimizingthe objetivefuntion (1.1) for

λ = 1

λ = 0 . 75

Figure1.2illustratesanotherfeatureofthesimulatedportfolios. Itshowstheplot

of the kernel estimation of the densityof the empirial distribution of the mean

yearlyreturnof thetrakingportfolio(left panel). Thevertialline indiatesthe

meanyearlyreturnoftheindex. TherightpanelinFigure1.2showstheempirial

distribution of the orrelation of the optimized portfolios with the market. The

dotted line orrespondsto the orrelation of theindex with the market. For the

trakingportfoliosweobservehighervalues. Table1.1summarizestheperformane

fortheportfoliosobtainedwithobjetivefuntion (1.1).

Table1.1: Resultsforthemedianpathofthesimulatedportfoliosforobjetivefuntion(1.1).

TE S r vol ρ r P ,r M

λ = 1

λ = 0.75

CST 1.04 7% 7% 0.54

The medianpath of theportfolio for

λ = 1

despitethefatthatithasalowtrakingerror. Thisresultmighthurtintuitionbut

2 4 6 8 10 12 14 16 18 0

0.2 0.4 0.6 0.8 1

λ=1.00 λ=0.75 CST

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 2 4 6 8 10 12 14

λ=1.00 λ=0.75 CST

Figure1.2:Densityofmeanyearlyreturn

r

ρ r P ,r M

the trakingerrorompares the returnsand not portfolio values. Twoportfolios

with a dierent rst return but idential returns for the remaining periods will

translateinto o-evolvingtrajetories. The portfolio for

λ = 0.75

returninreased by50%omparedwith

λ = 1

trakingperformaneandsmallinreaseofvolatility. Furthermoretheorrelation

oftheportfoliowiththemarketdereases.

Optimization oftraking error, exess return and

ρ r P ,r M

Thehigherorrelationofthetrakingportfolioswiththemarketisnotadesirable

feature. To improve on this we now turn to the objetive funtion (1.2) whih

inludestheorrelationofthetrakingportfoliowiththemarket. Figure1.3gives

the orresponding median pathsfor valuesof

λ = 1, 0.75, 0.7

λ = 1

noexessreturnenterstheoptimization,weobserveapartiularlysmoothmedian

pathalmostnotaetedbythedropin theS&P500attheendof2008.

Results in the right panel of Figure 1.4 are remarkablewhen ompared with the

ones in Figure1.2 as thedistributions indiate that, while maintainingthe same

levelofreturns,orrelationisnowwellontrolled.

Table1.2: Resultsforthemedianpathofthesimulatedportfoliosforobjetivefuntion(1.2).

TE S r vol ρ r P ,r M

λ = 1

λ = 0.75

λ = 0.70

CST 1.04 7% 7% 0.54

Jan00 May01 Oct02 Feb04 Jul05 Nov06 Apr08 Aug09 0.5

1 1.5 2 2.5 3

λ=1.00 λ=0.75 λ =0.70 CST S&P500

Figure1.3:Medianpathsforportfoliosminimizingtheobjetivefuntion(1.2)ontrollingorre-

lation

ρ r _P ,r _M

2 4 6 8 10 12 14 16 18

0 0.2 0.4 0.6 0.8 1

λ=1.00 λ=0.75 λ=0.70 CST

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 2 4 6 8 10 12 14

λ=1.00 λ=0.75 λ=0.70 CST

Figure1.4: Densityofmeanyearlyreturn

r

ρ r P ,r M

theobjetivefuntion(1.2)ontrollingorrelationwiththemarket.

Optimization oftraking error, exess return,

ρ r P ,r M

ρ r P ,r I

Wenowhekwhetherobjetivefuntion (1.3)whihonsiderstrakingerrorand

orrelationbetweentheportfolioandthetrakedindexenhanesperformane. No-

tie thattheeet of

λ

tionsduetotheimpatoftheadditionalterms. Theresultsindiatenosigniant

hangeinoverallperformane,weratherobserveashifttoimprovedSharperatios

for allvaluesof

λ

λ

leads to portfolios with highernal wealth but of ourseat the ostof inreased

volatility.

As visible in Figure 1.6, from the point of view of returns

λ = 0.6

trativeportfolioswhihmoreovershowquiteloworrelationwiththemarketand

Jan00 May01 Oct02 Feb04 Jul05 Nov06 Apr08 Aug09 0.5

1 1.5 2 2.5 3

λ=0.68 λ=0.65 λ=0.60 CST S&P500

Figure1.5:Medianpathsforportfoliosminimizingtheobjetivefuntion(1.3)ontrollingorre-

lation

ρ r _P ,r _M

ρ r _P ,r _I

highSharperatios.

2 4 6 8 10 12 14 16 18

0 0.2 0.4 0.6 0.8 1

λ=0.68 λ=0.65 λ=0.60 CST

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 2 4 6 8 10 12 14

λ=0.68 λ=0.65 λ=0.60 CST

Figure1.6: Densityofmeanyearlyreturn

r

ρ r P ,r M

theobjetivefuntion(1.3)ontrollingorrelationswiththemarketandtheCST.

Optimization oftraking error, exess return,

ρ r P ,r M

ρ r P ,r I

D max

Wenowexploretheimpatoftheintrodutionofdrawdowninourobjetivefun-

tionontheoverallperformaneoftheportfolios. Figure1.7 belowreports results

for pure maximum drawdown minimization (dark line) as well as results for ob-

jetivefuntion(1.4) withdierentweightingoftrakingerrorandexessreturn.

As previously,higherweightsforexessreturn(low

λ

portfolios.

Lookingat thenalwealth distributiongivenin Figure1.8 wesee thatfor

λ = 1

the expetation of the mean yearly returns is approximatively that of the index

Table1.3: Resultsforthemedianpathofthesimulatedportfoliosforobjetivefuntion(1.3).

TE S r vol ρ r P ,r M

λ = 0.68

λ = 0.65

λ = 0.60

CST 1.04 7% 7% 0.54

Jan00 May01 Oct02 Feb04 Jul05 Nov06 Apr08 Aug09

0.5 1 1.5 2 2.5 3

D max λ =1.00 λ =0.70 λ =0.00 CST S&P500

Figure1.7:Medianpathsforportfoliosminimizingtheobjetivefuntion(1.4)ontrollingorre-

lations

ρ r _P ,r _M

ρ r _P ,r _I

howeverfor

λ = 0

Table1.4: Resultsforthemedianpathofthesimulatedportfoliosforobjetivefuntion(1.4).

TE S r vol ρ r P ,r M

DD max

λ = 1

λ = 0.7

λ = 0

CST

Inthelightoftheseresultsportfoliosobtainedfromthislastmodeloerproperties

suitable tosubstitute theCreditSuisse/Tremonthedge fund index. Inpartiular

for

λ = 0.7

orrelation with the market. However the portfolio an be modulated hoosing

dierentvaluesfor

λ

investor.

Itmightbeinterestingtoshowhowthedierentassetlassesarerepresentedinthe

2 4 6 8 10 12 14 16 18 0

0.2 0.4 0.6 0.8 1

D max λ=1.00 λ=0.70 λ=0.00 CST

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 2 4 6 8 10 12 14

D max λ=1.00 λ=0.70 λ=0.00 CST

Figure1.8:Densityofmeanyearlyreturn(leftpanel)anddensityoforrelationwiththemarket

(right panel) for the objetive funtion (1.4) ontrolling orrelations

ρ r _P ,r _M

ρ r _P ,r _I

maximumdrawdown.

median portfolios. This isplotted in Figure1.9 andwenotie howtheoptimizer

reats to market onditions. For instane, in periods of distress, the weight of

xed inome instrumentsgains importane at the expenseof equities. Also, the

portfoliowhereexessreturnisfavoredhasrelativelyhigherweightedommodities

andequities.

1.4 Conluding remarks

Thegoalofthisresearhistoinvestigatetheuseofasetofmodelsforthelassial

problem of index traking. Thespeiationof themodels is madewithout on-

sideringtheonstraintsimposedbythelassialoptimizationframework. Instead,

weresorttoheuristioptimizationmethodsapableofeientlyhandlingomplex

nononvexproblems. Westartfromasimplespeiationombiningtrakingerror

andexessreturn,whihgivensomeoftheonstraints,isalreadynottratablewith

anylassialmethod. Thismodelisthensuessivelyenrihedbyaddingelements

to ontrol for the orrelationof the portfolio with themarket and laterwith the

index. Ourresults onrmthat weare able to derease signiantlythe orrela-

tionwiththemarket. Finally,weexploretheimpatofminimizing themaximum

drawdown as well as a ombination of it with traking error, exess return and

orrelations. Thislast speiationyieldsportfoliosappearingto beanattrative

substitute to the traked index. Indeed, theirSharperatios ompare to those of

the index, while their average mean return is higher and their orrelation with

the market is lower. These results are notpoint estimates but are supported by

0 0.5 1

0 0.5 1

0 0.5 1

Jan00 May01 Oct02 Feb04 Jul05 Nov06 Apr08 Aug09

0 0.5 1

bonds

equities

commodities

Figure 1.9: Medianportfolioompositionintermsofassetlasses(lowerband equities,middle

ommoditiesandupperbandbonds)fortheobjetivefuntion(1.4). Fromtoptobottom

D ^max

λ = 1, 0.7

0

empirialdistribution obtainedfromintensivebaktesting.

FX trading: An empirial

study

∗†

2.1 Introdution to foreign exhange

The foreign exhange (FOREX) market was reatedin 1971, year in whih the

oating exhangerates began to appear andeversineitis onsidered to be one

of the largest and most important markets. The high liquidity, the guaranteed

quantity, theround-the lok business hoursworldwidetradingativity, two-way

market,aessibilitytoalltradersandlowtransationostsarefeaturesthathar-

aterizetheFOREXmarket. Partiipantsinthismarketareentralbanks,banks,

brokers,ommerialompanies,hedge funds,investorandspeulators.

Historially,theurrenymarketoriginatesin theearly20th entury. Inthe'30s,

LondonwastheenterofnaneandtheBritishpoundthemainurrenyusedin

tradingas wellas,thereserveurreny. AfterWorldWarIItheUnitedStatesbe-

ame theleadingenter andtheU.S.dollarthe"vehile"urreny. Allurrenies

arequoted againsttheU.S.dollarandthemajoronesaretheeuro,Japaneseyen,

BritishpoundandSwissfran.

The foreignexhange hasexperiened spetaular growthin its volume. In1988

the turnoverreah the value of $600 billion, rises up to 1 trillion in September

1992and inApril2010thedailyaverageforeignexhangemarketturnoverhit$4

trillion. Currenyandinterestratevolatility,advanesintehnologyandteleom-

†

ThishapterisbasedonGilli,Cabej,Lula,andShumann(2012).

†

WethankEvisKëllezifordisussionsandomments.

muniations had a inuene in the growth of the ativity in this market. This

is also due to theintrodution, in the early 1990s, of a tehnologial innovation:

eletroni trading. Nowadays,eletroni tradingrepresentssome50%ofthetotal

volume (BIS, 2010) and it has ompletely hangedthe struture of the markets

failitatingtheinterationamongators andtheintegrationofnewinformation.

The foreignexhange market ishighly eient ompared to other nanialmar-

kets,oeringsigniantlyreduedprotopportunities. However,urrenymarkets

are strongly aetedby monetary poliies and entral bank interventions. These

are thesoure ofinformationasymmetrywhih leadsto, at leasttemporary,inef-

ienies.

2.2 Tehnial analysis in urreny markets

Tehnialanalysis(TA)isbasedonDow'stheory 1

,thatpriesmoveintrends,the

trendishistoriallyrepetitiveandthevolumeonrmsthetrend.

Tehnialanalysisusesinformation regardingthehistorial prieof aurreny or

seurityand triesto foreastfuture priebehavior. Kaufman(2005)denesteh-

nial analysis as the systemati evaluation of prie, volume, breadth and open

interest, for the purpose of prie foreasting. Aording to Lo, Mamaysky, and

Wang (2000),tehnialanalysisdiersfrom quantitativenane. Andthisdier-

eneliesinthefatthattherstismainlyvisual,resortingtopatternreognition,

while the latter is mainly algebrai and numerial, using tools from probability

andstatistis. Osler(2000)examinesfromaneonomiperspetivehowtehnial

analysis anpredittheexhangerates.

Tehnial analysis is one of the rst attempts to take advantageof the market's

ineienies. For this reason many eonomists dene tehnial analysis as irra-

tional and despite the undeniable widespreadof TA among investors, aademis

havebeenquite septialon thesubjet as TA protability goesagainst thee-

ientmarkethypothesis(EMH).Indeed,literatureoftenreferstoTAasatestfor

EMH.

Fromthebeginnings,tehnialanalysisplayedadominantroleinurrenytrading

and still remains an important tool in the investors' deision making proess.

2

Aording to Henderson (2006) the tehnial analysis has an important role in

prediting exhange rates beause the urreny market follows a trend in short

term and there are speulators in this market. The rst examples of tehnial

1

writtenbyCharlesH.DowontheWallStreetJournal

2

AordingtoTaylorandAllen(1992)about90%ofurrenydealersusesomehartistinput.

trading rules (TTR) an be traed bak to Alexander (1961, 1964) who applied

lterrules 3

totestfortrendsin thestokmarkets. FamaandBlume(1966)show

that,inthisframework,ifdividendsandtradingostsaretakenintoaount,lter

rules do not perform well. However historially, in the urreny markets, lter

rulesyieldedenouragingresults. DooleyandShafer(1983)provideevideneofthe

outperformaneofsmall lters,i.e. hanges inthe orderof1to 5%,with respet

to buy-and-holdstrategies. Further, Sweeney (1986)shows that, for lter below

2%, strategies remain protable after adjusting for transation osts. Nowadays,

thereexistsalargesetofmoresophistiatedtehniques,inludingstrategiesbased

onstop-lossorders,movingaverages,hannels,momentumosillatorsandrelative

strengthindies. Toidentifytradingrules,algorithmiapproahes,suhasGeneti

Programminghavebeenlargelyexplored(seeDaorogna(1993)forearlyworkand

Neely,Weller,andDittmar(1997)foramoreompletesurvey).

A quite thorough literaturereview an be foundin Menkho and Taylor(2007),

ParkandIrwin(2007). Inanattempt toexploretheprotabilityofthestrategies

mentionedabove,ParkandIrwin(2007)lassifytheempirialevideneinto'early'

and'modern'studies. Therstonestendto showthattehnialtradingrulesare

protable in urreny markets alone, whilethe'modern' onesrevealprotsin all

markets,atleastuntilthebeginningofthe1990s. Apopularpaperinthisategory

isBrok,Lakonishok,andLeBaron(1992)whouseabootstrapproedureandtwo

tradingsystems(one basedonmovingaverageosillators andanotheron trading

range break)in the stokmarket.

4

Marsh (2000)and Olson(2004)highlightthe

delinein protabilityforpreviouslysuessfulstrategies. Thisiswhyweexplore

if anasset alloation approah to the FXmarket mightproveprotable. Toour

bestknowledgelittleliteraturerelatedtosuh anapproahexists.

Thehapterisorganizedasfollows: Setion2.3desribestheassetalloationmod-

elsandapartiularlterruletestedforthetehnialtradingapproah. Empirial

resultsaregiveninSetion2.4wherewealsopresentthedataaswellasthebak-

testingsheme. ArobustnesstestispresentedinSetion2.5. Setion2.6onludes.

3

Alterruleonsistsinabuyorsellsignalresultingfromagivenperentage hangeinthe

prieproess.

4

For an appliation of bootstrap in the foreignexhange markets see Levih and Thomas

(1993).

2.3 The models

2.3.1 Asset alloation approah

Inthis asethebuying andsellingdeisionsare taken asin ativeportfolio man-

agement. Thelassialassetalloationapproahonsistsinmaximizingameasure

of reward for agiven risk level. In the early work of Markowitz (1952)portfolio

rewardis measuredbyexpeted returnand portfoliorisk byitsvariane, this, in

partiular, foromputationalonveniene. Theaveatsof suh portfoliosare well

knownand havebeen extensivelydisussed in theliterature(Mihaud,1989; Co-

hen and Pogue,1967; Jobson and Korkie, 1980;Best and Grauer,1991). Having

explored,withsomesuess,avarietyofalternativeobjetivefuntions (Gilliand

Shumann, 2011b;Gilli et al.,2011b, 2010)we usefuntions suh as theOmega

ratio, drawdown,momentum and ombinations ofthese. The objetivefuntions

are omputed from portfolio returns, either historial or simulated. The models

are speied in inreasing order of omplexity by adding objetives whih might

beonurringorompeting. Thisorrespondstowhat isknownasmultiobjetive

optimization.

Theassetalloationproblemisformalizedintheusualway,i.e.

min x Φ(x)

X

j ∈J

x j p 0j = v 0

′

)

x ^inf _j ≤ x j ≤ x ^sup _j j ∈ J

^′′

where

Φ(x)

x

′

)andonstraint(2.1

′′

)limitsmin-

imumandmaximumholdingsize. Notethat

v 0

A rstobjetivefuntion usedfor oururrenyportfoliosismaximumdrawdown

dened as

max(D t )

D t = v _t ^max − v t

with

v t

t = 0, 1, 2, . . .

v ^max _t

runningmaximum,i.e.

v ^max _t = max { v s | s ∈ [0, t] }

Aseond objetivefuntion weinvestigatedarepartial momentsdened as

M lo = Z r d

−∞

(r d − r) ^{m lo} f (r)dr

M up = Z ∞

r d

(r − r d ) ^{m up} f (r)dr

with

r d

m lo

m up

f ( · )

simulatedportfolioreturns

r

M lo = 1 n S

n S

X

s=1

(r d − r s ) ^{m lo} 1 _{ r s <r d }

M up = 1 n S

n S

X

s=1

(r s − r d ) ^{m up} 1 _{ r s >r d }

where

1

r d

momentsareoforderone,thustheobjetivefuntionbeomes

Omega = M lo

M up

= − P

r s 1 { r s <0 }

P r s 1 _{ r s >0 }

.

Athirdobjetivefuntiononsidersmomentumdierentialsforthehistorialport-

folios. Thefuntion tobemaximizedis

Mom t i = (v t i − v t i − s )

| {z }

shortmomentum

− (v t i − v t i − ℓ )

| {z }

longmomentum

where

v t i

t i

intervalsmeasuringthelongandshortmomentsare

ℓ

s

The kind of problems presented above generally annot be solved with lassial

methods, beause of nononvexities in the objetive funtion and/or the on-

straints. Therefore we resort to heuristi tehniques, in partiular threshold a-

epting, whih is a loal searh algorithm that only performs objetive funtion

evaluationsand doesnotrequiretheevaluation ofderivatives. Ageneraldesrip-

tionofthresholdaeptinganbefoundin Winker(2001),implementationdetails

arepresentedinGilliandWinker(2009);Gillietal.(2011a)andadisussionabout

thestohastisofthesolutionispresentedin GilliandShumann(2011a).

Heuristis are omputationally intensive and in order to redue exeution times

omputationshavebeendistributedoveralusterof32mahines 5

usingMatlab's

ParallelComputingToolbox.

5

MyrinetClusteroftheUniversityofGeneva(http://sp.unige.h/m yrin et) .

2.3.2 Tehnial trading approah

As mentionedin Setion 2.2 there exist avariety oftehnial tradingrules. This

appliationislimitedtotheuseoflterrulesgeneratingbuyandsellsignalsbased

ontrendidentiation. Thealibrationof themodelisnotbasedonoptimization

but the parameters are rather seleted through intuition, observation and expe-

riene. Indeed in most ases optimization of tehnial trading models leads to

overtting,i.e. goodin-sampleperformanebutpoorout-of-samplepredition. As

a onsequene of the absene of optimization, these tehniques neessitate little

omputationaleort.

Inthefollowingwewillmodelthetrendfortheomingholdingperiod. Todothis

we investigate the empirial distribution of the urreny pairs identifying three

states, i.e. positive,negativeand neutralor zero return. Theneutralstate fora

urrenyisdenedbyaurrentreturnlyingbetweenagivenupperandlowerem-

pirialquantile. Returnbelowthelowerquantileharaterizeanegativestateand

thoseaboveapositiveone. Inourpatternreognitionexperimentweusesequenes

Q_L Q_U

0.25 0.75

neutral

negative positive

Figure2.1:Empirialdistributionofpooledurrenypairsreturns.

of3statestodenepartiulartrends. Aneutralstateinthesequenewillbeinter-

pretedasaontinuationofthepreviousstate. Thisyieldsthefollowing8possible

sequenes. Thebuy signalsforeah urrenypair translateinto atransation of

axedamountin theorder ofseveralperentoftheinitialwealth. Thereforethe

portfoliowillbeonlygraduallyinvested. Similarlyasellsignalwillleadto axed

redution of the invested position. This allowsthe portfolio to gradually swith

betweensituationsoffullandpartial investment.

Table2.1:Setofpossiblesequenes.

t 0 − 2 t 0 − 1 t 0 t 0 + 1

ր ր ր

ր ր ց

ր ց ր

ր ց ց

ց ր ր

ց ր ց

ց ց ր

ց ց ց

2.4 Empirial results

2.4.1 Data and baktesting sheme

The data have been downloadedfrom www.oanda.omand omprisetik-by-tik

time series of two sets of ve urreny pairs, whih are EUR/USD, GBP/USD,

CAD/USD, CHF/USD and JPY/USD. One set overs the period from 31-De-

2007to 31-De-2008andtheseondtheperiodfrom01-Jan-2010to31-De-2010.

Thenumberofdatapointsexeedssome10millionobservationsforeahurreny

pair. Inoursimulationthemidquoteshavebeenonsidered.Thedataarenotused

tik-by-tikbutathourlyordailyfrequeny. Forthemodellingproessweonsider

themedianoftheintrahour,respetivelyintradaytikswhileforrebalaningthe

last tik is used. Figure 2.2 shows the plot of the mid prie time series for all

urreny pairs of the rst set (year 2008). For better readability alltime series

startatone. Figure 2.3showstheintra hourvolatilityforthesameseries.

Dec07 Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08 Jan09 EUR 1

GBP 1 CAD 1 CHF 1 YEN 1

Figure2.2: Saledmidquotesofurrenyrates.

Dec07 Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08 Jan09 EUR

GBP CAD CHF YEN

Figure2.3: Intrahourlyvolatilityofurrenyrates.

Theperformaneofthesuggestedportfoliosisbaktestedoverthewholedata set

using a rolling window tehnique. The window is partitioned into an historial

segment of length

H

length

F

Theshemebelowsummarizesthetehnique.

Period

1

t 1 − ^H t 1 t 1 + ^F

F

H

invest

Period

2

t 2 − ^H t 2 t 2 + ^F

rebalane

In order not to rely on simple point estimates of the simulated performane we

alwaysompute a set of 100 simulationsfrom jakknifedhistorial observations.

The jakkning onsist in removing randomlysome 10%of datapointsfrom the

historialwindow. Theresultsobtainedfromthe100simulationsaresummarized

withthemedianpathwhihorrespondstothewealthpathoftheportfoliohaving

themediannalwealth. Wealsogivetheempirialdistributionsofthenalwealth.

Simulation results are ompared with a benhmark portfolio whih is the result

of a buy-and-hold strategy. The initial alloation of the benhmark portfolio is

proportional to the turnover of the urrenies in 2007 (BIS (2007)) and results

into the following weight vetor

[0.44 0.20 0.07 0.08 0.21]

1 / ^N

buy-and-holdbenhmarkandthereforeitisnotpresentedin theresults.

2.4.2 Simulation result for asset alloation approah

This approah lends itself to a large spetrum of speiations onsidering the

many possibilities of hoies for the aggregation level of the data, the length of

thehistorial window,rebalaningfrequenyand soon. An optimization overall

theseriteriaisnotfeasible(andnotwise)andoneisforedtoproeedwithsome

pragmatism.

Havingtested afewoptionsforthefrequenyofrebalanementsandtheaggrega-

tion levelof the prieswe retaineddaily rebalanementsand hourlyaggregation.

Inidentally,longerholdingperiods(foreasthorizons)implyfewerrebalanements

forthebaktesting. Thistranslatesintolowertransation ostsand,from aom-

putational pointof view,in asmallernumberofoptimizations.

To begin weompare the median wealth path and the nal wealth densities for

the models presented above, whih are maximum drawdown (

DD

momentum driven portfolios (

Mom

hoursand rebalanementsourdailyat 9am. All positionsin theportfoliosare

onstrainedtoabuy-inthresholdof5%andamaximumsizeof70%. Theminimum

trading size for eah position is 1000 USD whih implies that the portfolio may

remainunhangediftherebalanementssuggestedbytheoptimizationareinferior

to this threshold. Transation osts are set to 2.5 bp. Figure 2.4 illustrates the

resultsthatatthis stagearenotappealing.

The wealth pathsin Figure 2.4 reveal an alternationbetweenprotable and un-

derperformingperiods. This motivatesthe introdutionof the possibility forthe

portfolio toget partially orompletely divested, ahievedbyintroduing anarti-

ialpositionin USD.Forthisposition weallowamaximumweightof100%,i.e.

Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08 0.8

0.85 0.9 0.95 1 1.05 1.1

DD Omega Mom Bench

−15 0 −10 −5 0 5 10 15

0.1 0.5 0.9 1

Period returns: 15−Jan−2008 08:00:00 −− 31−Dec−2008 DD Omega Mom Bench

Figure2.4:Wealthpathsandnalwealthdistributionfor

DD

Mom

the portfolio mightbe fully or partially in ash. The resulting performanes are

shownin Figure2.5. Weobservethat drawdown andOmegaimprovein termsof

nal wealth densities. Fordrawdownin partiular, weobserveverylowvolatility

eventhough its behavior might look trivial, as we stay in ash for long periods

(the wealth path appears as straight line for most of the time). However mini-

mizing drawdown preserves the initial apital even in periods of adverse market

onditions, where the benhmark and all other strategiesare nonprotable, thus

drawdownseemstobeausefulomponentof aninvestmentmodel.

Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08

0.8 0.85 0.9 0.95 1 1.05 1.1

DD Omega Mom Bench

−15 0 −10 −5 0 5 10 15

0.1 0.5 0.9 1

Period returns: 15−Jan−2008 08:00:00 −− 31−Dec−2008 DD Omega Mom Bench

Figure2.5: Wealthpathsandnalwealthdensitiesfor

DD

Mom

Inthefollowingwewilloptforlessonservativestrategies,thatiswewillnotallow

ourportfoliostobeompletelyoutofthemarket. Thusashwillbesubjettothe

sameholding size onstraintsas theother assets. As aonsequenetheportfolio

willbeatleastinvested30%.

Onemightwonderwhyaportfoliowhihminimizesdrawdownanenteraposition

otherthanash,asthereisnodrawdownwithash. Thereexisttworeasonsforthis,

one is market dependent, i.e., historial diversied portfolios without drawdown

mayexistandbehosenbytheoptimization;theotherdependsontheoptimization

tehnique: weinterrupttheoptimization proedure afteragivennumberofsteps

whihdoesnotneessarilyproduetheglobaloptimum. Thisis doneonpurpose

in ordertoavoidovertting.

Filtering out noise

It isawellknownfat thatnanialdata ontainmuh noise. Inafurther inves-

tigation we analyzethe impat that ltering outnoise from the data has onthe

performane of the portfolios. Filtering is done by approximatingthe matrix of

observedquotes byasumof rank-one matriesomputedfrom the leftand right

singular vetors of the singular value deomposition. Comparing the results for

fully invested portfolios in the upper panels of Figure 2.6 with 2.4 reveals that

ltering datasigniantlyenhanesoverall performane forallstrategies. This is

alsotheaseforthepartiallyinvestedportfolios(seelowerpanelin Figure2.6).

Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08

0.8 0.85 0.9 0.95 1 1.05 1.1

DD Omega Mom Bench

−15 0 −10 −5 0 5 10 15

0.1 0.5 0.9 1

Period returns: 16−Jan−2008 08:00:00 −− 30−Dec−2008 DD Omega Mom Bench

Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08

0.8 0.85 0.9 0.95 1 1.05 1.1

DD Omega Mom Bench

−15 0 −10 −5 0 5 10 15

0.1 0.5 0.9 1

Period returns: 16−Jan−2008 08:00:00 −− 30−Dec−2008 DD Omega Mom Bench

Figure2.6: Wealthpathsand nalwealth densitiesfor

DD

Mom

Upperpanel: portfoliofullyinvested.Lowerpanel: maximumash70%.

Combining objetives

The nal wealth densities show that drawdown dominates. However when in-

spetingthe wealth path,theother strategiesshow,forertain periods, desirable

properties. Therefore, ombiningseveralobjetivesshould resultinbetteroverall

performane. Inthefollowingourobjetivefuntion

Φ(x)

singlestrategiesdesribedearlier,i.e.

Φ(x) = Ψ( D t , Omega, Mom t ) .

where

Ψ

illustratesresultsforsuh ombinations. Onlydrawdowninonjuntionwithmo-

mentumperformswell. ThereasonwhyOmegadoesnotontributeinperformane

mightresideinthefatthatOmegaisnotpathdependentbut

Mom

are. Also,Omegadoesnotusetheinformationprovidedbythewealthpath, that

drawdownandmomentumdo.

Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08

0.8 0.85 0.9 0.95 1 1.05 1.1

DDMom DDOm DDOmMom Bench

−15 0 −10 −5 0 5 10 15

0.1 0.5 0.9 1

Period returns: 15−Jan−2008 08:00:00 −− 31−Dec−2008 DDMom

DDOm DDOmMom Bench

Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08

0.8 0.85 0.9 0.95 1 1.05 1.1

DDMom DDOm DDOmMom Bench

−15 0 −10 −5 0 5 10 15

0.1 0.5 0.9 1

Period returns: 15−Jan−2008 08:00:00 −− 31−Dec−2008 DDMom

DDOm DDOmMom Bench

Figure 2.7: Wealth paths and nal wealth densities for ombined objetivesand ltereddata.

Upperpanel: fullyinvestedportfolio. Lowerpanel: partiallyinvestedportfolio.

We havetwoalternatives,either bealwaysfullyinvested or allowash positions.

The rstoptionyieldshigher average nal wealth,see upperpanelof Figure 2.7,

theseond ismoreonservative,yieldinglessreturn,but alsolowervolatilityand

almostalwayspreservestheinitialwealth(lowerpanelofFigure2.7).

Inreasing holdingperiods/short positions

Retaining the best performingstrategies from above weontinue to enhane our

portfoliosbyinreasingthelengthoftheholdingperiods (i.e.reduingrebalane-

ment frequeny), introduing simulated priesfor the optimization and allowing

for short positions up to 30 perent of the value of the portfolio. The numeri-

alproedureforomputingsuhportfoliosremainsunhanged. Notethat,unlike

stokmarkets,urrenymarketsdonotriskshortsqueezesbuttheinvestorisstill

exposedto interestratemovements. Thisis why alimithasbeenimposed tothe

short book. Thepriegeneration proessfollowsideasfromDembo(1991)andis

explainedin moredetailin GilliandShumann(2011).

Altogether this leads to a signiant improvement for nal wealth as well as to

lowernal wealth volatility. Figure 2.8 summarizesthese resultsfor thestrategy

ombiningdrawdownwithmomentum. Bestresultsareobtainedfor theportfolio

allowingshortpositionswhihreahesamediannal wealth ofabout12(beating

thebenhmarkby17).

Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08

0.9 0.95 1 1.05 1.1

1.15 DDMom

Bench

−15 0 −10 −5 0 5 10 15

0.1 0.5 0.9 1

Period returns: 15−Jan−2008 08:01:00 −− 31−Dec−2008 DDMom

Bench

Feb08 Apr08 May08 Jul08 Aug08 Oct08 Dec08

0.9 0.95 1 1.05 1.1

1.15 DDMom

Bench

−15 0 −10 −5 0 5 10 15

0.1 0.5 0.9 1

Period returns: 15−Jan−2008 08:00:00 −− 31−Dec−2008 DDMom

Bench

Figure2.8:Wealthpathsandnalwealthdensitiesforombinedobjetives,simulatedpriesand2

rebalanemetsperweek. Upperpanel:Longonlyportfolios. Lowerpanel: Long/shortportfolios.

2.4.3 Simulation results for tehnialtrading

As explained in Setion 2.3.2 the model generates `buy', `sell' and `hold' signals

whiharetranslatedinto xed-sizetradingorders. Asin theassetalloationase

thereisaminimumtradingsize.Theminimumandthemaximumholdingsizeare

thoseusedin theoptimizationapproah. Thereisnoadditionalonstraintforthe