Growth optimal investment in discrete-time markets with proportional transaction costs

Contents lists available atScienceDirect

Digital

Signal

Processing

www.elsevier.com/locate/dsp

Growth

optimal

investment

in

discrete-time

markets

with proportional

transaction

costs

N.

Denizcan Vanli

a

,

∗

,

Sait Tunc

b

,

Mehmet

A. Donmez

c

,

Suleyman

S. Kozat

a a_{DepartmentofElectricalandElectronicsEngineering,BilkentUniversity,Ankara,Turkey}

b_{DepartmentofIndustrialandSystemsEngineering,UniversityofWisconsin-Madison,WI,USA} c_{DepartmentofElectricalandComputerEngineering,UniversityofIllinoisatUrbana-Champaign,IL,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline19October2015 Keywords:

Growthoptimalportfolio Thresholdrebalancing Proportionaltransactioncost Discrete-timestockmarket

Weinvestigatehowandwhentodiversifycapitaloverassets,i.e.,theportfolioselectionproblem,froma signalprocessingperspective.Tothisend,wefirstconstructportfoliosthatachievetheoptimalexpected growth ini.i.d. discrete-timetwo-asset markets under proportionaltransactioncosts. We thenextend our analysis to cover markets having more than two stocks. The market is modeled by a sequence of pricerelative vectorswith arbitrarydiscrete distributions,which canalso beused toapproximate a wideclassofcontinuous distributions.To achieve the optimal growth, weuse threshold portfolios, where weintroduce arecursive update to calculate the expectedwealth. We thendemonstrate that underthethresholdrebalancingframework,theachievablesetofportfolioselegantlyformanirreducible Markovchainundermildtechnicalconditions.Weevaluatethecorrespondingstationarydistributionof thisMarkovchain,whichprovidesanaturaland efficientmethodtocalculatethecumulativeexpected wealth.Subsequently,thecorrespondingparametersareoptimizedyieldingthegrowthoptimalportfolio underproportionaltransactioncostsini.i.d.discrete-timetwo-assetmarkets.Asawidelyknownfinancial problem, wealso solve the optimal portfolio selection problem indiscrete-time markets constructed bysamplingcontinuous-timeBrownianmarkets.Forthecasethattheunderlyingdiscretedistributions of the price relative vectorsare unknown,we provide a maximum likelihoodestimator that is also incorporatedintheoptimizationframeworkinoursimulations.

©2015ElsevierInc.All rights reserved.

1. Introduction

The problem of how and when an investor should diversify capital over various assets, whose future returns are yet to be realized,isextensivelystudiedinvariousdifferentfieldsfrom sig-nal processing[1,2,12,28–30,33,35]andfinancial engineering[25, 26]tomachinelearning[13,34]andinformationtheory[9]. Natu-rally,thisisone ofthemostimportantfinancialapplications due to the amount of money involved. However, the recent financial crisis demonstrated that there isa significant room for improve-ment in this field by sound signal processing methods [12,30], whichisthemaingoalofthispaper.Inthispaper,weinvestigate howandwhentodiversifycapitaloverassets,i.e.,theportfolio se-lectionproblem,froma signalprocessingperspectiveandprovide portfolioselectionstrategiesthat maximizethe expected

cumula-*

Correspondingauthor.

E-mailaddresses:vanli@ee.bilkent.edu.tr(N.D. Vanli),

stunc@wisc.edu

(S. Tunc), donmez2@illinois.edu(M.A. Donmez),

kozat@ee.bilkent.edu.tr

(S.S. Kozat).

tive wealth in discrete-timemarkets under proportional transac-tioncosts.

Inparticular,we studyan investmentprobleminmarketsthat allows tradingatdiscreteperiods,wherethediscreteperiodis ar-bitrary,e.g.,itcanbeseconds,minutesordays[24].Furthermore, the market levies transactionfeesforboth selling andbuyingan asset proportional to the volume of trading at each transaction, whichaccuratelymodelsabroadrangeoffinancialmarkets[3,24]. Inourdiscussions,wefirstconsidermarketswithtwoassets. Two-stockmarketsareextensivelystudiedinfinancialliteratureandare shownto accurately modelawide rangeoffinancial applications

[24] such asthe well-known “Stock and Bond Market” [24]. We thenextendouranalysistomarketshavingmorethantwoassets, i.e.,m-stockmarkets,wherem isarbitrary.

Following the extensive literature [9,19,24–26,33], the market ismodeled byasequenceofpricerelativevectors,say

{

X

(

n

)

}n

≥1,

X

(

n

)

∈ [

0

,

∞)

m, whereeach entryof X

(

n

)

,i.e., Xi

(

n

)

∈ [

0

,

∞)

,is theratiooftheclosingpricetotheopeningpriceoftheithstock per investment period. Hence, each entry of X

(

n

)

quantifies the gain (or the loss) of that asset at each investment period. The sequenceofpricerelativevectorsisassumedtohaveani.i.d.

“dis-http://dx.doi.org/10.1016/j.dsp.2015.08.009 1051-2004/©2015ElsevierInc.All rights reserved.

crete” distribution [24–26,33],however, the discrete distributions on the vector of price relatives are arbitrary. In this sense, the correspondingdiscretedistributions canapproximatea wideclass ofcontinuous distributionson thepricerelativesthat satisfy cer-tain regularity conditions by appropriately increasing the size of thediscretesamplespace.Wefirstassumethatthediscrete distri-butions onthe pricerelative vectorsare knownand thenextend ouranalysistocoverthecase, wheretheunderlyingdistributions areunknown.We emphasizethatthei.i.d.assumptiononthe se-quenceofpricerelative vectorsisshowntoholdinmostrealistic markets[14,24].

At each investment period, the diversification of the capital over the assets is represented by a portfolio vector b

(

n

)

, where

∀

i

∈ {

1

,

. . . ,

m

}

,bi

(

n

)

representstheratioofthecapitalinvestedin theithassetatinvestmentperiodn, i.e.,wehave

m

_i=1bi

(

n

)

=

1, where 0

≤

bi

(

n

)

≤

1. As an example, if we invest using b

(

n

)

, we earn (or lose) bT

(

n

)

X

(

n

)

at the nth investment period, af-ter X

(

n

)

is revealed. Given that we start with one dollar, af-teran investment periodof N days,we have the wealthgrowth

N

n=1bT

(

n

)

X

(

n

)

.Underthisgeneralmarketmodel,weprovide al-gorithmsthatmaximizetheexpectedgrowthoveranyperiodN by

using“thresholdrebalancedportfolios”(TRPs),whichareshownto yieldoptimalgrowthingenerali.i.d.discrete-timemarkets[14].

In [9], Cover et al. showed that the portfolio that achieves the maximal growth is a constant rebalanced portfolio (CRP) in i.i.d.discrete-timemarkets, undercertain assumptions onthe se-quenceofpricerelativesandwithoutanytransactioncosts.ACRP isaportfolioinvestmentstrategy,wherethefractionofwealth in-vestedin each stock is kept constant ateach investment period. A problemextensivelystudiedinthisframeworkistofind sequen-tialportfolios that asymptotically achieve the wealthof the best CRP tuned to the underlying sequence of price relatives. Several sequentialalgorithms are introduced to achieve the performance ofthebest CRP (suchas[9,13,16,34]) withdifferentconvergence ratesanddifferentperformancesonhistoricaldatasets.In[3], se-quentialalgorithms thatachieve theperformanceofthebest CRP under transaction costs are introduced. However, we emphasize that keeping a CRP may requireextensive trading due to a pos-sible rebalancing at each investment period which deems CRPs (eventhebestCRP)ineffectiveinrealisticmarketsevenundermild transactioncosts[19].

Incontinuous-time markets, however,it has beenshown that under transaction costs, the optimal portfolios that achieve the maximal wealth are certain class of “no-trade zone” portfolios

[7,11,32]. In simple terms, a no-trade zone portfolio has a com-pactclosed set and a rebalancing occurs ifthe current portfolio breaches this set, otherwise no rebalancing occurs. Clearly, such ano-tradezone portfoliomayavoid heftytransactioncosts since itcanlimitexcessiverebalancingbydefiningappropriateno-trade zones.Analogous to continuous time markets,it hasbeenshown in[14] that in two-asset i.i.d. markets under proportional trans-action costs, compact no-trade zone portfolios are optimal such thatthey achievethe maximalgrowthundercertain assumptions onthesequenceofpricerelatives.Intwo-assetmarkets,the com-pactno-tradezone isrepresentedby thresholds,e.g., ifat invest-mentperiodn,theportfolioisgivenbyb

(

n

)

= [

b

(

n

),

(

1

−

b

(

n

))

]

T, where0

≤

b

(

n

)

≤

1,thenrebalancingoccursifb

(

n

)

∈ (

/

α

,

β)

,given thethresholds

α

,

β

,where 0

≤

α

≤ β ≤

1.Similarly, the interval

(

α

,

β)

canbe representedusinga target portfoliob anda region aroundit,i.e.,

(

b

−



,

b

+



)

,where0

≤



≤

min

{

b

,

1

−

b

}

suchthat

α

=

b

−



and

β

=

b

+



.ExtensionofTRPstomarketshavingmore thantwostocksisstraightforwardandexplainedinSection3.2.

However,howtoconstructtheno-tradezoneportfolio,i.e.,how toselectthethresholdsthatachievethemaximalgrowth,hasnot yetbeensolvedexceptinelementaryscenarios[14].In[15],a uni-versalalgorithm that asymptotically achieves the performance of

the best TRPtuned to the underlyingsequence of pricerelatives is introduced. This algorithm leverages Bayesian type weighting from [9] inspired from universal source coding and requires no statisticalassumptionsonthesequenceofpricerelatives.In simi-larlines,variousdifferentuniversalalgorithmsareintroducedthat achieve the performance ofthe best expertindifferent competi-tion classes in[1,2,17–20].Although the performance guarantees in[1,2,15,18,19]) are derivedwithout anystochasticassumptions, thesemethodsarehighly conservativeduetotheworst case sce-nario optimization, i.e., they are only optimalin an asymptotical sense.However,theorderofsuchperformanceupperboundsmay not be negligible in actual financial markets [6,20], even though theymaybeneglected insourcecodingapplications(wherethese algorithms are inspired from). We demonstrate that our algo-rithmreadilyoutperformstheseuniversalmethodsoverhistorical data.

Ourmain contributions are asfollows.We first consider two-asset markets and recursively evaluate the expected achieved wealth ofa threshold portfoliofor anyb and



overany invest-mentperiod.Wethenextendthisanalysistomarketshavingmore than two-stocks. We next demonstrate that under the thresh-old rebalancing framework, the achievable set ofportfolios form an irreducible Markovchainunder mild technicalconditions.We evaluate thecorresponding stationary distributionof thisMarkov chain,which providesa naturalandefficientmethodto calculate the cumulativeexpected wealth.Subsequently, thecorresponding parameters are optimized using a brute force approach yielding

thegrowthoptimalinvestmentportfoliounderproportionaltransaction costsini.i.d.discrete-time two-asset markets. We note thatfor the casewiththeirreducibleMarkovchain,whichcoverspracticallyall scenariosintherealisticmarkets,theoptimizationofthe parame-tersisofflineandcarriedoutonlyonce.However,forthecasewith recursivecalculations, thealgorithmhasanexponential computa-tionalcomplexityintermsofthenumberofstates.However,inour simulations,weobservethatareducedcomplexityformofthe re-cursivealgorithm thatkeepsonlya constantnumberofstatesby appropriately pruning certain states provides nearly identical re-sultswiththe“optimal”algorithm.Furthermore,asawellstudied problem,wealsosolveoptimalportfolioselectionindiscrete-time markets constructedby samplingcontinuous-timeBrownian mar-kets [24].When theunderlyingdiscrete distributions oftheprice relative vectors are unknown, we provide a maximum likelihood estimatortoestimatethecorrespondingdistributionsthatis incor-poratedintheoptimizationframeworkintheSimulationssection. Foralltheseapproaches, wealsoprovidethecorresponding com-plexitybounds.

The organization of the paper is as follows. In Section 2, we briefly describe our discrete-time stock market model with dis-cretepricerelativesandsymmetricproportional transactioncosts. In Section 3, we start to investigate TRPs, where we first intro-duce a recursive update in Section 3.1 for a market having two-stocks. Generalization of the iterative algorithm to the m-asset

market case is provided in Section 3.2. We then show that the TRP framework can be analyzedusing finite state Markovchains in Section 3.4 and Section 3.5. The special Brownian market is analyzedinSection3.6.Themaximumlikelihoodestimatoris de-rivedinSection4.Wesimulatetheperformanceofouralgorithms inSection5andconcludethepaperwithcertainremarks in Sec-tion6.

2. Problem description

We consider discrete-time stock markets under transaction costs. We first consider a market with two stocks and then ex-tend the analysis to markets having more than two stocks. We modelthemarketusingasequenceofpricerelativevectors

X

(

n

)

.

A vector of price relatives X

(

n

)

=

[ X1

(

n

), . . . ,

Xm

(

n

)

]T in a mar-ket ofm assetsrepresents thechange inthe prices ofthe assets overinvestment periodn, i.e.,forthe ith stock Xi

(

n

)

isthe ratio oftheclosingtotheopeningpriceoftheithstock overperiod n. For a market having two assets,we have X

(

n

)

=

[ X1

(

n

)

X2

(

n

)

]T. Weassumethat thepricerelativesequences X1

(

n

)

and X2

(

n

)

are i.i.d. over with possibly different discrete sample spaces

X

1 and

X

2,i.e., X1

(

n

)

∈

X

1 and X2

(

n

)

∈

X

2,respectively [14].For techni-cal reasons,inourderivations, we assume thatthe sample space is

X  X

1

∪

X

2

= {

x1

,

x2

,

. . . ,

xK}forboth X1

(

n

)

and X2

(

n

)

where

|

X |

=

K isthecardinalityoftheset

X

.Theprobabilitymass func-tion(pmf)of X1

(

n

)

is p1

(

x

)



Pr

(

X1

=

x

)

andthepmfof X2

(

n

)

is p2

(

x

)



Pr

(

X2

=

x

)

. Wedefine pi,1

=

p1

(

xi

)

and pi,2

=

p2

(

xi

)

for

xi

∈

X

andtheprobability massvectors

p

1

=



p1,1p2,1

. . .

pK,1

T

andp2

=



p1,2p2,2

. . .

pK,2

T

,respectively.Here,we firstassume that the corresponding probability mass vectors p1 and p2 are known.Wethenextendouranalysistothecase,where

p

1 and

p

2 areunknown, andsequentiallyestimate p1 andp2 using a maxi-mumlikelihoodestimatorinSection4.

Anallocation ofwealth overtwo stocks isrepresentedby the portfoliovector

b

(

n

)

=

[b1

(

n

),

b2

(

n

)

]T,whereb1

(

n

)

andb2

(

n

)

rep-resentstheproportionof wealthinvested inthefirst andsecond stocks,respectively,atinvestmentperiodn.Intwo stockmarkets, we haveb2

(

n

)

=

1

−

b1

(

n

)

,thus b

(

n

)

is completelycharacterized by the proportion ofthe total wealth invested in the first stock, i.e., b1

(

n

)

. For notational clarity, we use b

(

n

)

instead of b1

(

n

)

throughout the paper, hence our portfolio vector is denoted by

b

(

n

)

= [

b

(

n

),

1

−

b

(

n

)

]

T.

We denote a threshold rebalancing portfolio with an initial and target portfolio b and a threshold



by TRP

(

b

,



)

. At each market periodn, an investor rebalances the asset allocation only if the portfolio leaves the interval

(

b

−



,

b

+



)

. When b

(

n

)

∈

/

(

b

−



,

b

+



)

, the investor buys andsells stocksso that the as-setallocation is rebalanced tothe initial allocation,i.e., b

(

n

)

=

b,

wheretheinvestorhastopaytransactionfees.Weemphasizethat therebalancing canbe madedirectly totheclosest boundary in-steadoftob as suggestedin[14],however,werebalancetob for

notationalsimplicity,whereasourderivationsalsoholdforthe re-balancingtotheboundarycase.Wemodelthetransactioncostof rebalancingbya fixedproportionalcost c

∈ (

0

,

1

)

[3,14,19].Asan example,iftheinvestor buysorsells S dollarsofstocks,then he

pays c S dollars of transaction fees.Although we assume a

sym-metrictransactioncostratio,alltheresultscanbecarriedoverto marketswithasymmetriccosts[14,19].

We let S

(

N

)

denote the achieved wealth at investment

pe-riod N and assume, without loss of generality, that the initial

wealth of the investor is 1 dollar. Then, as an example, if the portfolio b

(

n

)

doesnot leave the interval

(

b

−



,

b

+



)

andthe allocation of wealth is not rebalanced for N investment peri-ods,then theachievedwealthisgivenby S

(

N

)

=

b

N

_n=1X1

(

n

)

+

(

1

−

b

)

N

_n=1X2

(

n

)

andthe currentproportionofwealthinvested in the first stock is given by b

(

N

)

=



b

N

_n=1X1

(

n

)

 

S

(

N

)

. On the other hand, ifthe portfolio leavesthe interval

(

b

−



,

b

+



)

at period N, i.e., b

(

N

)

∈ (

b

−



,

b

+



)

, then the investor rebal-ances the asset distribution to the initial distribution and pays

c S

(

N

)

|

b

(

N

)

−

b

|

dollars fortransactioncosts [3].In thenext sec-tion, we first derive a compact form for the expected achieved wealth E

[

S

(

N

)

]

so that we can optimize b and



to maximize

E

[

S

(

N

)

]

.

3. Threshold rebalanced portfolios

Inthissection, we first investigateTRPs indiscrete-time two-assetmarketsunderproportionaltransactioncosts.Wefirst intro-duceaniterativemethodtocalculatetheexpectedachievedwealth

Fig. 1. Block diagram representation of N period investment.

atagiveninvestmentperiod.Wethenpresentanupperboundon the complexity ofthisalgorithm. Wenext calculate theexpected achieved wealth of markets having more than two assets, i.e.,

m-assetmarketsforanarbitrarym.Wethenprovidethenecessary andsufficientconditionssuchthattheachievableportfoliosare fi-nite such that thecomplexity of the algorithmdoes notgrow at anyperiod.Wealsoshowthattheportfoliosequenceconvergesto astationarydistributionandderivetheexpectedachievedwealth. Based on the calculation of the expected achieved wealth, we optimize b and



using a brute-force search. Finally, with these derivations, we consider the well-known discrete-time two-asset Brownian marketwithproportional transaction costsand investi-gatetheasymptoticexpectedachievedwealth.

3.1. Aniterativealgorithm

Before introducing the iterativealgorithm, we first define the set of achievable portfolios at each investment period since the iterative calculationof theexpected achievedwealth isbased on theachievableportfolioset.Wethen introducetheportfolio tran-sitionsets andthe transitionprobabilitiesofachievableportfolios at successive investment periods in order to iteratively find the probabilityofeachportfoliostateandtocalculateE

[

S

(

N

)

]

.

We define the set ofachievable portfolios at each investment periodasfollows.As

|

X |

=

K

<

∞

,thesetofachievableportfolios atperiod N canonlyhavefinitelymanyelements. Wedefine the setofachievableportfoliosatperiodN as

B

N

= {

b1,N

,

. . . ,

bMN,N

}

,

whereMN

 |

B

N

|

.AsillustratedinFig. 1,foreachachievable port-foliobm,N

∈

B

N,m

=

1

,

. . . ,

MN,thereisacertainsetofportfolios in

B

N−1 that are connected to bm,n. At a given investment pe-riod N,thesetofachievableportfolios

B

N isgivenby

B

N

= {

b

} ∪

bm,N

:

bm,N

=

bk,N−1u bk,N−1u

+ (

1

−

bk,N−1

)

v

∈ (

b

−



,

b

+



),

u

,

v

∈

X

.

(1)

Since b is an achievable portfolio at each N (due to a possible rebalancing), withoutloss of generality, we let b1,N

=

b for each

N

∈ N

.NotethatinFig. 1,thesizeofthesetofachievable portfo-liosmaygrowasN increases.

Having constructeda state transition diagram, we next repre-sentthetransitionprobabilitiesfrombk,N−1 tobm,N asfollows qk,m,N



Pr



b

(

N

)

=

bm,N

|

b

(

N

−

1

)

=

bk,N−1



,

(2)

where k

=

1

,

. . . ,

MN−1 and m

=

1

,

. . . ,

MN. Given that b

(

N

)

=

bm,N,forsomem

=

1

,

. . . ,

MN,thereexistsacorrespondingsetof portfolios

N

m,N

⊂

B

N−1 fromwhichbm,N isachievable,i.e.,

N

m,N

 {

bk,N−1

∈

B

N−1

:

qk,m,N

>

0

,

k

=

1

, . . . ,

MN−1

}

(3) wherem

=

1

,

. . . ,

MN.Then,theprobabilityofeachportfoliostate canbecalculatedasfollows

Pr



b

(

N

)

=

bm,N



=

bk,N−1∈Nm,N qk,m,NPr



b

(

N

−

1

)

=

bk,N−1



(4)

wherem

=

1

,

. . . ,

MN.Bythe definitionof

B

N andusingthelaw oftotalexpectation[31],wecanwrite

E

[

S

(

N

)

] =

MN

m=1 Pr



b

(

N

)

=

bm,N



E



S

(

N

)

|

b

(

N

)

=

bm,N



.

(5)

Toobtainaniterativeformulationusing(5),wenextfindthe tran-sitionprobabilitiesbetweenachievableportfolios.

Toaccomplishthis,welet

U

k,m,N denotethesetofpricerelative vectorsthatconnectbk,N−1tobm,N,i.e.,

U

k,m,N



w

= [

w1

,

w2

]

T

∈

X

2

:

w1bk,N−1 w1bk,N−1

+

w2

(

1

−

bk,N−1

)

=

bm,N

,

for k

=

1

,

. . . ,

MN−1 and m

=

2

,

. . . ,

MN. We consider the price relative vectorsthat connect bk,N−1 to b1,N

=

b separately since, in this case, there are two subcases depending on whether the portfolio leaves the interval

(

b

−



,

b

+



)

or not. We let

U

k,1,N

=

V

k,1,N

∪

R

k,1,N, where

V

k,1,N is the set of price rela-tive vectorsthat connect bk,N−1 tob1,N

=

b withoutrebalancing, i.e.,

V

k,1,N

=

w

=

[w1

,

w2]T

∈

X

2

:

w1bk,N−1 w1bk,N−1

+

w2

(

1

−

bk,N−1

)

=

b

,

(6) and

R

k,1,N isthesetofpricerelativevectorsthatconnect bk,N−1 tob1,N withrebalancing,i.e.,

R

k,1,N

=

w

=

[w1

,

w2]T

∈

X

2

:

w1bk,N−1 w1bk,N−1

+

w2

(

1

−

bk,N−1

)

/

∈ (

b

−



,

b

+



)

.

(7)

Withthesedefinitions, we can obtain the transitionprobabilities asfollows qk,m,N

=

Pr



X

(

N

)

∈

U

k,m,N



=

w∈Uk,m,N p1

(

w1

)

p2

(

w2

),

(8) wherek

=

1

,

. . . ,

MN−1 andm

=

1

,

. . . ,

MN.

Havingderived a recursiveformulationforthestate probabili-ties, we can calculatethe termin thesum in(5) by considering twoseparatecasesasfollows.

i) As the first case, we consider b

(

N

)

=

bm,N, where m

=

2

,

. . . ,

MN,i.e.,we knowthat theportfoliodoesnotleave the in-terval

(

b

−



,

b

+



)

atperiodN. Therefore,notransactioncostis paidandwehave

Pr



b

(

N

)

=

bm,N



E

[

S

(

N

)

|

b

(

N

)

=

bm,N

]

=

bk,N−1∈Nm,N E



S

(

N

)

|

b

(

N

)

=

bm,N

,

b

(

N

−

1

)

=

bk,N−1



×

Pr



b

(

N

)

=

bm,N



Pr



b

(

N

−

1

)

=

bk,N−1

|

b

(

N

)

=

bm,N



(9)

=

bk,N−1∈Nm,N E



S

(

N

)

|

b

(

N

)

=

bm,N

,

b

(

N

−

1

)

=

bk,N−1



×

Pr



b

(

N

−

1

)

=

bk,N−1



qk,m,N

,

(10)

=

bk,N−1∈Nm,N E



S

(

N

)

|

b

(

N

)

=

bm,N

,

b

(

N

−

1

)

=

bk,N−1



×

Pr



b

(

N

−

1

)

=

bk,N−1



Pr



X

(

N

)

∈

U

k,m,N



,

(11)

=

bk,N−1∈Nm,N

w∈Uk,m,N E



S

(

N

)

|

b

(

N

)

=

bm,N

,

b

(

N

−

1

)

=

bk,N−1

,

X

(

N

)

=

w



×

Pr



b

(

N

−

1

)

=

bk,N−1



Pr



X

(

N

)

=

w

|

X

(

N

)

∈

U

k,m,N



×

Pr



X

(

N

)

∈

U

k,m,N



,

(12)

where(9)and(12)followfromthelawoftotalexpectation[31],

(10)followsfromBayes’theorem[31],and(11)followsfrom(2). Asnorebalancingoccurs,wealsohave

E



SN

|

b

(

N

)

=

bm,N

,

b

(

N

−

1

)

=

bk,N−1

,

X

(

N

)

=

w



=

E



S

(

N

−

1

)(

bk,N−1w1

+ (

1

−

bk,N−1

)

w2

)

|

b

(

N

−

1

)

=

bk,N−1



.

(13)

Putting(13)backin(12),weobtain Pr



b

(

N

)

=

bm,N



E

[

S

(

N

)

|

b

(

N

)

=

bm,N

]

=

bk,N−1∈Nm,N

w∈Uk,m,N Pr



b

(

N

−

1

)

=

bk,N−1



Pr

(

X

(

N

)

=

w

)

×

E



S

(

N

−

1

)(

bk,N−1w1

+ (

1

−

bk,N−1

)

w2

)

|

b

(

N

−

1

)

=

bk,N−1



(14)

=

bk,N−1∈Nm,N Pr



b

(

N

−

1

)

=

bk,N−1



×

E



S

(

N

−

1

)

|

b

(

N

−

1

)

=

bk,N−1



×

w∈Uk,m,N

(

bk,N−1w1

+ (

1

−

bk,N−1

)

w2

)

p1

(

w1

)

p2

(

w2

),

(15) wherethelastlinefollowssincePr

(

X

(

N

)

=

w

)

=

p1

(

w1

)

p2

(

w2

)

.

ii) In the second case, we have b

(

N

)

=

b1,N. For thiscase, if

X

(

N

)

∈

V

k,1,N,then the portfoliois not rebalanced andno trans-actionfee ispaid; whereasif

X

(

N

)

∈

R

k,1,N,thenthe portfoliois rebalancedandtransactioncostispaid.Thus,wehave

Pr



b

(

N

)

=

b1,N



E

[

S

(

N

)

|

b

(

N

)

=

b1,N

]

=

bk,N−1∈N1,N E



S

(

N

)

|

b

(

N

)

=

b1,N

,

b

(

N

−

1

)

=

bk,N−1



×

Pr



b

(

N

−

1

)

=

bk,N−1



qk,1,N (16)

=

bk,N−1∈N1,N

w∈Vk,1,N Pr



b

(

N

−

1

)

=

bk,N−1



Pr

(

X

(

N

)

=

w

)

×

E



SN

|

b

(

N

)

=

b1,N

,

b

(

N

−

1

)

=

bk,N−1

,

X

(

N

)

=

w



+

w∈Rk,1,N Pr



b

(

N

−

1

)

=

bk,N−1



Pr

(

X

(

N

)

=

w

)

×

E



SN

|

b

(

N

)

=

b1,N

,

b

(

N

−

1

)

=

bk,N−1

,

X

(

N

)

=

w



,

(17)

where(16) followsfrom(10) andthe last linefollowsin similar linesto(14).Here,when

X

(

N

)

=

w

∈

V

k,1,N,norebalancingoccurs. Thus,wehave E



SN

|

b

(

N

)

=

b1,N

,

b

(

N

−

1

)

=

bk,N−1

,

X

(

N

)

=

w



=

E



S

(

N

−

1

)(

bk,N−1w1

+ (

1

−

bk,N−1

)

w2

)

|

b

(

N

−

1

)

=

bk,N−1



.

(18)

Onthe other hand,when X

(

N

)

=

w

∈

R

k,1,N,rebalancing occurs. Hence,wehave E



SN

|

b

(

N

)

=

b1,N

,

b

(

N

−

1

)

=

bk,N−1

,

X

(

N

)

=

w



=

E



S

(

N

−

1

)(

bk,N−1w1

+ (

1

−

bk,N−1

)

w2

)

×



1

−

c





bk,N−1w1 bk,N−1w1

+ (

1

−

bk,N−1

)

w2

−

b





 



b

(

N

−

1

)

=

bk,N−1



.

(19)

Putting(18)and(19)backin(17),weobtain Pr



b

(

N

)

=

b1,N



E

[

S

(

N

)

|

b

(

N

)

=

b1,N

]

=

bk,N−1∈N1,N Pr



b

(

N

−

1

)

=

bk,N−1



w∈Vk,1,N Pr

(

X

(

N

)

=

w

)

×

E



S

(

N

−

1

)(

bk,N−1w1

+ (

1

−

bk,N−1

)

w2

)

|

b

(

N

−

1

)

=

bk,N−1



+

w∈Rk,1,N Pr

(

X

(

N

)

=

w

)

×

E



S

(

N

−

1

)(

bk,N−1w1

+ (

1

−

bk,N−1

))

×



1

−

c





bk,N−1w1 bk,N−1w1

+ (

1

−

bk,N−1

)

w2

−

b





 



b

(

N

−

1

)

=

bk,N−1



.

Aftersomealgebra,weobtain Pr



b

(

N

)

=

b1,N



E

[

S

(

N

)

|

b

(

N

)

=

b1,N

]

=

bk,N−1∈N1,N Pr



b

(

N

−

1

)

=

bk,N−1



×

E



S

(

N

−

1

)

|

b

(

N

−

1

)

=

bk,N−1



×

w∈Vk,1,N

(

bk,N−1w1

+ (

1

−

bk,N−1

)

w2

)

p1

(

w1

)

p2

(

w2

)

+

w∈Rk,1,N

(

bk,N−1w1

+ (

1

−

bk,N−1

)

w2

)

p1

(

w1

)

p2

(

w2

)

×



1

−

c





bk,N−1w1 bk,N−1w1

+ (

1

−

bk,N−1

)

w2

−

b







.

(20)

Hence,wecaniterativelycalculate(5)i) using(15),forthecase where b

(

N

)

=

bl,N and l

=

2

,

. . .

MN, ii) using (20), for the case whereb

(

N

)

=

b1,N

=

b.Since we havethe recursive formulation, wecanoptimizeb and



by abruteforcesearch asshowninthe Simulationssection.Inthefollowingsection,we extendour anal-ysestothem-assetmarketcase.

3.2. Generalizationoftheiterativealgorithmtothem-assetmarketcase

In thissection, we generalize ouriterativemethod introduced in Section 3.1 for a market with m assets where m

∈ Z

+. We model the market as a sequence of i.i.d. price relative vectors

X

(

n

)

= [

X1

(

n

),

. . . ,

Xm

(

n

)

]

,where Xi

(

n

)

∈

X

andthepmfof Xi

(

n

)

is pi

(

x

)

=

Pr

(

Xi

(

n

)

=

x

)

. For the m-asset case, the portfolio vec-torisgivenby

b

(

n

)

= [

b1

(

n

),

. . . ,

bm

(

n

)

]

,where

m

i=1bi

(

n

)

=

1 and

bi

(

n

)

≥

0,thetarget portfoliovector isdefinedas

b

= [

b1

,

. . . ,

bm] andthethresholdvectorisgivenby



= [



1

,

. . . ,



m].Similartothe two-assetcase, TRP

(

b

,



)

rebalancesthewealthallocation

b

(

n

)

to

b only when

b

(

n

)

∈

/

b

 (

b1

−



1

,

b1

+



1

)

× (

b2

−



2

,

b2

+



2

)

×

. . .

× (

bm

−



m

,

bm

+



m

)

.Inthiscase,ifthewealthallocationisnot rebalanced forN investmentperiods,thentheachievedwealthis givenbyS

(

N

)

=

m

_k=1bk

N

n=1Xk

(

N

)

andtheproportionofwealth invested in the ith asset becomes bi

(

N

)

=

bi N n=1Xi(N) m k=1bknN=1Xk(N) . We definethesetofachievableportfoliosatperiodN as

B

N

= {

b

} ∪

bm,N

:

bk,N

=

bm,N−1

◦

x xT_b m,N−1

∈

b

,

x

∈

X

m

,

(21)

where b

◦

x

 [

b1x1

,

. . . ,

bmxm]T is an elementwisemultiplication operation. In accordance with the definitions given in two-asset marketcase,thedefinitionsoftheportfoliotransitionsetsandthe transitionprobabilitiesofachievableportfoliosfollow.Then,similar to the iterativealgorithm introduced inSection 3.1,i.e., (15)and

(20),we caniteratively calculatethe followingexpectedachieved wealth E

[

S

(

N

)

] =

MN

l=1 Pr



b

(

N

)

=

bl,N



E



S

(

N

)

|

b

(

N

)

=

bl,N



.

(22)

We emphasize that the complete iterations for the m-asset case can be obtained by changingthe scalars in (15)and(20) by the correspondingvectors.Fortherestofthepaper,weconsider two-assetmarketsforeaseofexposition.

3.3. Complexityanalysisoftheiterativealgorithm

In thissection, we investigatethenumber ofachievable port-folios at a given market period to determine the complexity of the iterativealgorithm.We show that theset ofachievable port-foliosatperiod N isequivalenttotheset ofachievable portfolios when theportfoliob

(

n

)

doesnotleave theinterval

(

b

−



,

b

+



)

forN investmentperiods.Wefirstdemonstratethatifthe portfo-lioneverleavestheinterval

(

b

−



,

b

+



)

forN periods,thenb

(

N

)

is given by b

(

N

)

=

1

 

1

+

1−_bbeNn=1Z(n)



,where Z

(

n

)



lnX2(n)

X1(n)

withasamplespace

Z 



z

=

lnu_v

|

u

,

v

∈

X



and

|

Z|

=

M. Then, we argue that the number of achievable portfolios at period N,

i.e., MN,isequal tothe numberofdifferentvalues that thesum

N

n=1Z

(

n

)

can take when theportfolio doesnot leave the inter-val

(

b

−



,

b

+



)

forN investment periods.Wethenobservethat wehaveM

≤

K2

₋

_K

₊

_{1 asthepricerelativesequencesX}

1

(

n

)

and X2

(

n

)

areelementsofthesamesamplespace

X

with

|

X |

=

K . Us-ingthisinequality,wefinallyfindanupperboundonthenumber ofachievableportfolios.

Lemma 1. ThenumberofachievableportfoliosatperiodN,i.e., MN,

isequaltothenumberofdifferentvaluesthatthesum

N

_n=1Z

(

n

)

can takewhentheportfoliob

(

n

)

doesnotleavetheinterval

(

b

−



,

b

+



)

for N investmentperiodsandisboundedby



N+K_N2−K



,i.e.,MN

= |

B

N

|

≤



N+K2_−K

N



.

Remark 1. Note that the complexity of calculating E

[

S

(

N

)

]

is bounded by

O

N

_n=1



n+K_n2−K



/

N



since at each period n

=

1

,

. . . ,

N, we calculate E

[

S

(

n

)

]

as a summation of Mn terms in

(5)andMn

≤



n+K2_−K

n



.

Ascan beobserved fromRemark1,the numberofachievable portfolios tendsto go to theinfinity as N increases.However, in thenextsection,weshowthatthenumberofachievableportfolios isfiniteundermildtechnicalconditions,hencethecomputational complexityofouralgorithmisconstant.

3.4.Finitelymanyachievableportfolios

In this section, we investigate the cardinality of the set of achievableportfolios

B 



∞_n=1

B

n and demonstratethat

B

is fi-niteunder certainconditions.Consequently, when

B

is finite,we canderive a recursive update witha constant complexity to cal-culate the expected achieved wealth E

[

S

(

n

)

]

at any investment period.Toshow this, wedemonstratethat theportfoliosequence formsa Markov chainwitha finite state spaceandconverges to astationarydistribution.Then,we investigatethelimiting behav-ioroftheexpectedachievedwealthusingthisupdatetooptimize

b and



.Beforeprovidingthemaintheorem,wefirststatea cou-pleoflemmasthatareusedinthederivationofthemainresultof thissection.

We first show that the portfolio b

(

n

)

does not leave the in-terval

(

b

−



,

b

+



)

for N periods iff the sum

k

_n=1Z

(

n

)

∈

(

α

−

,

α

+

)

fork

=

1

,

. . . ,

N,where

α

−



ln₍b₁₋(1_b−₎₍b_b−₊_)₎

<

0 and

α

+



ln₍b₁₋(1_b−₎₍b_b+₋_)₎

>

0. We then prove that the number of achievable portfoliosisequaltothecardinalityoftheset

Y ∩(

α

−

,

α

+

)

,where

Y



⎧

⎨

⎩

M+

i=1 yizi

:

yi

∈ Z,

zi

∈

Z

+

,

i

=

1

, . . . ,

M+

⎫

⎬

⎭

,

(23)

Z

+

_{ {}

_z

_∈

_{Z :}

_z

_≥

₀

_}

_{, and M}+

_{ |}

_Z

+

_|

_{. Note that}

_Z

+ _{is the set} of positive elements of the set

Z

and any value that the sum

N

n=1Z

(

n

)

cantakeisanelementof

Y

.Hence,ifwe can demon-stratethattheset

Y ∩ (

α

−

,

α

+

)

isfiniteundercertainconditions, thenityieldsthatthecardinalityoftheset

B

isalsofinite.

Lemma 2. Theportfoliob

(

n

)

doesnotleavetheinterval

(

b

−



,

b

+



)

forN investmentperiodsiffthesum

k

_n=1Z

(

n

)

∈ (

α

−

,

α

+

)

,wherek

=

1

,

. . . ,

N.

Proof. The proofisinAppendix B.

2

In the following lemma, we demonstrate that the set of dif-ferent values that

N

_n=1Z

(

n

)

can take for any N

∈ N

, when the portfolio never leaves the interval

(

b

−



,

b

+



)

for N

in-vestment periods is equivalent to the set

Y ∩ (

α

−

,

α

+

)

, when

|

z

|

<

min

{|

α

−

|,

|

α

+

|}

. Hence, weshow that thecardinality ofthe set ofachievable portfolios is equal to the cardinality of the set

Y ∩ (

α

−

,

α

+

)

.

Lemma 3. If

|

z

|

<

min

{|

α

−

|,

|

α

+

|}

,

∀

z

∈

Z

+,then

∀

y

∈

Y ∩ (

α

−

,

α

+

)

,

∃{

Z

(

n

)

=

Z(n)

_}

N

n=1

∈

Z

forsomeN

∈ N

suchthaty

=

N

n=1Z(n)and

k

n=1Z(n)

∈ (

α

−

,

α

+

)

,

∀

k

=

1

,

. . . ,

N.

Proof. The proofisinAppendix C.

2

Thislemmaillustrates that thesetofdifferentvaluesthat the sum

N

_n=1Z

(

n

)

can take forany N

∈ N

when the portfolio does

not leave the interval

(

b

−



,

b

+



)

for N investment periods is equivalenttotheset

Y ∩ (

α

−

,

α

+

)

.Thus,thenumberofachievable portfoliosisequaltothecardinalityoftheset

Y ∩ (

α

−

,

α

+

)

.Inthe following theorem, we demonstrate that if

|

z

|

<

min

{|

α

−

|,

|

α

+

|}

,

∀

z

∈

Z

+, and the set

Y

has a minimum positive element, then

Y ∩ (

α

−

,

α

+

)

is finite. Hence, the set of achievable portfolios is also finite under these conditions. Otherwise, we show that the set

Y ∩ (

α

−

,

α

+

)

containscountablyinfiniteelements.

Theorem 1. If

|

z

|

<

min

{|

α

−

|,

|

α

+

|}

,

∀

z

∈

Z

+,andtheset

Y

hasa minimumpositiveelement,i.e.,

∃δ =

min

{

y

∈

Y :

y

>

0

}

,thenthesetof achievableportfolios

B = ∪

∞_n=1

B

nisfinite.Ifsuchaminimumpositive

elementdoesnotexist,then

B

iscountablyinfinite. Proof. The proofisinAppendix D.

2

InTheorem 1we presenta sufficient andnecessarycondition fortheachievableportfoliostobefinite.Weemphasizethatthe re-quiredcondition,i.e.,

|

z

|

<

min

{|

α

−

|,

|

α

+

|}

,

∀

z

∈

Z

+,isanecessary required technical condition which assures that the TRP thresh-olds are large enough to prohibit constant rebalancings at each investmentperiod.Inthissense,thisconditiondoesnotlimit the generalityoftheTRPframework.

ByTheorem 1,weestablishtheconditionsforaunique station-arydistributionoftheachievableportfolios.Withtheexistenceof a unique stationarydistribution, in the next section, we provide the asymptotic behavior of the expected wealth growth by pre-senting the growth rate. However, before proceedingfurther, we notethatalthoughTheorem 1statesthatthenumberofachievable portfoliosisfiniteundercertainconditions,itdoesnotspecifythe exactnumber.Inthefollowingcorollary,wedemonstratethat the numberofachievableportfoliosis



α+−_δα−



ifthesetofachievable portfoliosisfinite.

Corollary 1. If

|

z

|

<

min

{|

α

−

|,

|

α

+

|}

,

∀

z

∈

Z

+,and

∃δ =

min

{

y

∈

Y :

y

>

0

}

,thenthenumberofachievableportfoliosis



α+−_δα−



.1

Proof. The proofisinAppendix E.

2

In Theorem 1, we introduce conditions on the cardinality of the set of all achievable portfolio states,

B

, and showed that

B

is finiteundercertain conditions.Using thisresult, we next ana-lyzetheasymptoticbehavioroftheexpectedachievedwealth,i.e., wedemonstratethatwhen

B

isfinite,theportfoliosequence con-verges to a stationary distribution.Hence, we can determine the limitingbehavioroftheexpectedachievedwealthsothatwecan optimizeb and



.Toaccomplish this,we firstpresentarecursive updatetoevaluate E

[

S

(

n

)

]

.Wethenmaximize

g

(

b

,



)



lim n→∞

1

nlog E

[

S

(

n

)

]

(24) over b and



witha brute-force search, i.e., we calculate g

(

b

,



)

fordifferent

(

b

,



)

pairsandfindtheonethatyieldsthemaximum expectedwealth.

3.5. FinitestateMarkovchainforthresholdportfolios

When

B

is finite, it follows from Corollary 1 that there are exactly L

 

α+−_δα−



achievable portfolios. Then, we let

B =

{

b1

,

. . . ,

bL}, where b1

=

b without loss of generality. We define the probability mass vector of the portfolio sequence as

π

(

n

)

=

1 _{Here,x/yisthelargestintegerlessthanorequaltox/y andx/yisthe} smallestintegergreaterorequaltothex/y.