Tracking Error

8/11/2019 Tracking Error

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1

Tracking

Error

and

Active

Portfolio

Management

NadimaEl-HassanSchool

of

Finance

and

Economics

University

of

Technology,

Sydney

and

PaulKofman#DepartmentofFinance

TheUniversityofMelbourne

Parkville,

VIC

3010

[email protected]

Persistentbearmarketconditionshave ledtoashiftoffocusin thetrackingerrorliterature.Until recently theportfolioallocation literature focusedon trackingerrorminimizationasaconsequence ofpassivebenchmark management underportfolio weights, transaction costsandshortsellingconstraints.Abysmalbenchmarkperformanceshifted the literaturesfocustowardsactiveportfoliostrategiesthataimatbeatingthebenchmarkwhilekeepingtrackingerror within acceptablebounds. We investigate an active (dynamic)portfolio allocationstrategythatexploitsthepredictabilityintheconditionalvariance-covariancematrixofassetreturns. To illustrate our procedure we use Jorions (2002) tracking error frontier

methodology.

We

apply

our

model

to

a

representative

portfolio

of

Australian

stocks

over

theperiodJanuary1999throughNovember2002.

June25,2003

1.

Introduction

Persistent

bear

market

conditions

on

stock

markets

worldwide

have

generated

a

bull

market

in the tracking error literature. Practitioner-orientedjournals (inparticular the Journal of

Portfolio Management and the Journal of Asset Management, see our references) recently

devoted whole issues to implementation andperformance measurement of tracking error

investment strategies. More technical mathematically-inclined journals (e.g., the

International Journal of Theoretical and Applied Finance)publish ever-faster optimization

Wearegratefulforhelpfulcommentsandsuggestionsfromtwoanonymousreferees,StanHurnand

participants

at

the

Stock

Markets,

Risk,

Return

and

Pricing

symposium

at

the

Queensland

University

ofTechnology.

#Correspondingauthor.


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2

and risk measurement algorithms for increasingly realistic portfolio dimensions. Not

coincidentally, this surge in academic interest tracks the global stock market slump with

fundmanagerperformancecomingunderintensescrutinyfrominvestors.

With absolute return performance as a first-order condition of investors utility

preferences,trackingthebenchmarkascloselyaspossibleisnormallysufficientduringbull

marketconditions.Whenthebenchmarkfailstodeliver,however,fundmanagerswillhaveto

prove relative returnperformance against thebenchmark. The aim is then topersistently

outperformthebenchmark.Ofcourse,suchperformancewillonlybefeasibleifthemanager

ispreparedtoacceptactiverisk,andhenceincurariskpenalty.Mostinvestmentfundsaccept

that investorswanttocap thispenaltyand thereforesetamaximumportfolio trackingerror

accordingly. Thus, tracking error can either be the investment goal, or an investment

constraint.Thisleadstothefollowingtwointerpretationsofindextracking:

Apassivestrategy thatseeks to reproduce as closely aspossible an index or benchmark

portfolio

by

minimizing

the

tracking

error

of

the

replicating

portfolio;

or,

an

active

strategy

that

seeks

to

outperform

an

index

or

benchmark

portfolio

while

staying

withincertainriskboundariesdefinedbythebenchmark.

Whatdistinguishesthesestrategiesisthecompositionoftotalriskexposure.Bothactiveand

passive strategies will incur incidental risk, while the active strategy will also incur

intentionalrisk.Intentionalriskmayconsistofstockspecificrisk(activestockselection)or

systematic risk (activebenchmark timing). Interestingly, some index fund managers claim

that ex antepassive indexing generatespersistent above average returns i.e., an active

portfolio outcome. Of course, what they really mean is that passive indexing often

outperforms

the

average

active

strategy

(which

is

more

likely

a

reflection

of

the

poor

active

outcomes).Astandardmeasureusedtotradeoffactiveperformanceagainstintentionalriskis

the information ratio (a.k.a. appraisal ratio), defined as theportfolios active return the

alphadividedbytheportfoliosactiverisk.Thisstandardizedperformancemeasurecanbe

used to assess ex ante opportunity but it is more frequently used to assess ex post

achievement. Ex ante opportunity is definedby the maximumpossibleIR given a set of


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forecast stock returns (and their forecast risk measures) and an inefficientbenchmark1. A

passivemanager(minimizingtrackingerror)willhaveanexanteIRclosetozero.Anactive

manager (maximizingexcess returns)willhave a much largerIR.TheexpostachievedIR

will depend on the realized excess return over thebenchmark and as such depend on the

Information Coefficient (IC). The managersIC measures the correlationbetween forecast

excess returns and realized excess returns. Whereas the ex anteIR will (by necessity)be

strictly non-negative, the ex post IR can of course be negative. For a comprehensive

discussion of theseperformance measures, we refer to Grinold and Kahn (1999). Active

managersperceive that theyneed to frequently reallocate theirportfolioseither tocapture

excess returnsor tostaywithina trackingerrorconstraint.At times thismay lead toa less

thanperfectly diversifiedportfolio, and will incur substantial transaction costs and assume

hightotalrisk.Ofcourse,fewpassiveindexfundmanagersholdportfoliosthatexactlymatch

the index, e.g., due to liquidity constraints. Just as active management incurs transaction

costs,thepassivemanagerthenalsohastorebalancetheindexportfoliotomatchtheactual

indexreturnsascloselyaspossible.Thissuggeststhatthereisafairlyclosesymmetryinthe

treatment of active and passive tracking error strategies. The key difference in the

interpretation ofpassive and active expost IR is that the best active manager willbe

characterizedbyapersistentlylargepositiveIR,whereasthebestpassivemanagerwillhave

an

ex

post

IR

close

to

zero.2

Thepassive trackingpractitioners havebeen well servedby the academic literature.

Rudd(1980),ChanandLakonishok(1993)andChan,Karceski,andLakonishok(1999)are

butafewofthemanyexamplesofthisliterature.Theactivetrackingpractitionershavenot

yetattractedsimilarattention3.Tothebestofourknowledge,Roll(1992)andJorion(2002)

are the firstpapers that comprehensively derive and interpret activeportfolio allocation

1

If

the

benchmark

happens

to

be

an

efficient

portfolio,

the

ex

ante

maximum

possible

IR

will

be

zero!

Ofcourse,thisishighlyunlikelyinpractice.TypicalbenchmarksliketheS&P500ortheASX200arecommonly

foundwellbelowtheefficientfrontier.2However,themeasurecanbepoorlydefined.Considerthepassivemanagerwhoactuallyholdsthebenchmarkportfolio.Forthismanager,theIRwillnotbedefined.

3Moststandardinvestmenttextbooksstilldiscusstrackingerrorinapassiveportfoliomanagementcontext,seee.g.,Eltonetal.(2003,p.677).Trackingerroristhentypicallydefinedasthestandarderrorofaregressionofthepassiveportfolio returns onbenchmark returns, see also Treynor and Black (1973). This regressionmeasureisappropriateifthe beta intheregressionequals1(asitwouldforpassiveportfolios),butitwilloverstate trackingerrorwhen thisisnot thecase(as itwould foractiveportfolios).Thesameapplies tothecorrelationmeasuresuggestedinAmmannandZimmermann(2001).Wethereforedefinetrackingerrorasthesquarerootof thesecondmomentof thedeviationsbetweenactiveportfolioreturnsandbenchmarkreturns.Alternatively, one can define tracking error as the mean absolute deviationbetween activeportfolio and

benchmark

returns,

see

e.g.,

Satchell

and

Hwang

(2001).

Both

definitions

can

be

used

for

ex

ante

trackingerror(usingforecastactiveandbenchmarkreturns)aswellasexposttrackingerror(usingrealizedactiveand

benchmarkreturns).


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solutionswithinatrackingerrorcontext.Thereareafewpapers(e.g.,Clarkeetal.,2002)that

investigate different active strategies with or without constraints on weights and/or risk.

UnlikeRollandJoriontheydonotanalyticallytracethetrade-offbetweenactiverisk-taking

andexpectedexcessreturns.

In this paper we apply Jorions approach to activeportfolio management within a

trackingerrorconstrainedenvironment.Weextendthemethodologybytakingacareful(and

practical) approach to compute the input list. We investigate the impact of seriously

inefficient benchmarks (which Jorion excludes) and the introduction of short selling

constraints.Weapplythemethodologytothetop-30stocksoftheAustralianStockExchange

duringathreeyearsampleperiodcharacterizedbyastrongbullmarketfollowedbyasharp

and persistent bear market. We find, not surprisingly, that market conditions have a

substantial impact on the active portfolio allocation and its ex post performance. This

becomesevenmoreapparentwhenweallowforshortsellingconstraints.

The next section briefly describes our methodology, by reviewing the well-known

portfolio optimization algebra and the lesser-known tracking error analytical solutions. We

also describe how we operationalize the general portfolio allocation model. Section 3

summarizes the data from the Australian Stock Exchange and illustrates typical

implementation

issues

that

confront

portfolio

managers.

Section

4

discusses

the

empirical

resultsofourtrackingerroroptimization.Weconcludewithlessonslearntfromthisexercise

andpossiblevenuesforfurtherresearch.

2.

Methodology

Our methodology isbased on Jorion (2002) to derive a constrained tracking error frontier.

Defineanobservation,orestimation,period[t-j,t]fromwhichwederivetheinputlist.Based

on

this

input

list

we

first

compute

the

global

efficient

frontier

without

restrictions

on

risk

or

weights(exceptfortheusualfullinvestmentconstraint).Wetheninvestigatethereductionin

investmentopportunitieswhenwe introducea trackingerrorconstraint,followedbyashort

selling constraint. The introduction of abenchmark leads to a tracking error frontier, from

which we derive the (conditionally) optimal activeportfolio allocation. We then track this

activeportfoliosperformanceoverasubsequent trackingperiod[t+1,t+k]andcompute the

realized tracking error. We dynamically update the activeportfolio allocation at different

frequencies

(daily,

weekly,

monthly).

Each

time

we

update

the

investment

opportunity

set,


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wealsolocatethenewexantepositionofthepreviousperiodsactiveportfoliorelativeto

theupdatedtrackingerrorfrontier.

Computation of the input list (arguably the most important stage of portfolio

management, see Zenti and Pallotta, 2002) tends tobe inconsistent inpractice. Return

forecasting is a strictly separate exercise from risk forecasting. Stock analystsprovide the

portfoliomanagerwithforecastreturns.Thesearetypicallypointestimateswithoutmatching

confidenceintervals,i.e.,stockanalystsdonotgeneratepredictionintervals,seeBlair(2002).

A stock analysts information set typically comprises accounting information, economic

information,managementinformation,etc.Itwouldbeextremelycomplicatedtocombinethe

uncertainty surrounding each information variable into a single confidence interval for the

stock return forecast. That is, the accuracy of the forecast is hard to define and measure.

Econometric forecast models are commonlybased on a much smaller information set of

fairlyhomogeneousvariables(likehistoricalreturns,dividends,growthrates).Evenforthese

models it is still a challenge to find thejoint confidence interval. In the absence of an

analytical solution, simulation techniques or scenario analysis maybe used to generate the

uncertaintymeasure.ThehighestdensityforecastregionproposedinBlascoandSantamaria

(2001) or thebootstrappedprediction densities in Blair (2002) wouldbe morepromising

candidates tosolve thisproblem.In theabsenceofanalystforecasts(as inourapplication),

the

portfolio

manager

may

apply

some

version

of

the

beta

pricing

model

along

the

lines

of

RosenbergandGuy(1976),andRosenberg(1985).Wedonotusefundamentalinformation

as theportfolio manager would but instead opt for the following simplification of the

BARRAmodeltoforecastbetas:

( ) htatf += ,, 1 (1)

wherea is thebeta computed over the estimationperiod (effectively derived from the

forecastvariance-covariancematrixasdiscussedbelow)andh is thelong-runbetaused to

smooth the forecast,4 with a smoothing parameter . Of course, there are different

techniques to choose the smoothing parameter (e.g., Bayesian updating, Maximum

LikelihoodEstimation)andthehistoricallong-runbeta.Wedonotfocusontheselectionof

orh,butdo investigate thesensitivityofour results todifferentvaluesof. We thenuse

stocki'ssmoothedforecastbeta,f,togenerateitsforecastreturn

4

In

our

application,

h

is

computed

as

an

expanding

average,

i.e.,

at

each

optimisation

period

we

compute

thelong-runbetaoverthefullsampleperioduptothatdate(alternativelyonecouldusetheaveragebeta

measuredovertheyearprecedingtheobservationperiod).


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6

( ) ( ) fttBti

tf

f

ttit RRERRE += ++ 1,,1, (2)

where theforecastreturnon thebenchmark isitsaveragereturnover theestimationperiod.

Ofcourse,ourchoiceofasinglefactormodelisafurthersimplification.GrinoldandKahn

(1999)

discuss

more

appropriate

multifactor

generalizations

(including

the

BARRA

risk

factormodel).Chan,KarceskiandLakonishoks(1999)multifactormodelthoughrestricted

toforecast(co)variancesispotentiallysuitabletogeneratethecompleteinputlist,bothrisk

andreturnforecasts.GiventhatwedonotintendtomaximizetheexpostInformationRatio,

wedonotpursuethosemoreelaboratemodelsatthisstage.

As suggested in Blair (2002) and discussed above, portfolio managers often apply

forecast riskmodels incomplete isolationfromforecastreturnmodels.Theymightadopta

number

of

modeling

specifications.

Most

of

these

are

based

on

the

premise

that

a

stocks

volatility changes over time (as does itsbeta). The ARCH modelby Engle (1982) and its

many offspring have dominated the academicportfolio literature for thepast two decades.

Muchprogress hasbeen made in achieving everbetter fitting specifications for univariate

time series. Multivariate extensions quite crucial forportfolio applications have not

witnessed similar advances, the main obstacle being the dimensionality problem. A

completely unrestricted time-varying variance-covariance matrix is almost impossible to

estimateforrealisticportfoliodimensions(letaloneachievingconvergenceinarealistictime

frame)5.Pragmaticsolutionsarethereforeneededandweoptfortheapproachadvocatedby

RiskMetrics (1995). Forecast volatility is an exponentially weighted moving average

(EWMA)ofpastsquaredreturnsforstocki:

( )

==

0

2,

2, 1

j

jti

j

ti R (3)

where, thedecay factor,dependsonhow fast the mean levelofsquared returnschanges

over

time.

The

more

persistent

(autocorrelated)

the

squared

returns

will

be,

the

closer

should

be

to

1.

For

highly

persistent

time

series

of

financial

returns,

we

find

values

of

between

0.9

and

1.

Given

the

choice

of

decay

factor,

we

then

determine

how

many

past

observations

should

be

used

to

compute

forecast

volatility.

For

a

tolerance

level

of

0.01

(when

we

consider

the

observations

impact

on

forecast

volatility

to

have

sufficiently

decayed),

and=0.9, we need about 40 historical observations. As in RiskMetrics , we

apply

(3)

to

each

element

of

the

variance-covariance

matrix.

Theoretically,

with

a

universe

of

n

stocks

we

should

have

as

many

as

n+(n)x(n-1)/2 unique s and matching estimation


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periods.

In

our

application

this

implies

465

individual

variance

forecast

processes.

While

still

computationallyfeasible(andcertainlyfaster thanamultivariateGARCHestimationof the

same

dimension),

for

our

purposes

we

impose

a

singleonallvariance-covarianceelements.

The

penalty

for

this

choice

is

reduced

precision

in

the

forecast

variance-covariance

matrix.

It

would

be

worthwhile

to

further

investigate

whether

there

is

a

diversification

effect

in

this

additional

forecast

error

across

stocks,

which

would

effectively

reduce

this

penalty.

Which

brings

us

to

the

next

step,

portfolio

optimization.

Rudd

and

Rosenberg

(1980)

comprehensively

derive

the

investment

allocation

problem

for

a

restricted

portfolio

universe

(that

is,

for

example,

only

the

top

50

liquid

stocks

on

a

particular

exchange)

to

match

empirical

practice.

Jorions

(2002)

derivation

is

similar

in

style

and

we

follow

his

notation.

First,

the

standard

constrained

portfolio

variance

minimization

problem

=

=

Ew

wts

ww

P

p

PPw

1..

min

(4)

whereE isavectorof forecast returnsand is the forecastvariance-covariance matrixof

returnsandisatargetportfolioreturn.OptimizationoverportfolioweightswPleadstothe

wellknownhyperbolainmeanstandarddeviationspace:

C

BEED

C

EB

CCB

D PP

21

1

1

2

2 11

=

=

=

+

=

(5)

The weights of the efficientportfolios that fall on the hyperbola are readily obtained for

different values of the target return constraint. Easy to apply,but of course, as soon as

additional

constraints

are

added

on

(e.g.,

a

short

selling

constraint),

we

lose

this

analytical

resultandhavetonumericallyoptimizetofindthefeasibleinvestmentopportunityset.Now,

ifweslightlyrefocustheoptimizationproblemtoreflectasearch forportfoliovalueadded

(inexcessofabenchmarkportfoliovalue),weget

=

PP

P

Pa

aa

ats

Ea

0..

max

(6)

5ArecentexceptionisTimmermannandBlake(2002)buttheirportfoliodimensionsaresmall.


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8

where

the

active

weights

aP (indeviation from thebenchmark weights,wB)have to add to

zero

to

satisfy

our

original

full

investment

constraint.

This

guarantees

that

total

active

portfolio

weights,

wB+aP,stilladduptoone.Activerisk(indeviationfrombenchmarkrisk)

may

not

exceed

tracking

error

target

variance.Jorionderivestheactiveportfoliosolutionif

we

set

the

tracking

error

constraint

exactly

equal

to

the

target

excess

variance

as

= C

BE

DaP

1

(7)

where

B,

C

and

D

are

as

defined

in

(4).

As

Jorion

notes,

these

are

solutions

in

excess

return

versus

excess

risk

space.

For

ease

of

comparison

with

the

standard

portfolio

setup,

we

would

rather

have

a

representation

in

the

mean

(total

return)

versus

standard

deviation

(total

risk)

space:

( ) ( ) 2

0..

max

PPBPB

PP

P

Pa

awaw

aa

ats

Ea

=++

=

(6)

whichuponmaximizationresultsinthefollowingellipsoidalsolutions(P,P)

( ) ( ) ( )( )

01

4

4

1

42

2

2222222

=

+

C

B

CD

C

B

CD

BB

BPBPBBPBBP

(8)

as long as thebenchmark (B,B ) lies within the efficient set. The ellipse is vertically

centred around thebenchmark expected return,but it is horizontally centred around the

benchmarkvarianceplustrackingerrorvariance.Theellipsesprincipalaxis ishorizontalif

benchmark expected return coincides with the expected return of the global minimum

variance

portfolio

(MVP=B/C).

If

B>MVP

typical

in

a

bullish

market

it

will

have

a

positive slope; ifB


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for the meanandvarianceof the maximum expectedexcess returnportfolio.Note that the

portfoliomanagercanonlygenerateexcessreturnsbyassumingtrackingerror.Notealsothat

thepenaltyforthisactiveportfolioallocationnotonlyincreaseswithtrackingerrorvariance

butalsowithamoreefficientbenchmarkportfolio(anincreasinggapbetweenB

andB/C).

Themoreefficientthebenchmark,theharderitwillbetobeatitsperformance.

WetesttheJorionsmethodologyonrealdata.Twoempiricalfeaturesstandoutforour

application.Thebenchmark is frequentlyso inefficient, that itsexpected return fallsbelow

theexpectedreturnoftheglobalminimumvarianceportfolio(B/C).Also,theunconstrained

weights take completely unrealistic values duringbear market conditions with huge short

sellingimplications.Wethereforeaddashortsellingconstraint.Thiscomplicatesmatters,as

Jorionsuggests.Thenumericaloptimizationof(4)withsuchaconstraintposesnoparticular

problems.Unfortunately,wecannotsimplydistort the trackingerrorellipseby taking the

intersectionofthetrackingerrorellipseandtheconstrainedmean-varianceefficientfrontier.

Wehavetoperforman integratednumericaloptimizationof(6) including theshortselling

constraint. The solution set thenbecomes considerably thinner. Ultimately we look for the

maximum excess return active portfolio (a single point) that satisfies all constraints

simultaneously. There is no analytic solution for thisproblem.Numerical optimization is,

however,quitefeasible.

3.

Data

Issues

Ourapplicationconsidersaportfolioof30AustralianstockswithareducedAustralianAll

Ordinaries Index(XAO)asourbenchmark.Thestockpricedataandrisk-free interestrates

areobtainedfromDatastreamandIRESSMarketTechnology.Thetop-30stocksaccountfor

about62%of theXAO index6.Westandardize the top-30weights toaddup to100% and

generate a new top-30benchmark accordingly. We choose a sample that covers the rather

turbulent

period

from

4

January

1999

until

29

November

2002,

a

total

of

991

trading

days.

Whatmakesthissampleparticularlyappealingisthestrongbullmarketfromthestartofour

sample until April 2000, followedby a sequence of collapses andpersistentbear market

conditionsuntiltheendofoursample.

Figure1indicatesthatourtop-30benchmarktrackstheXAOindexquitecloselyforthe

first year and a half of our sample. Mid 2000, however, the top-30 starts to diverge

6Thetop-10stocksalreadyaccountforover40%oftheXAO.Forcomparison,thetop-10stocksintheS&P500

account

for

about

25%.

This

dominance

of

a

few

stocks

should

make

it

theoretically

easy

to

keep

the

tracking


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10

substantially from the XAO index. The underperformance of the XAO index is easily

explainedbytheunderrepresentationoftechnologystocksinthetop-30,relativetotheXAO.

This is even more apparent when we include the S&P500 in the comparison. Whereas the

S&P500outperformsbothtop-30andXAOindicesuntilaboutJune2000,ithasbeenmuch

moreexposedtothetechstockscollapse.AcloserlookatthekeymarketdropsinFigure1

highlightsthisphenomenon.

Figure

1.

All

Ordinaries

versus

Top-30

Benchmark

Index

and

S&P500

60

85

110

135

160

1/1/1999

5/1/1999

9/1/1999

1/1/2000

5/1/2000

9/1/2000

1/1/2001

5/1/2001

9/1/2001

1/1/2002

5/1/2002

9/1/2002

S&P500

Benchmark

XAO

Note:Indiceshavebeenstandardized at100onthe1stofJanuary1999.ThebenchmarkconsistsoftheTop-30

stocks

in

the

All

Ordinaries.

RankedbythesizeoftheS&P500collapsetheyare:theNasdaqcollapseonthe14thof

April2000, the17thofSeptember2001(followingSeptember11), the12thofMarch2001,

the3rdofSeptember2002,themillenniumbugonthe4thofJanuary2000andthe19thofJuly

2002. TheinitialNasdaqcollapseismirroredintheXAOcollapseonthefollowingtrading

day,buttoalesserextentforthetop-30.Subsequenttechstockspilloversarelessapparent

indicating the smaller presence of tech stocks in the Australian market. These US-led

collapseshaveamoderateimpactontheXAOindex,buthavevirtuallynoimpactonthetop-

30index.Marketwidecollapses(likeSeptember11,and themillenniumbug),ontheother

hand, are similarly reflected in the XAO and the top-30 index. The lesser exposure to

extremalprice changes of the top-30 index does not imply less volatility. The standard

deviationofreturnsisalmostidenticalforbothAustralianindices(13%perannum),butboth

aresubstantiallylessvolatilethanS&P500returns(astandarddeviationof22%perannum).

error withinacceptablebounds. Our expost trackingerror resultsshow that this is true withashort-selling


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11

Figure2stressestherelevanceoftime-variationintherisk-freerateofreturn(the90-day

billrate).Itsuggeststhatthebullmarketreturnsweresomewhattemperedbyarisingrisk-

freerateofreturn.Similarlybearmarketnegativereturnswereoffsetbyadeclineintherisk-

freerateofreturn(at leastafterOctober2000).Thisoffset lastsuntilApril2002,when the

risk-free rate again starts to increase and risk-takingperformance was doublypenalized

(squeezingtheexcessreturns).

Figure

2.

Risk-Free

Rate

of

Return

4%

5%

6%

7%

1/4/1999

5/4/1999

9/4/1999

1/4/2000

5/4/2000

9/4/2000

1/4/2001

5/4/2001

9/4/2001

1/4/2002

5/4/2002

9/4/2002

90DayBill%

Table1summarizesthefullsampleinputlist.Thetop-30stocksarelistedbydescending

top-30

weight.

Annualised

mean

returns

and

annualised

standard

deviations

are

given

in

columns3and4,respectively.Volatilityvariesbetweenalowof15%(WFT)andahighof

53%(QBE),butisgenerallyaround30%perannum.Riskcompensationisrathermeagreata

low of 20% (LLC) mean return to a high of 19% (WES) mean return. Ourbenchmark

comparesreasonablywellwithameanreturnof5%pa.(medianreturnof11%pa.)againsta

standard deviation of 13%pa. Taking into account that the risk-free rate of return varied

between4%and7%pa.,thisdoesnotsuggestagenerousexcessreturn.

Table

1

also

gives

daily

(not

annualised)

maximum

return

and

minimum

return

in

columns 5 and 6, respectively. The maxima varybetween 4% and 25%, while the minima

vary between 4% and 35%, truly extremal values. Not surprisingly, the empirical

distributionsofdailyreturnsarehugelykurtotic(fat-tailed).Thiscausesproblemswhenwe

wanttooperationalisetheinputlistforportfoliooptimisationpurposes.Thenon-normalityof

stock returns suggests that mean-variance optimization might not reflect investors utility

tradeoffbetweenriskandreturn.

constraint,butisemphaticallynottruewhenshortsellingispossible.


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Table

1.

Descriptive

Statistics

for

Top

30

ASX

Stocks

Code Top-30

weight

Annualized

Average

return

Annualized

Standard

deviation

Daily

Maximum

return

Daily

Minimum

return

Beta

Ljung-Box

(p=12)

Test

NAB

12.60

7.31

24.10

5.00

-13.87

1.18

25.87

CBA 9.37 4.06 20.80 3.97 -7.13 0.95 12.27

BHP 8.94 16.43 29.93 6.87 -7.62 1.21 16.72

TLS 7.75 -13.09 25.67 8.34 -9.86 0.91 20.49

WBC 6.96 6.75 21.12 4.87 -5.21 0.92 11.00

ANZ 6.81 14.42 21.99 4.71 -5.60 0.99 22.07

NCP 4.85 4.18 46.49 24.57 -14.89 1.95 9.18

AMP 4.14 -11.55 26.61 7.65 -8.03 1.03 17.16

RIO 4.14 14.88 30.02 7.18 -5.78 1.13 16.19

WOW 3.11 18.17 23.41 5.16 -8.34 0.53 28.93

WSF 2.38 13.38 29.62 20.92 -6.55 0.86 16.35

WES 2.31 19.03 26.78 10.58 -6.42 0.71 17.49

FGL 2.29 1.35 21.95 4.83 -7.06 0.49 36.79

WMC 2.28 13.41 34.32 16.32 -10.40 0.97 46.10

SGB 2.18 14.57 18.93 7.17 -5.02 0.53 6.92

WPL 2.05 12.55 27.02 7.57 -11.13 0.62 13.06

BIL 1.92 -19.41 33.86 9.54 -35.25 0.83 25.12

CML

1.90

-6.72

27.77

12.83

-18.18

0.61

10.62

WFT 1.67 -1.01 14.98 3.64 -4.38 0.38 20.28

CSR 1.54 11.46 27.07 9.16 -6.37 0.74 9.91

PBL 1.47 3.51 29.22 10.41 -10.83 0.82 21.62

GPT 1.31 -1.55 16.60 4.55 -3.98 0.42 19.58

CCL 1.20 -1.16 35.65 12.42 -10.80 0.69 13.43

MBL 1.20 10.35 26.65 7.11 -11.46 0.91 37.72

TEL 1.13 -13.72 29.44 9.63 -9.12 0.77 13.88

CSL 1.13 10.84 39.41 26.76 -12.01 0.76 22.29

LLC

1.05

-20.15

29.68

5.32

-17.11

0.72

36.05

QBE 0.98 4.67 52.83 41.85 -52.62 1.25 131.26

AGL 0.92 -3.61 22.54 4.42 -8.48 0.38 33.24

AXA 0.79 -4.04 29.61 7.89 -6.12 0.78 17.97

Index 100.00 4.82 13.28 2.65 -5.14 1.00 11.83

Note: Columns 2-6 are inpercentages. The top-30 weights are fixed as ofNovember 2002 and are thestandardized(theysumto100%)XAO-weightsonthatdate.Weignoreanyweightchangesduringoursampleperiod. Betas are computed against the top-30 index. The Ljung-Box column gives the teststatisticforserialcorrelationinthereturnsupto12thorderlaglength,witha95%criticalvalueof21.03.


13/24

13

Campbelletal.(2001)illustrateanalternativeoptimisationprocedurethatbettercaptures

thekurtotic(andperhapsskewed)natureofstockreturns.Asamatterforfutureresearch,we

couldincorporateanequivalent trackingerrorconstraint intheiroptimisationprocedure.As

longastheempiricaldistributionsaresymmetric,weexpectverylittledifferenceintermsof

activeweightselection.

Unlike stock analysts, we do not have the fundamentals (growth forecasts, accounting

information,etc.)tovalueandthenrankstocksbyforecastreturntogeneratebuy/sellsignals.

Our informationset is restricted tohistorical returns.Forsimplicity,consider thefollowing

examplewhereourforecastreturnisasimpleaverageofthepast5tradingdaysreturns.If

wehappentoencounterasingleextremalreturninourinformationsample(say,10.58%)and

fourzero returns,wegenerateanannualised forecast returnof535%.Clearly, theextremal

returnisnon-representativeforforecastpurposes.Evenforlongerinformationsamples(saya

month, or 20 trading days) this inflation of annualised forecast returnsbased on extremal

observationsremainsaproblem.Itreflectsthefactthatdailystockreturndistributionsarenot

normallydistributed,butarebettercharacterizedbysomefat-taileddistribution(like,e.g.,a

Student-t).Unlikethenormaldistribution,aStudent-tisnotclosedunderaddition, i.e.,its

properties (e.g., the variance) cannotbe simply scaled to derive equivalents at a different

samplingfrequency(sayfromdailytoannualised).

This

problem

highlights

the

difficulties

that

one

encounters

when

optimising

portfolios

of individualstocksbasedonsimplistic input listrules. Itmightexplainwhy theacademic

literaturepreferstobuildefficientportfoliosfromportfoliosofindividualstocks7.Indexing

clearlynormalizes theempiricaldistributions (justconsider thedescriptivestatistics for the

top-30 index), which makes theseportfolios much more suitable forportfolio optimisation

purposes. This may be feasible for academic purposes, but it will not typically be a

satisfactory solution forpracticalpurposes. To somehow moderate the impact of extremal

returns

on

our

input

list,

we

therefore

choose

to

forecast

returns

based

on

a

smoothed

beta

modelasexplainedintheprevioussection.

Ofcourse,thevalidityofusingpastaveragereturnstoforecastfuturereturns(eitherasa

simple average or as a market expectation) depends crucially on the stationarity of stock

returns.TheLjung-Boxportmanteaustatistic(autocorrelationuptolaglength12)incolumn

8,Table1, indicates thatquitea fewseriesdisplaysignificantautocorrelation(95%critical

7Theauthorsarewellawareofmoreimportantreasons(liketheerrors-in-variablescorrection)forthischoice.


14/24

14

value 212 =21.03).Assuggestedby Lawton-Browne (2001), thismay lead toadownward

biasinexantetrackingerror.Weinvestigatethisbelow.

4.

Empirical

Results

Tostartouranalysis,wefirstcomputeourinputlist.Toupdatethevectorofforecastreturns,

E,weuseasimplifiedversionoftheBARRAbetapricingmodelencapsulatedinequations

(1) and (2). There is no realprecedent for thisprocedure, and it obviously lends itself for

future improvement. We choose =0.34 for our application, but also investigate the

sensitivityofour results fordifferentvaluesof.Thisproceduregeneratesfairlysmoothly

evolvingforecastreturns.

To

update

the

forecast

variance-covariance

matrix,

we

use

the

RiskMetrics

TM

EWMA

methodology. It is simple to understand, straightforward to implement, and generates

GARCH-likevarianceprocesses.Toillustratethispoint,considertheGARCH(1,1)outputin

Figure3forNAB.TheGARCH(1,1)parameterswereestimatedat1=0.17,2=0.74,while

the average decay factor in the EWMA was estimated tobe =0.91 (with an effective

estimationsamplelengthof40days).TheEWMAprocessissomewhatsmoother,butthere

seemslittletoseparatethetwoprocessesasimilarpointismadeinRiskMetrics (1995).

Figure

3.

NAB

Conditional

Volatility

GARCH(1,1)

versus

EWMA

0

0.25

0.5

0.75

1/4/1999

5/4/1999

9/4/1999

1/4/2000

5/4/2000

9/4/2000

1/4/2001

5/4/2001

9/4/2001

1/4/2002

5/4/2002

9/4/2002

Note: Annualized standard deviations for NAB are based on fitting a GARCH(1,1) model:

21,2

21,10

2 ++= titiit R ,respectivelyanEWMAmodel: ( )

==

0

2,

2 1j

jti

j

it R .

Whereas it is relatively straightforward to estimate a univariate GARCH(1,1)process

like

the

one

above,

the

computational

burden

becomes

excessive

for

a

multivariate

GARCH

process involving30stocks.ScowcroftandSefton (2001) investigate theperformanceofa

EWMA

GARCH(1,1)


15/24

15

number of time-varying risk matrix specifications including the EWMA and GARCH

modelsandfindthatthetrackingerrorpredictionsagreedreasonablywell.

Figure 4 gives some insight in the dynamically updated input list for three stock

components in the top-30benchmark:NAB (the largest weight, 12.6% and a full-sample

unconditionalbetaof1.18),NCP(thehighestfull-sampleunconditionalbeta,1.95)andAGL

(the lowestfull-sampleunconditionalbeta,0.38).Figure4showsthetimevariationin their

betas,accordingtoequation(1).

Figure

4.

Time

Variation

in

Betas

0

0.5

1

1.5

2

2.5

3/3/1999

6/3/1999

9/3/1999

12/3/1999

3/3/2000

6/3/2000

9/3/2000

12/3/2000

3/3/2001

6/3/2001

9/3/2001

12/3/2001

3/3/2002

6/3/2002

9/3/2002

Note:

Unconditional

betas

are

respectively

1.18

(NAB),

1.95

(NCP),

and

0.38

(AGL);

Betas

are

measured

againstthetop-30benchmarkindex.

NABhasthemorestablebeta,whereasNCPandAGL(thestockswithmoreextremehigh

and low betas) have much more volatile intertemporal betas. This has obvious

repercussionsfortheforecastreturnsaccordingtoequation(2),wherethemorevolatilebeta

forecastswillgeneratemorevolatilestockreturnforecasts.

Figure

5.

Time

Variation

in

Correlation

-1

-0.5

0

0.5

1

3/3/1999

6/3/1999

9/3/1999

12/3/1999

3/3/2000

6/3/2000

9/3/2000

12/3/2000

3/3/2001

6/3/2001

9/3/2001

12/3/2001

3/3/2002

6/3/2002

9/3/2002

-1

-0.5

0

0.5

1

3/3/1999

6/3/1999

9/3/1999

12/3/1999

3/3/2000

6/3/2000

9/3/2000

12/3/2000

3/3/2001

6/3/2001

9/3/2001

12/3/2001

3/3/2002

6/3/2002

9/3/2002

Note:NAB-WBCunconditionalcorrelation=0.46;NCP-AGLunconditionalcorrelation=0.06;

AGL

NCP

NAB

NAB-WBC

NCP-AGL


16/24

16

Figure 5 illustrates the time-varying intra-sector correlationbetweenNAB and WBC,

twobankingstocks inouruniverse;respectivelybetweenNCPandAGLforan inter-sector

illustration.Intra-andinter-sectortime-varyingcorrelationareequallyvolatile,butobviously

haveadifferentmean.Theintra-sectorcorrelationdisplaysswitchingbehaviourwithperiods

ofhighcorrelationalternatingwithperiodsoflowcorrelationandnotmuchinbetween.The

inter-sectorcorrelationdoesnotsharethisfeature.

Havingcompletedtheinputlist,wecanproceedtocomputetheefficientfrontiersolving

(4)with=0.91.Thenwesolve(6)toobtaintheactivetrackingerrorfrontierwithatracking

errortargetof5%.Ourtop-30benchmarkisfoundtobeseriouslyinefficient(throughoutour

sampleperiodwithafewexceptionswhen it isclose to theglobalefficient frontier)which

suggestsactiveinvestmentopportunities.Or,inexanteInformationRatioterms,theyoffera

positive IR for activeportfolio managers. To illustrate this, consider the following two

representative optimizationperiods. Figures 6a and 6b are representative for abull market

episode, respectivelyabear marketepisode.Asexpected inabull (bear)market theactive

investmentopportunitysettheellipseisupward(downward)sloping.Thebenchmark(the

squaresymbol)typicallyhaslowerriskthantheindividualstocks(thediamondsymbols)but

ofcourse,onlyaverageexpectedreturns.

Figure

6.

Bulls

and

Bears

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4

EfficientFrontier ConstrainedActivePortfolio

ConstantTrackingErrorFrontier UnconstrainedActivePortfolio

NoShortSellingEfficientFrontier BenchmarkPortfolio

The locationof theactiveportfolioschosen in thepreviousperiodoptimisation isalso

indicated inboth graphs. Without rebalancing, the unconstrained activeportfolio (circle

symbol

outside

ellipse)

needs

no

longer

be

on

the

updated

ellipse.

As

it

turns

out,

the

constrainedactiveportfolio(circlesymbolinsideellipse)isalwaysinsidetheupdatedellipse,


17/24

17

but the unconstrained active portfolio is almost without exception outside the updated

tracking error frontier. That does not necessarily imply a violation of the tracking error

constraint over that period, but does imply an ex ante tracking error violation for the

subsequentperiod.

We summarize our dynamic optimisation exercise in Figure 7, which gives the active

portfolioweightsforthreestocks(NAB,NCP,AGL)inthe(un)constrainedactiveportfolios.

Perhaps surprisingly the unconstrained weights theblack lines are as volatile (inboth

positive and negative direction) during thebull market as they are during thebear market.

There is limited evidence of short-livedpersistence in the active weights, suggesting that

portfolios need to be rebalanced fairly frequently (at least monthly). The short selling

constrained weights the grey lines are obviously much smoother which implies

substantiallylessrebalancingcosts.

Figure7.ActiveportfolioweightsforNAB,NCPandAGL

-1.0

-0.5

0.0

0.5

1.0

3/3/199

9

6/3/199

9

9/3/199

9

12/3/199

9

3/3/200

0

6/3/200

0

9/3/200

0

12/3/200

0

3/3/200

1

6/3/200

1

9/3/200

1

12/3/200

1

3/3/200

2

6/3/200

2

9/3/200

2

-1.0

-0.5

0.0

0.5

1.0

3/3/199

9

6/3/199

9

9/3/199

9

12/3/199

9

3/3/200

0

6/3/200

0

9/3/200

0

12/3/200

0

3/3/200

1

6/3/200

1

9/3/200

1

12/3/200

1

3/3/200

2

6/3/200

2

9/3/200

2

-1.0

-0.5

0.0

0.5

1.0

3/3/1999

6/3/1999

9/3/1999

12/3/1999

3/3/2000

6/3/2000

9/3/2000

12/3/2000

3/3/2001

6/3/2001

9/3/2001

12/3/2001

3/3/2002

6/3/2002

9/3/2002

Note: Weights are given as fractions. Weights in the top-30benchmark are respectively 0.126 (NAB), 0.049(NCP),and0.09(AGL).Greylinesgivenoshortsellingweights;Blacklinesgiveunconstrainedweights.

A comparison across stocks is interesting. The stock with the highest top-30benchmark

weightandamoderatebeta(NAB),switchesmostfrequentlyfrompositivetonegativeactive

positions. The stock with the lowest weight in the top-30 and the lowestbeta (AGL), has

predominantlypositiveactive weights (althoughwithsubstantialvolatility).Thestockwith

the

highest

beta

(NCP)

not

surprisingly

the

least

Australian

of

the

30

stocks

has

the

NAB NCP

AGL


18/24

18

most stable active weights (although they switch frequently frompositive to negative and

viceversawithoutmuchpersistence).

Sincewerebalanceeverytimeperiod,ourexantetrackingerrorisalwaysexactlyequal

tothetarget.Thisisclearlynotthecaseexpost.Theproblemnowishowtomeasuretheex

post tracking error. We choose to measure expostperformance using a quadratic tracking

errormeasure,8thatisthesquareddeviationofportfolioreturnsfrombenchmarkreturns.We

firstmeasuretheexpostdifferencebetweenportfolioreturnandbenchmarkreturn,i.e., the

activereturn.Thisactive/excessreturnperformanceisillustratedinFigure8.Acomparison

suggests that theshortsellingconstrainedactiveportfolioonly rarelywandersaway from

the benchmark (Figures 6a and 6b are fairly typical of thisphenomenon). In fact, the

constrainedportfolioseemstohug thebenchmarkoverthetrackingperiod.

Figure

8.

Ex

Post

Active

Returns

0

10

20

30

40

50

60

70

80

-0.135

-0.118

-0.102

-0.085

-0.068

-0.052

-0.035

-0.019

-0.002

0.01

4

0.03

1

0.04

8

0.06

4

0.08

1

0.09

7

0

10

20

30

40

50

60

70

80

-0.020

-0.018

-0.015

-0.013

-0.011

-0.008

-0.006

-0.003

-0.001

0.00

1

0.00

4

0.00

6

0.00

9

0.01

1

0.01

3

Note: the histograms have frequency count on the vertical axis and daily (fractional) excess returns on thehorizontalaxis.

Thestandarddeviationofshortsellingconstrainedexcessreturnsisalmostexactly5%

perannum.Notentirelysurprisinggiven thevisualevidenceinFigure8,theunconstrained

standarddeviationis20%perannum.Aniceillustrationindeedofthetheoreticalresultofthe

downwardbiasinexantetrackingerrorasaforecastofexposttrackingerror,Satchelland

Hwang

(2001).

Further

empirical

evidence

for

this

bias

is

given

in

Zenti

and

Pallotta

(2002).

Among thepossible reasons for thisbias, Lawton-Browne (2001) and Scowcroft and

Sefton (2001) suggest autocorrelation in returns and volatility clustering. We therefore

investigatetheimpactofbothviolationsofthemean-varianceoptimizationassumptions.We

alsolookatthesensitivityofourresultstovariationsintrackingduration,andtrackingerror

target.TheresultsoftheseexercisesaregiveninTables2a,2b,and2c.

8Rudolfetal.(1999)suggestthatameanabsolutedeviationmeasurewouldbepreferablesinceitmatchesfundmanagerscompensationschedules.Weagreethatthisisworthinvestigating.

unconstrained noshortsales


19/24

19

Table

2a.

Ex

Post

Tracking

Error

Varying

tracking

duration

Tracking

Duration(days)

Constrained

TrackingError

Unconstrained

TrackingError

1 5.27% 19.67%2 5.27% 19.93%3

5.02%

19.68%4 5.23% 19.77%5 4.93% 20.54%10 4.87% 19.76%20 5.11% 19.85%40 5.04% 19.79%60 4.77% 18.20%125 5.28% 22.25%Note:trackingerrortarget,=0.05;volatilitypersistenceparameter=0.91.

Table

2b.

Ex

Post

Tracking

Error

Varying

tracking

error

target

Tracking

Error

Target(st.dev)

Constrained

TrackingError

Unconstrained

TrackingError

1% 1.08% 3.97% 2.5% 2.59% 9.93% 5% 5.11% 19.85%10% 7.57% 39.70%20% 12.52% 79.41%Note:trackingduration=20days;volatilitypersistenceparameter=0.91.

Table

2c.

Ex

Post

Tracking

Error

Varying

volatility

decay

factor

EWMA

decay

Factor

(

)

Constrained

Tracking

Error

Unconstrained

Tracking

Error

0.85 4.66% 50.84%0.87 4.84% 36.40%0.90 4.67% 15.25%0.91 5.11% 19.85%0.95 4.69% 12.62%0.97 4.32% 11.66%0.99

4.67% 15.25%Note:trackingduration=20days;trackingerrortarget,=0.05.

Table2agivesexposttrackingerrors(asastandarddeviationofactivereturns)forarangeof

tracking durations (k=1, ,125 days). Apparently, the duration of tracking does not

materiallyaffecttherealizedtrackingerror.Perhapssurprisingly,thereisverylittlevariation

in expost tracking error when rebalancing of the activeportfolio occurs more or less

frequently. Short selling constrained activeportfolio management generates tracking errors


20/24

20

that stay within the target tracking error. Unconstrained active portfolio management

generatestrackingerrorswellinexcessofthetarget.

Table 2b gives expost tracking error for a range of tracking error targets (=1%,

,20%).

We

observe

that

the

ex

post

short

selling

constrained

tracking

error

marginally

exceeds(issubstantiallylessthan)theexantetrackingerrorfortargetsbelow(above)5%per

annum.Expostunconstrainedtrackingerror,however,generallyexceedstheexantetracking

erroratanincreasingratewithincreasingtargettrackingerror.

Table2csuggeststhesourceofthebiasinexantetrackingerrorexpectations.Itgivesex

post tracking errors for a range of EWMA decay factors (=0.85, ,0.99). For this

(admittedly limitedinscope)experiment,wefindthatat=0.97,thebiasisminimized.We

suspect

that

individual

optimisation

of

the

parameter

for

each

and

every

element

in

the

variance-covariancematrixwouldallowanevenbetteroutcome.

We also experimented with the serial correlation in stock returns by varying the

smoothingparameterin(1).Thisdidnotaffectthebiasinexantetrackingerror.Itdoes,of

course,affectthelocationoftheexcessreturndistribution,butnotthescale!

Bringing the two results (time-variation in weights and expostperformance) together

suggeststhatthetypicalshortsellingconstraintactsasasafeguardonexposttrackingerror

whilesimultaneouslyreducing thecostofrebalancing.Ashortsellingconstrainteffectively

turns active portfolio management into something very close to passive portfolio

management. So why do we not observe the matching reduction in active returns? Simply

becausewemadenorealattempttoactivelyforecaststockreturns.Infact,wehavetakena

ratherpassive approachwhengeneratingtheinputlist9.Wecanimaginethatstockanalysts

(orsophisticatedeconometriciansmodelsforthatmatter)areabletomanipulatetheseactive

performancedistributions totheirbenefitandhencechangethe locationof theactivereturn

distributionsinFigure8accordingly.

5.

Conclusions

Althoughstraightforwardincontent,properimplementationofportfoliooptimizationtheory

canbe notoriously complicated. Choices have tobe made with regard to the input list,

constraints,estimationprocedure,andimpliedactions.Thispapergivesaflavourofsomeof

themanyissuesthathavetobedealtwithbyportfoliomanagers.Themainadvantageofour

9Weuseamechanisticrule as inequation (2) togenerate expected returns.Ourchoiceof thesmoothingparameterisnotbasedonpropermodelselectioncriteria,sincethiswasnotthemainfocusofourpaper.


21/24

21

procedureisitstransparentnaturewhichconsiderablyfacilitatescommunicationwithanever

moresophisticatedclientele.

Jorions (2002) main contribution is the visualization of the active investment

opportunityspace. Weillustratethattheintroductionofashortsellingconstrainteliminates

mostof thisopportunitysetpartlydrivenbypersistentbearmarketconditionsduringour

sampleperiod.However,asaninvestmentadvicetool,successiveintroductionofinvestment

constraints clearly identifies the location of (and the reduction in) the relevant investment

opportunity set. The methodology also highlights the tradeoffbetween riskpenalty and

excessreturngainwhenviolatingtheinvestmentconstraints.Fromthisperspective,Jorions

methodologyisaninvaluablecontributiontothepracticalinvestmentliterature.

Wefind(asdomanyothers,seee.g.,PlaxcoandArnott,2002)thatfrequentrebalancing

isanabsolutenecessitytokeepsomecontrolovertotalrisk(thoughnotnecessarilytracking

errorrisk)whenactivelymanagingportfolios.LarsenandResnick(2001)considerarangeof

optimization and holdingperiods,but do not consider transaction cost constraints. Clearly,

thecostsofrebalancinghavetobeoffsetagainstthegainsinriskcontrol,butitseemstous

thatcertain(threshold)levelsofriskwillsimplybeunacceptable.Theissue,ofcourse,ishow

tooptimallyrebalancesoastominimizethecontrolcosts.

Notsurprisingly,wealso find thatexante trackingerrorexpectationsdonotmatchex

post

realizations,

see

also

Rohweder

(1998).

Satchell

and

Hwang

(2001)

show

that

we

can

reasonably expect a worse expost tracking error outcome due to the stochastic nature of

portfolioweights.Theyreportthatthisupwardbiasisnotrestrictedtoactiveportfolios,but

can also be found in passive portfolios where the weights are not stochastic (due to

rebalancing).Asimilarupwardbiasinexposttrackingerror(butduetoadifferentsource)is

causedbytheapparentserialcorrelation,notjustintheunderlyingstockreturns,butalsoin

theexcessreturnsandsquaredexcessreturns.FrinoandGallagher(2001),e.g.,findevidence

of

seasonality

in

tracking

error

(partly

driven

by

seasonality

in

dividend

payments

on

the

benchmark).PopeandYadavs(1994)resultsindicatethat thiswill leadtoabiasedexante

estimateoftrackingerror.

Where to from here? There isplenty of scope to improve the selection of optimal

observationperiodandforecastperiodduration.Ultimately, thisisanempiricalmatter.We

hinted at the possibility (and Table 2b underlines its importance) to individualize the

stochasticprocesses for every element of the variance-covariance matrix. A trade-off will

then

have

to

be

made

between

the

improvement

in

goodness-of-fit

and

the

increased


22/24

22

computationalburdenofsuchanexercise.Despitethis,wearguethatthereisnourgentneed

toresorttocomputationallyburdensomemultivariateGARCHspecifications.

Anotherconstraintworthinvestigatingisacaponthenumberofstocksinthemanaged

portfolioortheminimumnumberofstocksinanactiveportfolio.JansenandvanDijk(2002)

illustrate the small portfolio constraint for a passive tracking portfolio. Ammann and

Zimmermann (2001) investigateadmissible active weight ranges, which wouldguarantee a

limitonindividualstockweights.Alternatively,wecouldinvestigateacaponthenumberof

stocksinwhichtheportfoliomanagertakesactivepositions,whiletakingneutralpositionsin

the remaining benchmark component stocks. Yet another approach could be a factor-

neutrality constraint as in Clarke et al. (2002), which would fit typical style-typeportfolio

constraints. It isquitepossible thatsomeoftheseconstraintsare internallyconflicting.The

long-only constraint, e.g., tends to favour small capitalization active stock weights, which

wouldobviouslyclashwithalargecapitalizationstyleconstraint.

Another issue is the composition of thebenchmark. Fund managers can frequently

choosetheirbenchmarks(withinreasonableboundaries,i.e.,amongapeergroup).Smallcap

fundmanagerswould typicallychoosea representativesmallcapbenchmark, like theASX

SmallOrds.AsshowninLarsenandResnick(1998),themarketcapitalizationofcomponent

stocks in thebenchmark index has a non-trivial impact on the trackingperformance of

enhanced

benchmark

portfolios.

Though

their

analysis

quantifies

the

impact,

they

are

not

explicit on the source. It couldbe a liquidity constraint, orperhaps the (related) excessive

non-normalityofthereturnsofthesestocks.

Afewpapershaverecentlyfocusedontrackingerrormeasurementthatbetterreflectsthe

incentive structure of theportfolio manager, see e.g., Kritzman (1987) and Rudolf et al.

(1999). Roll (1992) suggests that diversification of an investorsportfolio across managers

reducestheimpactofexcessivelyriskyactiveportfolios.Jorion(2002)showsthatthisisnot

a

satisfactory

solution

and

instead

favours

additional

constraints

on

total

risk

to

better

control

forthefreeoptionprovidedtoportfoliomanagerswhoareonlyconstrainedbyactiverisk(or

tracking error). A closer look at investors utility functions andbetter integration of these

utility functions with constrainedportfolio optimizations seems a worthwhile extension of

thispaper.

Perhapsmostimportantly,thereneedstobemoreresearchtowardsproperintegrationof

return and risk forecasts. Though it seems obvious that stock analysts do make risk

assessments

when

computing

forecast

returns,

it

is

much

less

obvious

to

extract

and

combine

theseriskassessmentsintoajustifiableriskmeasure.


23/24

23

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