Lancaster University Management School Multifactor_models

8/3/2019 Lancaster University Management School Multifactor_models

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Lancast er Un ivers i ty Management Sc hoo l Workshop

Quant i t a t ive Equi t y Por t fo l io Managem ent : An Indust ry

Perspect i ve

Presentat ion 1:Issues Relat ing t o Const ruc t ing Mul t i fac t or Models for Equi ty

30th October, 2009

Xavier Gerard

Ron Guido

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Contents

1 I nt r od uc t i on : M ul t if ac t o r M od el s a nd t h ei r u se i n t h e I nv es t m en t Pro c es s2 Overv iew o f ex is t ing m et h ods t o form al pha fo rec as t s

Equal Weighting

Performance Based Weighting Schemes

Weighting Schemes Based investor utility functions

3 Deve lopment o f a fac t or w e ight ing schem e that incorpora tes tu rnover

The basic problem - no transaction costs (no turnover penalty)

Incorporating Turnover in the Portfolio Managers Objective Function

Decomposing the alpha and IR in terms of turnover

The Turnover Adjusted Alpha

4 Si m ul at i on St u dy : T es t in g t h e f ra m ew o rk i n a si m ul at e d e nvi ro nm en t

Generating controlled data

Testing the framework: Performance Turnover Adjusted Alphas versus Alpha that does not incorporateturnover

5 Conc lusions and Appendic es

Calculating supporting risk adjusted IC

Algorithm to generate factors and returns

A new objective function: net IR

6 References

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I n t roduc t ion

Mot ivat ion o f St udy Following Grinold (1989, 1994), an active portfolio managers objective can be characterized as

maximizing the Information Ratio (IR) or value added of her portfolio in the presence of variousportfolio constraints.

Faced with multiple information sources, a portfolio manager can employ linear factor models to

generate return forecasts for the set of securities in his/her investment universe.

Given a set of signals and portfolio constraints, optimal portfolio construction can therefore bereduced to the problem of selecting an optimal weighting scheme for these factors that will providea forecast for any given security

It is well known however, that choosing factor weights to maximize a factor models forecastingability is not the same as selecting weights so that a portfolios Information Ratio (IR) is maximized.In fact it has been demonstrated that least squares regression of linear factor models will notgenerally lead to estimators that maximize unconditional Sharpe Ratios (Sentana, 2005)

In this presentation we attempt to formally derive a framework that produces optimal factorweights which exploits the differential characteristics of the factors themselves to arrive at afactor weighting scheme that generates optimal valued added portfolios under variousportfolio constraints, namely portfolio turnover

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I n t roduc t ion

In Ac t ive Quant i ta t ive por t fo l io management , mu l t i -fac t o r m ode ls a re deve loped

and appl ied t o

Generate risk estimates (covariance matrix of returns) to control portfolio risk in the optimisationprocess

Examples include Barra and Northfield Risk Models

Linearly combine multiple sources excess return forecasts (Alpha Modelling)

In th is present a t ion w e w i l l focus on the la t t e r and cons ider how to fo rm op t im a lcom binat ions o f re tu rn fo recas t s fo r genera t ing va lue added w i th in an equ i ty

por t fo l io

Risk Estimates(Covariance Matrix of

Return)

Return Forecasts

(Alpha)

Optimiser

Constraints/Penalties

Portfolio Weights

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In t roduc t ion : What i s a Mul t i fac t or Mode l?

Mul t i fac t o r mode ls a re used to ex p la in t he ex pect ed retu rns on a cross-sec t ion o f asse ts

The fac t o rs represent c omm on components o f the variance o f secur i t y re tu rns w h ich

con t r ibu te to t he expec t ed retu rn

Typ ica l ly , w e assume a l inear re la t ionsh ip be tw een a secur i ty s re tu rn and the

comm on fac to rs :

it

K

j

ijtjt eFb += =1

i

t

i

KtKt

i

tt

i

tt

i

t eFbFbFbr ++++= ...2211

jtb ijtF

i

teRepresents the specific (idiosyncratic)returnto security iat end of period t.

Represents the returnto Factorjat end of period t.

Represents the excess returntosecurity i at end of period t.

i

tr 0)( =i

teE

Represents the exposureofsecurity ito common Factor j (e.g.Book to Price), at the start of theperiod.

In Ac t ive Quant i ta t ive por t fo l io management , mu l t i -fac t o r mode ls a re used in

Generating Risk Estimates

Generating Excess Return Forecasts (Alpha)

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I n t roduc t ion: How do w e op t ima l l y com b ine a lpha fac t o rs?

Alpha mode ls a lmost a lw ays employ mu l t ip le fac t o rs inst ead o f a s ing le one

Such factors include

Sentiment Based (eg Price Momentum, Earnings Revisions)

Value Based (eg Book to Price, Dividend Yield)

Quality (Change in Working Capital, Growth in Capital Expenditures)

We def ine a Mul t i fac t or model as one t hat l inear ly com bines scores o f ind iv idua l

a lpha fac t o rs to c rea t e a compos i te fo recas t o f exc ess re tu rns

As an ac t ive manager , the m ost im por tan t e lements a t a por t fo l io leve l (in addi t ion t o

por t fo l io cons t ra in t s ) can be sum mar ised by

Returns

Risk

Transaction Costs and Turnover

I f t he p r imary ob jec t ive o f an ac t ive manager is to iden t i f y va lue added opportun i t ies ,

t hen a t an a lpha leve l these por t fo l io leve l cons iderat ions are summ ar ised by

Alpha Factor Performance (Factor Returns)

Alpha Factor Risk (Volatility in the Factor Returns)

Stability of Alpha Factors (Serial Correlation of Alpha factors)

Th is study looks a t how to op t ima l ly fo rm a l inear combina t ion o f a lpha fo recas ts t omeet a port fo l io managers investm ent ob jec t ives

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Overv iew o f ex i s t i ng met hods t o c onst ruc t a lpha

The a im in a lpha const ruc t ion is to de te rmine the opt im a l linear com binat ion o f an

ex is t ing set o f a lpha fac t o rs

i.e. we solve for the optimal weighting of a linear multifactor model

The alpha factors are typically re-scaled and represent excess return forecasts themselves

The factor weights are typically scaled to sum to unity

For portfolio construction, a scaled version of this alpha forecast is then used as the expectedexcess return input

Trad it iona l ly w e can c ons ider 3 b road approaches to de t e rmine the w e ight ing

scheme

Equal / Static Weighting

Weighting schemes based on factor performance

Weighting Schemes based maximising investor utility functions

We w i l l cons ider each in tu rn and high l igh t how the p roposed scheme res ts w i th in

one of these approaches

In so doing we highlight a continuum of alpha weighting scheme approaches that range in

complexity and richness

i

KtKt

i

tt

i

tt

i

t fff +++= ...2211

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The Equal ly Weight ed Approac hAdvantages: straghtforward to implement, no errors in estimation, no additional turnover caused

by time varying weights

Disadvantages: fails to take into account relative performance of factors, fails to considerpotential correlation between alpha factors

In the absence of any additional information, a scheme that equally weights each factor is themost sensible and intuitive and allows the greatest potential for alpha source diversification

This approach is often used a benchmark or base case with which to assess alternative weightingschemes

Kk /1=

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Weight ing sc hemes based on fac t o r per fo rmanceThis approach attempts to utilise the linear decomposition of returns into common factors andtheir factor returns

Factor return forecasts are first generated, which are then mapped into the alpha factor weights

Commonly, the forecasts of each factors performance can be generated via a number oftechniques including:

Macroeconomic based models

ARIMA Models

Markov Switching Models

Historical averages obtained from regression analysis [eg Fama-Macbeth regression methodology]

Advantages:

straghtforward to implement

intuitive

Disadvantages:

In the absence of forward looking factor return forecasting models, the factor return forecastsbased on historical averages are backward looking in nature and susceptible to marketturning points

Such approaches only consider factor performance (returns) as important when determiningfactor weights; they do not include the impact of the correlation or risk of the factorsthemselves

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Weight ing schem es based on investor u t i l i t y func t ions

Weight ing Sc hemes Based investor u t i l i t y func t ionsIssue with previous schemes is that they fail to consider the active portfolio managers true

objective function in deriving optimal alpha weighting scheme

Following Grinold (1989, 1994), an active portfolio managers objective can be characterized asmaximizing the Information Ratio (IR) or value added of her portfolio in the presence of various

portfolio constraints

There have been several approaches that attempt to align the use of factors with principle ofmaximsing the investors objective function

Brandt, Santa-Clara and Valkanov (2009)

Propose an approach whereby the weight in each stock is modeled as a linear function of thefirms characteristics (factors), such as its market capitalization, book-to-market ratio, and laggedreturn.

The coefficients of this function are found by optimizing the investors average utility (theirobjective function) of the portfolios return over a given sample period

Though intuitively appealing, the problem is not analytically tractable, and is reliant on numerical

procedures that are not guaranteed to be solvable. Solution can also potentially result inunbounded portfolio weights

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Sorenson , Qian, Hua, and Schoen (2004)Qian Hua and Sorenson (2004, 2005): Extend the framework of Qian and Hua (2004) and show how

one can derive the factor weighting scheme that maximises a portfolios Information Ratio (IR)

By specifying a straghtforward mean-variance objective function for the portfolio manager and as wellas simplifying assumptions about factor score correlations, their approach yields an analytically

tractable solution for the factor weights

In particular they highlight that the IR of an long-short mean variance optimised strategy is directlyrelated average performance of the alpha model (via the Information Coefficient) of a strategy as wellthe consistency of the forecasts over time

They further demonstrate that the active risk of a strategy is not simply the target tracking error ( )but it is also a function of the strategy risk of the investment strategy (captured by )

Using these insights they form optimal alpha weights so that the ratio of IC to its std deviation ismaximised

)( ptravg

)()( ta rdisNICstd =

)()( ta rdisNICavg =)( ptrstd

)(

)(

)(

)(

ICstd

ICavg

rstd

ravgIRa

t

at ==

a)(ICstd

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Sneddon (2008): Incorporating Transaction Costs in Optimal Alpha ConstructionSneddon considers the problem of optimal factor weighing schemes by considering an even more

realistic objective function that takes into account portfolio turnover and transaction costs

This approach derives a multiperiod IR in a semi-analytical framework under transaction costs overweighting tortoise factors that have slow but stable IC decay and underweighting hare

factors high but fast decaying

While posing a more realistic metric to maximise (net IR), the main drawback with this approachis that it fails to incorporate strategy risk in constructing optimal factor weights

Qian Sorenson and Hua, 2007 (QSH) Extension for transaction costs

Attempt to introduce realistic portfolio construction objectives by modelling the cost ofimplementation and by constructing optimal alphas n the presence of transaction costs.

The main drawback is that, unlike Sneddon (2008), transaction costs are introduced byconstraining the portfolio to achieve a target alpha autocorrelation which in principle determines

drives portfolio turnover.

This implementation is ad-hoc at best and not a realistic description of the investment manager'sobjective function

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In th is p resenta t ion , w e propose a new approach t ha t represents a syn thes is o f p revious methods to dea l w i th op t im a l a lpha const ruc t ion under more rea l ist ic

por t fo l io cons t ruc t ion se t t ings; namely t ransac t ion cos t s

To redress the shor tc omings o f the las t t w o approaches

We use the analytically elegant and tractable framework developed by QSH that link strategy

performance and strategy risk of the alpha model to optimal alpha weights

We then utilize the insights developed by Sneddon (2008) by introducing transaction costs into theobjective function of the portfolio manager. The key is to link transaction costs and portfolio turnoverto alpha factor serial correlation

We deve lop the f ramew ork in s teps ; f i s t c ons ider ing the o r igina l IR max imizat ion

prob lem developed by QSH and then ex t ending the prob lem for t he case of

t ransac t ion cos ts

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Opt im al Fac t or Weight ing:The bas ic p rob lem in t he absence o f t ransact ion c osts

Act ive Manager every per iod seeks to genera te a por t fo l io (h

) tha t m ax im ise va lueadded sub jec t t o cons t ra in ts :

The por t fo l io is do l lar neut ra l and neut ra l t o a l l r isk fac t ors (X) t h rough l inear equal i tyconst ra in ts and t a rgets an ac t ive r isk o f v ia the t rac k ing e r ro r c ons t ra in t .

represents the d iagona l mat r ix o f id iosyncra t ic var iances w i th t yp ica l e lement ,

Given a set of K a lpha fac t o r fo recas t s , fk (k = 1, K), t he manager s prob lem is toconst ruc t a compos i te a lpha fo rec as t fo r re tu rns by se lec t ing the fac t o r w e ights ( )

The ob jec t ive : selec t so that t he ex-post IR (over t he last T per iods) o f t he va lueadded por t fo l io h i s max im ised:

,01'.. =t

hts

=

==K

k

k

it

k

itit frE1

)(

)(

)(max

a

t

a

t

rstd

ravgIR =

tosubject 1=k

k 10 k

atttthh ='

and

The portfoliomanagers objectivefunction

ttttth

hhhU = '2

1'max

and0' =

ttXh

a

i

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15


Generat ing a so lut ion to the bas ic prob lem : decom posing the por t fo l io re turn

Given the objective function, the optimal portfolio weights can be solved analytically (dropping timesubscripts for notational ease)

Using this basic solution, we can express the expected excess return on the portfolio hof Nassets as

=ar

~11 =h where =~ X[ 11 )'( XX ]' 1X

2

1~

i

i

ih

=

=

=N

i

iirh1

excess return tosecurity i

=

=N

i i

i

i

i r

1

1 ~

=

N

i

iiRh1

mean adjusted residualreturn to security i

due to dollar neutrality and neutrality ofthe portfolio to risk factors X

ir = mean adjusted residualreturn to security isuch that in

the cross section avg(r/) =0

Employing the definition of covariance between the terms and yields the following (seeappendix 1)

We denote this correlation between the risk adjusted alpha and risk adjusted return as the RiskAdjusted Information Coefficient (IC) :

i

i

~

i

i

r

=ar

i

i

i

i

i

i

i

i rdispdispr

corrN

~,

~)1(

1

=

i

i

i

i rcorrIC

,

~


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Generat ing a so lu t ion t o the bas ic prob lem: decom posing the por t fo l io s IR

We can simplify the expression for the active portfolio return by substituting out the risk aversionparameter due to the fact the portfolio has a tracking error constraint set to

Assuming the cross sectional average of the risk adjusted alpha is approximately zero,

a

( )= =

N

i

iia h1

222

We therefore have the following expression for the realised single period excess return on theportfolio

=ar

i

ia

rdispICN

1

=

=

N

i i

i

a 1

2~1

=

=

N

i i

i

a 1

2~1

=

i

i

a

dispN

~1

1

From this we can obtain the expected excess return to the portfolio as

and the expected active risk as

Note the active risk of a portfolio is a function of both the (exante) tracking error ( ) AND thevolatility of the alpha models IC (strategy risk)

)(1)( iiaa rdispNICravg =

)(1)()( iiaa rdispNICstdrstd =

a

)()(

)(

ICstd

IC

rstd

ravgIR

a

a

=

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Generat ing a so lu t ion t o the bas ic prob lem

We can use the previous insight that IR of an optimal risk constrained portfolio is directly linked tothe ratio of the model IC and its variability over time

Given a set of Kforecasting signals, fk(k = 1, K), the managers problem is to construct a

composite alpha forecast for returns by selecting the factor weights,

The objective : select so that the ex-post IR is maximised

Based on the composite alpha definition, it can be easily shown that under the assumption that thefactors are standardised,

)(

)(

)(

)(max

ICstd

ICavg

rstd

ravgIR a

t

a

t== tosubject 1=k

k 10 kand

)1,0(~~

iidfkitwhere=

=K

k

k

itkit f1

~

=

= i

ik

j

j

ij

i

rfcorrIC

~,

~1

1

)'(1

1 =

=K

k

kk IC

=

i

i

i

k

ik

rfcorrIC

~,

~

where

and = '

= correlation matrix of factors

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Generat ing a so lu t ion t o t he bas ic p rob lem

Based on the composite alpha definition, we can therefore represent the maximisation problemas

The solution to this maximisation is straghtforward and tractable:

)(

)(

)(

)(max

ICstd

ICavg

rstd

ravgIR

a

t

a

t ==

tosubject 1=

kk 10 kand

where

)'()(1

1 =

=K

k

kk ICICavg

ICICstd =

')(1

IC = covariance matrix of factor ICs

IC

K

k

kk IC

=

=

'

'1

kIC

IC

IC

IC

1''

'1

1

=

The alpha weight of a factor depends not only

on its risk reward trade-off (based in IC) butalso on its performance correlation to otherfactors in the model

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Opt im al Fac t or Weight ing :Ex t ending the Bas ic Problem t o acc ount fo r t ransac t ion cos ts

Ac t ive Manager every per iod seek s to generate a long/shor t por t fo l io (h) t ha tmax imises va lue added sub jec t to cons t ra in ts and tu rnover pena lt ies :

We assume that turnover, often expressed as can be approximated with the quadratic term

effectively maps turnover to a transaction cost estimate

To make the analysis tractable we assume for simplicity that risk aversion () is stable over time andthat the residual risk is constant and identical across all firms, then we have the solution as

)(')(2

1'

2

1'max 11 ttttttttt

hhhhhhhh

t

t

htt

hh '

++=

2

1ttt

hh

Using this, we obtain an expression for the realised alpha (return) of the portfolio as

Recursively substitutingh, we obtain

which can be truncated at some appropriate lag, p*

i

ititpt rhr

1+= tt h

+=

i

ititit rh

1

=+

= 01 )(p i

itpitp

p

r

ptr

where + 2 It2=and

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Dec ompos ing t he a lpha

We assume t ha t t he alpha is a w e ighted c ompos i te o f k fac to r sc ores

The factors have been standardized and are uncorrelated with another. is a scalar that ensures the

composite has unit variance. We assume it is relatively stable over time.

Therefore at time tthe expression in can be written as

The portfolio return can therefore be expressed in terms of lagged factor ICs :

)1,0(~~

iidfkit

=

k i

it

k

itkit

i

it rfr~

)(

where

ptr

) =k

t

k

tkr rfcorrN ),~

()1(

ptr

( )

=+

=0

1)1(

p k

k

ptkp

p

r ICN

=

=K

k

k

itkit f1

~

k

ptIC ),~

( tk

pt rfcorr where

=+

=

01

)1(p

ptp

p

r ICN

where

),~

(

),~

(1

t

K

pt

tpt

rfcorr

rfcorrM

ptIC

5.0])1[(

= N

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Dec ompos ing t he ex -post IR

Using th is ex press ion w e can now cons ider the ob jec t ive of the va lue added fundmanager w hen se lec t ing opt im a l fac t o r w e igh ts

The objec t ive : se lec t so that t he ex-post IR (over t he last T per iods) o f t he va lueadded por t fo l io h i s max im ised:

The expression for the average return can be approximated by

The expression for the portfolio variance is given by

To simplify, we make the following assumptions

i.e. factor ICs are only contemporaneously correlated

)(

)(

maxpt

pt

rstd

ravg

IR =

tosubject 1=k

k 10 kand

)( ptravg =p

pp

r ICN )1( where 1+ p

pp

)(

k

pt

k

p ICavgIC = and

)( ptrvar ( )

= ppt

p

r ICvarN2

])1([

=K

p

p

p

IC

ICIC M

1

>

==

00

0),cov(

2

,

j

jICIC pkk jp

k

p

>

==

00

0),cov( ,,

j

jICIC plkl jp

k

p

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Dec ompos ing t he ex -post IR

The express ion fo r t he por t fo l io var iance can t herefo re be approx im ated by

The term represent the variance covariance matrix of the lagged factor ICs for lag p, which we will

assume can be consistently estimated using sample data,

Therefore the ex-post IR can be approx im at ed by

which is of the form

I f the ob jec t ive func t ion is Gross IR max imisat ion, w e c ou ld use th is ex press ion in

t he ob jec t ive func t ion to search fo r op t ima l fac t o r w e ights ,

In Appendix 2, we consider the case when the objective function is Net IR

( )

= =

*

1

22])1([

p

p

p

p

rN pjiji

p ,,

],[ where

p

p

)(

)(max

pt

pt

rstd

ravgIR =

( )

( )

=

=

=

*

1

2

*

1

p

pp

p

p

p

pp IC

*

*

p

p

=

kpp

pp

1''

'1

**

1

**

=

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Sim ulat ion St udy: Det a i ls of Dat a const ruc t ion

Using c on t ro l led exper imenta l da ta w e can be t te r unders tand the impact o f fac t o r tu rnover on opt im a l a lpha and port fo l io cons t ruc t ion

To do so, w e w ou ld l i ke t o c rea te da ta t ha t possess sal ien t charac t e r is t ics o f

observed re tu rns and fac t o rs typ ica l ly used in t he quant p rocess .

Key paramet ers

Marke t : N = 350, T = 100

Risk Model : Res idua l Risk es t im ated f rom a CAPM model based on FTSE 350 over 5 years

end ing 2008, c ross sec t iona l vo la t i l i t y es t im ated f rom re t urns

Alpha is const ruc ted f rom 4 typ ic a l quant fac t ors , w i th per formanc e:

IC cor re lat ions 20% across fac t o rs

Based on these ICs and Fact ors , re turns fo r the s t ock s are generated

avg(IC) s t d(IC) 1st order rho

Fac t or 1 (Qual i t y Type) 3.0% 4.0% 90.0%

Fac t or 2 (Earnings Rev is ion Type) 4.0% 6.0% 50.0%

Fac t or 3 (Event Type Signa l ) 8.0% 6.5% 30.0%

Fac t or 4 (Mom ent um Type Signa l) 5 .0% 8.5% 70.0%

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Sim ulat ion St udy: generat ing fac t ors and re t urns

Given cross sec t iona l re turn vo la t i l i t y , observed ICs and the 1st

order corre la t ion infac t o rs w e can produce s imu la ted fac t o rs and ret u rns

Generat ing t he fac t o rs

Each period, t, factors are generated using a non-linear least squares procedure which selectsfactor values that ensure: a cross sectional mean of zero, a cross sectional variance of 1,

orthogonal factors, and the specified 1st order serial correlation

Generat ing t he re turns

Using the history of simulated ICs generated from estimated mean and covariance, cross sectionalreturn volatility estimates and the generated factor scores; returns are generated based on thefollowing cross-sectional regression model:

is the NxK matrix of factor values at time, t

since the factors are orthogonal

See appendix 3 for details on algorithm

tttt ufr += ~

tf~

rfff tt '

~

)

~

'

~

(1

=rtIC =

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Sim ulat ion St udy: Observed Proper t ies o f Fact ors and Returns

Using t he ca l ib rated parameters w e can fu r ther inves t iga te how fac t o r tu rnover in f luences IC

Factor Decay: Autocorre la t ion in Factor

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 2 3 4 5 6 7 8 9 10 11 12

Lag ( in months)

Factor 1 (Quality Type)


-10%

0%

10%

20%

30%

40%

50%

60%

1 2 3 4 5 6 7 8 9 10 11 12

Lag ( in months)

Factor 2 (Earnings Revision Type)


-5%

0%

5%

10%

15%

20%

25%

30%

35%

1 2 3 4 5 6 7 8 9 10 11 12

Lag ( in months)

Factor 3 (Event Type Signal)


0%

10%

20%

30%

40%

50%

60%

70%

80%

1 2 3 4 5 6 7 8 9 10 11 12

Lag ( in months)

Factor 4 (Momentum Type Signal)

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Sim ulat ion St udy: Resul t s

Fact or decay (au toc or re lat ion) s t ruc tu re has a c lear impac t on the IC decay

Th is is cap t ured by the de l ta func t ion

w h ich w e ights t hese lagged ICs acc ord ing

to t he se lec t ed tu rnover pena lt y , e ta ( )

Lagged ICs for each Factor

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Lag ( in months)


Factor 2 (Earnings Revision T ype)



I t i s th is p ro f i le tha t w e w ish to inc orpora te

in to the fac to r w e igh t ing scheme w hencons t ruc t ing a lphas t hat acc ount fo r the

impac t o f tu rnover.

Delta at different horizons

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

horizon (P)

d

elt p

Eta = 0 Eta = 1 Eta = 5

Eta = 8 Eta = 10 Eta = 20

Eta = 100

26


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Sim ulat ion St udy: Per form anc e of Fac t or Weight ing Sc hem e

In underst and ing the impact o f the w e ight ing scheme in dea ling w i th t u rnover w e

cons ider the fo llow ing s t ra tegy :

We construct optimal factor weights over the sample period over a range of turnover penalties(eta) from eta = 0 (no turnover penalty) to 100 (high turnover penalty)

We then measure gross and net (IR) performance of the resulting alphas using the sameobjective function:

for a range of transaction costs : 0bps to 150bps

Fact or w e ight ing sc heme ou tc omes fo r a range of penal t ies

ttttth

hhht

'2

1'max

Eta = 0

(No Turnover

Penalt

Et a = 1 Et a = 7 Et a = 20 Et a = 50Eta = 100

(High Turnover

PenaltFac t or 1 (Qual i t y Type) 11.4% 12.9% 24.1% 40.4% 47.5% 48.8%

Fac t or 2 (Earn ings Revis ion Type) 28.0% 28.0% 26.0% 20.9% 18.5% 18.0%

Fac t or 3 (Event Type Signal) 45.2% 41.9% 25.7% 14.7% 11.0% 10.3%

Fac t or 4 (Mom ent um Type Signal) 15.4% 17.1% 24.2% 23.9% 23.0% 22.9%

27


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Fact or w e igh t ing scheme ou tc omes fo r a range of pena l t ies

As turnover penalties increase(transaction costs matter), factorswith slower decay progressively

obtain a higher weight (Factor 1)

Factors with high factor decayprogressively obtain a lower weight(Factor 3)

Opt im al fac tor w e ights account ing fo r tu rnover

0%

10%

20%

30%

40%

50%

60%

Eta = 0

(No

TurnoverPenalty)

Eta = 1 Eta = 5 Eta = 7 Eta = 8 Eta = 10 Eta = 20 Eta = 30 Eta = 50 Eta = 100

(High

TurnoverPenalty)Turnover Penal t y

FactorWeight


Factor 2 (Earnings Revision Type)



28


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Performanc e Resul ts

Net IR(TC in bps) 0 5 7 10 50 100 Dif ferenc e

0 4.50 4.47 4.42 4.35 4.07 4.04 -0.46

20 3.61 3.71 3.70 3.67 3.46 3.43 -0.18

40 2.71 2.95 2.98 2.99 2.84 2.81 0.10

60 1.81 2.19 2.26 2.31 2.22 2.19 0.38

80 0.91 1.43 1.54 1.63 1.60 1.58 0.66

100 0.02 0.68 0.82 0.94 0.98 0.96 0.95

120 -0.88 -0.08 0.10 0.26 0.36 0.35 1.23

140 -1.78 -0.84 -0.61 -0.42 -0.25 -0.27 1.51

150 -2.23 -1.22 -0.97 -0.76 -0.56 -0.58 1.65

Reduc t ion in Turnover (%) 72% 103% 142% 253% 259%

Note: N = 350, target risk = 9%, diagonal risk model

Turnover Penalty (Eta)

To gauge performance we look at relative net IR

In absence of any transaction costs, the mostoptimal scheme is one which sets eta to zero.

However as transaction costs increase, it is clearlysub-optimal to construct alphas that ignore eta;

At 100bps, the cost of ignoring turnover penalties(eta of 50) is almost 1.0 in terms of net IR

Net IR ( re lat ive t o average) across Eta

-1.30

-1.00

-0.70

-0.40

-0.10

0.20

0.50

0.80

0 1 5 7 8 10 20 30 50 100

Turnover Penaly (eta)

NetIR

0 bps 20 bps 40 bps

60 bps 80 bps 100 bps

120 bps 130 bps 150 bps

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Conc lus ions and Ex t ens ions

Summary

We have prescr ibed a f ramew ork w h ich is an ex tens ion o f ear l ie r mode ls

designed to form opt ima l fac t o r w e ight s in the p resence o f tu rnover and

t ransac t ion cos ts

By in t roduc ing tu rnover c ons idera t ions in the por t fo l io cons t ruc t ion

p rocess , w e exp l i c i t l y acknow ledge the impac t o f a lpha memory of the

por t fo l ios

I n par t i cu la r w e f ind tha t w hen the pena l t y to t u rnover i s h igh, w e p refe r

fac t o rs tha t de l iver no t s t ab le per formanc e in the shor t run bu t have s low

in fo rmat ion decay as c ap tured by the au toc or re lat ion o f the a lpha fac t o rs

We f ind tha t fa i l ing to incorpora te t he impact o f fac t o r based tu rnover

p roduces sub-op t im a l w e ight ing sc hemes and a lphas in rea l is t ic por t fo l io

se t t ings .

The f ramew ork adopted here and app l ied to t he p rob lem o f t ransac t ion

c os ts have poten t ia l l y many other app l ica t ions w here more general u t i l i t yfunc t ions a re w ar ranted in order t o cap ture real is t ic por t fo l io cons t ruc t ion

30


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Append ix 1 Ca lcu la t ion o f r isk ad jus t ed IC

Em ploy ing t he def in i t ion o f sample c ovar iance

Therefore

Given tha t by c ons t ruc t ion th rough arb i t ra ry sca l ing o f the re tu rns

i

i

i

i r

~~cov

===

=

N

i i

iN

i i

iN

i i

i

i

i r

N

r

N 111

~~1~~

1

1

=

N

i i

i

i

i r

1

~~

+

= ==

N

i i

i

N

i i

i

i

i

i

i rN

rN11

~~1

~,

~cov)1(

0~

1

=

=

N

i i

ir

=

i

i

i

i rN

~,

~cov)1(

= N

i i

i

i

i r1

~~

=

i

i

i

i

i

i

i

i rstdstdr

corrN

~~~,

~)1(

31


32/36

Append ix 2 : Fur ther Ext ens ions- Ex -post Net IR Formula t ion

I f t ransac t ion c os ts mat te r , t hen i t i s more na tura l to use Net IR ra ther than Gross

IR in the ob jec t ive func t ion :

To ob ta in an es t ima te w e approx im a te the t ransac t ion cos t o f t he port fo l i o a t t ime t

w i t h

Using once aga in t he de f in i t ion fo r t he op t im a l por t fo l io , h , and rec urs ive ly

subst i tu t ing

assuming c ross p roduct te rm s are neg l igib le

)(

)()(max

pt

ptpt

rstd

TCavgravgIR

=

varianceportfoliothesince

)()( 11 = tttt hhIhh =i

itit hh2

1)(

+=i

itititit hhhh )2( 12

1

2

)(2 12

2

=i

ititA hh

i

itA h222 =

i

Aith 2

22

= i itithh 1 + +

211 itit

i

itit hh

i

itit )(1

12

ptTC

32

A di 2 E t N t IR F l t i


33/36

Appendix 2 : Ex -post Net IR Form ulat ion

Not ing aga in tha t the a lpha is a compos i te o f K or thogonal fac t o r scores :

And assuming tha t the fac t o rs have the fo l low ing au toco r re la t ion s t ruc t u re

Then i t c an be show n tha t

The t ransact ion cos t o f the por t fo l io a t t im e t can t hen be approx ima t ed w i th

The average can be approx im ated by

=

=

=

i

K

k

k

itk

K

k

k

itk

i

itit ff1

1

1

2

1

~~

==00

)~

,~

( 1j

kjffcorr

ktj

it

k

it

=

K

k

ktk

i

itit N1

221 )1(

)(21

2

i

ititA

pt hhTC

k

ktkA N

2

2

22)1(

2

)(TCavg

ktkA avgN )()1(2 22

22

k

pt

33

A di 2 Th Fi l Obj t i F t i


34/36

Appendix 2 : The Fina l Objec t ive Func t ion

Inc lud ing tu rnover resul ts in an ob jec t ive func t ion w i th t h ree broad components :

Average Real ized Por t f o l io A lpha is an ex ponent ia l ly w eighted average of t he lagged

fac t or ICs

Average Ac t ive Risk o f t he Por t fo l io is also an ex ponent ia l ly w eighted average of

lagged fact or IC var iance-c ovar iance mat r ic es

Average Transac t ion/Mark et Im pact Cost o f t he Por t fo l io is governed by both t he

ac t ive r isk o f t he port fo l io and the degree o f the decay in t he fac t o r va lues (as

c aptured by the i r f i rst o rder au toc or re la t ion

)(

)()(max

pt

ptpt

rstd

TCavgravgIR

=

)( ptTCavg

k

ktkA avg

N)(

)1(2

2

2

22

)( ptravg =p

pp ICN )1(

( )

=

=

*

1

22)]1([

p

p

p

pN)( ptrvar

34

A di 3 A l i th t t f t d t


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Append ix 3 : A lgor i thm t o genera te fac t ors and re turns

Given cross sec t iona l re turn vo la t i l i t y , observed ICs and the 1 st order corre la t ion in

fac t o rs w e can produce s imu la ted fac t o rs and ret u rns

Generat ing t he fact ors (g iven )

Each period, t, using a non-linear least squares procedure, we select the set of factor valuesfor each factorj= 1, 4 so that the residuals from the following set of conditions are minimised

cross sectional mean is zero

cross sectional variance is 1

factors are orthogonal

factor has a serial correlation

Generat ing t he re t urns (g iven )

Using a nonlinear least squares procedure, returns are generated each period so that residuals

from the following set of conditions are minimised:

cross sectional regression residual average is zero

the residuals are uncorrelated with the factors and their lags

N

k

tf 1'~

)1(~

'~

Nff ktk

t

j

t

k

t ff~

'~

k

tf~

k

t

k

t fe 1~'

k

ttrtt ICfruwhere~

=Ntu 1'

j

ktt fu ~

'

ttr ICandf~

,

k

tk

k

t

k

t ffewhere 1~~

=

35

References


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References

Brandt, W., Santa-Clara, P., and Valkanov, R., (2009) Parametric Portfolio Policies:Exploiting Characteristics in the Cross-Section of Equity Returns, Review of FinancialStudies

Grinold, R.C. (1989), The Fundamental Law of Active Management, The Journal of PortfolioManagement, Spring, pp 30 37.

Grinold, R.C. (1994), Alpha is Volatility Times IC Times Score, The Journal of PortfolioManagement, Summer, pp 9 - 16.

Qian, E. and Hua, R., (2004) Active Risk and the Information Ratio, Journal of InvestmentManagement, Vol 2.

Sorenson, E., Hua, R., and Qian, E., (2005), Contextual Fundamental, Models and ActiveManagement, Journal of Portfolio Management, Vol 32, pp 23-36.

Sorenson, E., Hua, R., and Qian, E., and Schoen, R., (2004), Multiple alpha sources andactive management, Journal of Portfolio Management, Vol 31, pp 39-45.

Sneddon, L. (2005), The dynamics of active portfolios, Proceedings of Northfield Research

Conference

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