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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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Overv iew o f ex i s t i ng met hods t o c onst ruc t a lpha
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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Overv iew o f ex i s t i ng met hods t o c onst ruc t a lpha
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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Weight ing schem es based on investor u t i l i t y func t ions
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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Weight ing schem es based on investor u t i l i t y func t ions
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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Weight ing schem es based on investor u t i l i t y func t ions
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
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
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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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
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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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
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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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
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)
Factor Decay: Autocorre la t ion in Factor
-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)
Factor Decay: Autocorre la t ion in Factor
-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)
Factor Decay: Autocorre la t ion in Factor
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 1 (Quality Type)
Factor 2 (Earnings Revision T ype)
Factor 3 (Event Type Signal)
Factor 4 (Momentum Type Signal)
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
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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%
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Sim ulat ion St udy: Per form anc e of Fac t or Weight ing Sc hem e
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 1 (Quality Type)
Factor 2 (Earnings Revision Type)
Factor 3 (Event Type Signal)
Factor 4 (Momentum Type Signal)
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Sim ulat ion St udy: Per form anc e of Fac t or Weight ing Sc hem e
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
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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(
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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
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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
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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
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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.
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