integrated alpha modelling (2011)

Integrated Alpha Modelling

Xavier Gerard [email protected]

Independent

Ron Guido [email protected]

Fidelity International 25 Cannon Street, London EC4M 5TA

Peter Wesselius

[email protected] Independent

Abstract Alpha modelling typically refers to the selection and weighting of various information sources, which when combined are used by active portfolio managers to forecast security returns. It is traditionally seen as an exogenous input in the construction of the portfolio holdings. Instead, a number of authors have recently argued that alpha modelling should be integrated within the portfolio-construction process, to account for the active manager�’s objective and constraints. Building in particular upon the frameworks of Sneddon (2008) and Qian et al (2007), we present a parsimonious and analytically tractable alpha modelling approach that aims at maximising the typical objective function of an active manager. Our modelling scheme combines in one framework several salient features of previous methodologies. For instance, as transaction costs increase, allocating more weight to signals with lower information decay is shown to improve portfolio value added. Factors with higher return-to-risk ratios are also given a higher prominence since they help allocate strategy risk more efficiently. Keywords: Active Management; Alpha Modelling; Portfolio Construction

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1. Introduction

Whether or not an active manager follows a systematic process, he/she typically will have

to take into account various inputs when building his/her portfolio. For example, an active

manager who predicts future returns (alpha) using various information sources (factors) will

have to combine his/her views on factor performance with considerations of risk and

investment constraints, such as transaction costs. In order to manage complexity, the

portfolio-construction process is typically divided into a number of separate tasks that occur

sequentially. For example, active managers will usually build their alpha model by selecting

and appropriately weighting their information sources and then control for turnover and risk

at the portfolio-construction stage.

One limitation of this sequential approach is that it fails to recognise the interdependence

of the various inputs in the portfolio-construction process. If a manager�’s objectives and

constraints preclude him/her from utilising certain sources of alpha, there would be little

value in recognising these when modelling alpha. Instead, one should model the alpha that the

manager is best able to capture, in a manner that maximises his/her objective function.

Although this line of research is still in infancy, academics and practitioners have started to

formally tackle the issue, and a number of methodologies have recently been introduced to

align the use of factors with the principle of maximising the objective function of an active

manager.

In this paper, we first review the literature that shows that an integrated approach, which

accounts for a manager�’s objective and constraints in the alpha modelling, offers some

significant advantages. We then present a parsimonious factor-weighting scheme that is

intuitive and analytically tractable, and combines in one framework several salient features of

previous methodologies. In particular, our approach draws upon the work of Sneddon (2008)

and Qian et al (2007) who developed alpha modelling frameworks that address many of the

complexities of the active management process. For instance, in the presence of transaction

costs, it is generally no longer beneficial for the portfolio holdings to fully adjust to new

information. Maximising the portfolio manager�’s objective function becomes a multi-period

problem where one needs to consider factor performance over several stages. In this context,

as the turnover penalty increases, our approach gives more weight to those factors with lower

mean reversion, or alpha decay. Moreover, the mismatching between a proprietary alpha

model and a third-party risk model leads to strategy risk not being fully accounted for. One

can alleviate this problem, however, by recognising this source of active risk when modelling

3

alpha. Finally, we illustrate with an empirical example the mechanics behind our approach

and provide some evidence for its ability to add value to the active management process.

2. The Building Blocks of the Portfolio-Construction Process

Independent of whether an active manager uses a systematic quantitative approach to

portfolio construction or a heuristic one, he/she will typically consider each one of the

building blocks shown in Exhibit 1. The portfolio constraints will be a function of the

investment mandate of the active manager, which is largely externally driven. They will also

arise from internal considerations, such as the size of assets under management and the

resulting trading costs of the fund, which will lead to different levels of portfolio turnover

being prescribed.

[EXHIBIT 1 ABOUT HERE]

Another important part of the portfolio-construction process is to assess or model the risk

of each security to derive mean-variance efficient active holdings. Stock-specific factors,

macro factors, and statistical factors can be used to model the risk that a portfolio is exposed

to. Quantitative investors often rely upon third-party risk models at this stage.

The most important differentiating factor for an active manager is often his/her alpha

model, or forecast of future return. The alpha of stock i at time t is typically defined as a

linear combination of several factors ( jF ):

ti,m

tm

ti,

tti FwFw 11 (1)

Given this specification, the modelling involves finding a set of relevant factors and

determining an appropriate weighting scheme (w). This study focuses on the latter effort. An

active manager will usually choose the factor-weighting scheme that maximises the predictive

power of his/her model. Then, as described in Exhibit 1, he/she will control for risk and

turnover, at the portfolio-construction stage.

Unfortunately, this sequential approach is not guaranteed to maximise an active

manager�’s objective function for several reasons. For instance, Sentana (2005) points out that

least squares regressions of linear factors, which are often used in alpha modelling, will not

generally maximise unconditional Sharpe ratios. Lee and Stefek (2008) show how the

4

mismatching between a proprietary alpha model and a third-party risk model leads to strategy

risk not being fully accounted for. Moreover, when trading is costly, it is not usually

beneficial to fully adjust the portfolio holdings as new information arrives. Hence,

maximising the active manager�’s objective function is better described as a multi-period

problem where one needs to consider factor performance over several stages.

It is therefore important to consider the active manager�’s objective and constraints when

deriving a factor-weighting scheme. As a result, an integrated approach to alpha modelling, as

depicted in Exhibit 2, might be preferred to the sequential process in Exhibit 1. In this

context, several methodologies have been introduced to align the use of factors with the

principle of maximising an investor�’s objective function.


For instance, Brandt et al (2009) developed an approach where the portfolio holding in

each stock is modelled as a linear function of various characteristics (factors), such as market

capitalisation, book-to-market ratio and lagged returns. The coefficients on each characteristic

are then derived by maximising the investor�’s expected utility, defined in terms of his/her

portfolio returns. While the approach is intuitively appealing, the problem is not analytically

tractable, and is reliant upon numerical procedures.

The factor-weighting scheme of Sorenson et al (2004) also focuses on maximising the

investor�’s objective function. However unlike the work of Brandt et al (2009), this

methodology relies upon a set of assumptions and conditions that allow for an analytical

solution. Critically, the method builds upon the findings of Qian and Hua (2004) for the

importance of strategy risk as a source of risk unaccounted for by third-party risk models. The

authors use the fact that strategy risk enters a portfolio manager�’s objective function, which

they define as his/her information ratio (IR), to show that a factor-weighting scheme that

maximises the portfolio IR helps to efficiently allocate that risk across the factors of the alpha

model.

In addition to incorporating risk considerations when modelling alpha, several authors

have recently investigated the impact of transaction costs. In the presence of trading costs, it

is generally beneficial to adjust portfolio holdings only partially with the arrival of new

information. The portfolio-construction process becomes a multi-period problem where the

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portfolio holdings in one period influences the choice of portfolio holdings in subsequent

periods.

Qian et al (2007) show how the framework of Sorensen et al (2004) can be extended to

account for portfolio turnover. The authors first model the relation between the

autocorrelation of the scores of the alpha model and the turnover of the portfolio. They then

use an extended definition of the alpha model that includes not only current but also lagged

factor scores, which serve to control the autocorrelation of their alpha model. As in Sorensen

et al (2004), the weights on factors are chosen to maximise the IR of a long-short portfolio,

but the solution is also constrained to achieve an appropriate level of autocorrelation for the

alpha model, which matches a desired level of portfolio turnover.

Sneddon (2008) controls for the impact of trading costs in the �“value-added�” utility

function, which serves to derive the portfolio holdings, as well as the �“objective�” function,

which guides the selection of factor weights. This approach derives a multi-period IR in a

semi-analytical framework under transaction costs. In this scheme, the weighting of factors

with high and low mean reversion (alpha decay) varies depending upon turnover. Because the

author controls for turnover directly in the value-added utility function, this scheme does not

have to rely on the extended alpha model of Qian et al (2007). Instead, one can use a

parsimonious model with current factor scores only. However, in contrast to Qian and Hua

(2004), Sorensen et al (2004) and Qian et al (2007), this framework does not take into account

the strategy risk that arises from the mismatch between the proprietary alpha model and the

third-party risk model. Active risk is defined solely in terms of an ex-ante tracking error.

Like Sneddon (2008), Garleanu and Pedersen (2009) control for transaction costs in the

value-added utility function. However, instead of finding the weighting scheme that

maximises the portfolio IR, they attempt to maximise the present values of their utility

function. The optimal portfolio is a linear combination of the existing portfolio, the optimal

portfolio absent trading costs, and the optimal portfolio based on future expected returns and

transaction costs. In their framework, signals with slower mean reversion get more weight

since they lead to a favourable positioning both now and in the future.

In a recent article, Grinold (2010) presents an approach based on dynamic programming

to address an active manager�’s multi-period problem. Each signal is linked to an investment

strategy, so that finding the optimal weighting scheme becomes a search for the optimal mix

6

of strategies. In this search, three traits of signals are important: the predicted IR of each

strategy, their predicted return correlation and the rate of change of each signal.

Our methodology for deriving a factor-weighting scheme takes contributions particularly,

but not exclusively, from Sneddon (2008), Qian et al (2007), and Sorensen et al (2004). These

studies are of particular interest since they specifically address an active manager�’s

investment problem and do so in practical manners that have proven to be remarkably

relevant to practitioners. We preserve the analytical tractability of our results by relying upon

a set of simplifying assumptions, some of which are similar to those in Qian et al (2007), and

Sorensen et al (2004). We also control for strategy risk, and take into account the alpha decay

of our factors. Importantly, however, lagged factors only enter our framework because of the

turnover penalty that, as in Sneddon (2008), impedes portfolio holdings from fully adjusting

to new information. This crucially ensures that our alpha model remains parsimonious. In

short, and as will be discussed in more details next, our approach is tantamount to finding the

factor-weighting scheme that maximises the portfolio IR for an appropriate level of portfolio

turnover.

3. The General Portfolio-Construction Framework

Although our methodology for factor weighting should not be viewed as a prescriptive

rule, it is nonetheless important that we define a framework from which we can derive an

insightful set of guidelines. This section briefly introduces this framework and presents the

main steps involved in the derivation of our weighting scheme. To avoid crowding the report

with overly technical material, we intentionally skipped discussions of some of the steps and

assumptions in our approach. The important derivations of our methodology are described in

further details in the Appendix of this paper.

It is common to argue that, every period, a long-short active manager attempts to

maximise his/her value-added function to generate mean-variance efficient portfolio holdings:

sConstraint

tt

Variance

ttt

aversion RiskReturn

tt

h

X'h :to subject

h'h'h maxt

0

21

(2)

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where th and t are N dimensional vectors of portfolio holdings and forecasts of

returns, respectively. X is an arbitrary number of N dimensional vectors that could be used to

ensure neutrality to relevant sources of systematic risk (say, size, market beta, sectors,

countries, etc). t is a N x N positive definite variance-covariance matrix of stock returns.

Finally, is the risk aversion parameter, a positive scalar.

In the presence of transaction costs, however, one needs to further penalise for the

turnover of the strategy. We do this by including a cost in the value-added utility function.

Following Sneddon (2008) and Garleanu and Pedersen (2009), we define the cost of trading

as a quadratic function of the change in portfolio holdings from one period to the next, as

shown below:

sConstraint

tt

Variance

ttt

aversion Riskpenalty Cost

Trade

ttt

Trade

tt

Return

tt

h

X'h .t.s

h'h )hh(')hh('h maxt

0

21

21 1-1-

(3)

where the turnover penalty, , is a positive scalar. In line with Garleanu and Pedersen

(2009), we also assume that the trading cost is a function of the variance-covariance matrix of

returns. We will see later that choosing a trading cost that is proportional to risk significantly

improves the tractability of our results. This choice is also natural since the risk borne by the

liquidity provider, who takes the opposite side of the trade and asks to be rewarded for doing

so, is a function of h'h t (see Garleanu, Pedersen and Poteshman, 2009). In this

framework, the turnover penalty can be understood as follows. Trading h shares moves the

alpha by ht

21 , resulting in a total trading cost of h times the change in alpha.

Equation (3) sets the general rule that an active manager is expected to follow, each

period, when selecting his/her portfolio holdings. For ease of interpretation we impose that X

in (3) includes vectors of stock betas and ones, so that the strategy is market neutral and self-

financing. We also take X to be constant through time. In other words, we assume that, over

the test period, the relevant factors, which we want to control for, do not change significantly.

With these conditions and assumptions (market neutrality and constant X) in place we

then take t to be a diagonal matrix of stock-specific variances. For simplicity, we further

8

assume that although specific risk varies across securities, it is relatively stable over the test

period for each individual asset.

The last assumption regarding equation (3) concerns the turnover penalty ( ) and the risk

aversion ( ), which are also taken to be time invariant. We show next how these parameters

can be chosen to target a specific tracking error and achieve a desired level of turnover. This

also ensures that our assumptions regarding their stability through time make practical sense.

To conclude, our framework relies on a number of simplifying assumptions so that it is

ultimately an empirical question as to whether it will add value to an active manager�’s

investment process. We will show, however, that our assumptions for the stability of some of

the parameters in (3) should not prove overly constraining in practice. This is because for a

meaningful range of turnover penalties the impact of historical data should die out rapidly.

Moreover, it is our belief that these simplifying assumptions by making the approach

transparent help bring about a number of valuable insights on how to counter-balance

considerations of return, risk and transaction costs when modelling alpha.

4. The Solution to the Portfolio-Construction Problem

The solution to equation (3) can be readily found using the method of Lagrange

multipliers. It is given by (see Appendix 1 for more details):

1tt

t h~h (4)

where 2i

ti

ti /�ˆ~ . In this equation �ˆ is directly related to alpha in (1), and can be

thought of as the residual of a weighted cross-sectional regression of alpha on X, where the

weights are the inverse of stock-specific volatilities ( i/1 ).

The formula in equation (4) can be interpreted as follows. If costs are zero ( = 0), the

portfolio holdings are proportional to the alphas, and independent of last month�’s holdings. In

practice, however, transaction costs matter so that the portfolio holdings become a weighted

average of last month�’s holdings and this month�’s alphas.

Recursive substitution of h into equation (4) yields:

9

11

01

*pti

*ppt

i

*p

pp

pti h~

)(h (5)

For clarity, we follow Sneddon (2008) and present in this paper a steady-state, or

equilibrium, solution where we assume that p* is such that the second term in equation (5) is

negligible. Again, we will show that for a meaningful range of turnover penalties the weight

on past data should decay relatively rapidly. As a result, assuming that the impact of the

original choice of portfolio holdings is negligible should not prove by and large totally

unrealistic. In turn, equation (5) can be written as:

pti

*p

pp

pti

~)(

h0

1 (6)

For ease of interpretation, we introduce )p( , a scaled version of 1p

p

)(, such that

10

*p

p)p( . It is trivial to show that )p( is an exponentially decaying function in p. The

expression for the portfolio holdings in (6) may be expressed in terms of )p( as follows

(see Appendix 2 for more details):

2

0

11

i

tipt

i

*p

p

ti

�ˆ~)p(h (7)

where *p

p

pti

ti �ˆ)p(�ˆ

0. Equation (7) offers some of the same important insights as

those highlighted in Sneddon (2008). The portfolio holdings are a combination of current and

past alphas, since the turnover penalty impedes the active manager from fully adjusting to

new information. This expression resembles the solution to the maximisation of the value

added when trading costs are ignored. The important difference, however, is that in (7) alpha

is a weighted average of current and past values.

Finally, should be chosen such that we target a desired level of tracking error. Given

our assumption of stability for this parameter, it is important that we adjust factors

appropriately. In particular, we will seek to ensure that correlations between different factors

10

remain minimal, and will normalise factors so that they have a unit risk-adjusted standard

deviation.

5. The Calibration of the Portfolio-Turnover Constraint

In this framework, the main difficulty becomes finding the values of )p( in (7), which

achieve a desired and/or appropriate level of turnover. To this end, we first define (one-way)

turnover as follows:

i

ti

ti hhT 1

21 (8)

This definition assumes that the portfolio drift plays a secondary role to the turnover

induced by changes in the model forecasts. We then approximate turnover using the approach

of Qian et al (2007). Using their methodology, we write the expression in (8) as follows:

)�ˆ,�ˆ(corrNT ttm 11 (9)

where is the average stock-specific risk, m is the target tracking error, and �ˆ is a

weighted average of current and past alphas with )p( as a weighting function. This

expression for portfolio turnover follows from the definition of portfolio holdings in (7),

which are themselves weighted averages of current and past alphas. Equation (9) makes

intuitive sense, portfolio turnover increases with the risk that the manager is willing to assume

and the autocorrelation of portfolio holdings from one period to the next. In Exhibit 3, we plot

the level of turnover, calculated using the approximation in (9), for different degrees of

autocorrelation in �ˆ . In this example, we fix N (the number of assets) to 500, and the average

stock-specific volatility to 30%. Results for two different levels of target tracking error are

reported (5% and 2%).


As can be expected, the amount of turnover that is required to rebalance a portfolio

depends upon how similar the alphas are from one period to the next. By changing the weight

that we put on the historical alphas, we can reach the desired turnover level.

11

In Exhibit 4, we show how the steepness of )p( , the weighting function of current and

past alphas, will adjust to achieve different levels of (one-way) portfolio turnover, when the

tracking error is set at 5% and the autocorrelation of alpha is equal to 0.8. When turnover is

barely constrained at 800%, the weight of )p( lies predominantly on the latest alpha scores.

However, as the targeted level of portfolio turnover decreases, the function puts a heavier

weight on older alphas at the expense of recent ones.


To simplify the analysis, instead of searching for the optimal set of values for )p( , we

define the parameter , with and being the turnover penalty and the risk-

aversion parameter introduced in equation (3). It is easy to show that )p( can be written in

terms of as follows:

p)()p( 1 (10)

In Exhibit 5, we show how the steepness of )p( decreases as increases. In turn, this

graph illustrates how affects the portfolio turnover. A value of close to zero means that

the turnover penalty is loose and results in a steep decay function. In contrast, a value close to

one implies a tight turnover constraint and leads to a flat decay function. Portfolio managers

would typically wish to construct portfolios for meaningful values of . Although has in

theory a value that falls between zero and one, imposing that is no greater than a specific

threshold would make sense in practice. For example, by imposing that is inferior to say

0.9, a manager ensures that more than 70% of the weight of the decay function falls on the

previous 12 alphas. For investors who use fundamental information from financial reports to

form their information forecasts, such horizons are sensible. Too high a reliance on past

alphas would indeed be meaningless in most applications.

Such a practically motivated selection of the turnover penalty would also help validate the

use of our assumptions for the stability of some of the parameters in our framework. For

instance, we have derived a solution where portfolio holdings are neutral to certain

characteristics, X, assumed constant. In practice some characteristics are going to vary over

time, which could be an issue since equation (4) points out that portfolio holdings in one

period depend on previous periods holdings. However, it is sensible to assume that the

departure from the neutrality assumption will be larger for more distant portfolio holdings, so

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that as long as is selected appropriately this problem should only have a limited impact on

the neutrality of current holdings. The same argument applies to the assumption of constant

specific risk and the original choice of portfolio holdings.


Finally, we note that different factor-weighting schemes will lead to different degrees of

autocorrelation in alpha. As a result, different decay functions will be appropriate depending

upon the factor-weighting scheme. We discuss the selection of the factor-weighting scheme in

the next section.

6. The Selection of the Factor-Weighting Scheme

6.1 The Active Manager�’s Objective Function

The previous steps illustrated how one can derive the desired portfolio holdings under

transaction costs, given an alpha model and therefore a set of factor weights. The next step

has to do with the choice of the factor-weighting scheme.

Like Sorenson et al (2004) and Qian et al (2007), we select the weighting scheme that

maximises the information ratio (IR) of our portfolio, taking strategy risk into account. The IR

of a portfolio is the standard metric by which the performance of an active manager is

assessed. Moreover, Qian and Hua (2004) show how the active risk of a portfolio can be

decomposed into the static target tracking error estimated by the (often) third-party risk

model, and the strategy risk that is related to the proprietary alpha model. Their analysis

isolates an important cause for the typical under-estimation of realised tracking error by risk

models; it also offers some remedies. In particular, they show that strategy risk has a direct

impact on the IR of a portfolio, so that choosing factor weights to maximise IR helps allocate

strategy risk efficiently across factors.

To keep with our analysis that takes transaction costs into account, we follow Qian et al

(2007) and attempt to find the weighting scheme for the factors of the alpha model that

maximises the IR of the active portfolio subject to an appropriate level of portfolio turnover,

namely:

13

x%Turnover and 1,0 ,'w .t.s

]r[stdev]r[E

maxp

p

,w

11

(11)

In equation (11), pr is the portfolio return. The value of pr at time t+1 is given by:

N

i

ti

ti

tp rhr

1

11 (12)

Using the expression for the portfolio holdings in (7) and the definition of the raw alpha

in (1), it is easy to show, based on the analyses of Sorensen et al (2004) and Qian et al (2007),

that the portfolio return becomes a weighted average of the realised information coefficients

(ICs) of present and past factor scores, namely (see Appendix 3 for more details):

m

j

tp,j

*p

pj

tp IC)p(wr

1

1

0

1 (13)

where 1tp,jIC is defined as the coefficient of correlation between the scores of factor j at

time t-p and the returns of the stocks in our universe at time t+1. Equation (13) demonstrates

the importance of taking transaction costs into account when modelling alpha. As transaction

costs increase, it is generally no longer beneficial for the portfolio to fully adjust to new

information so that returns, and in turn ICs, need to be forecast over the longer term.

6.2 The Working Assumptions and Final Representation

To operationalise the approach, we make a number of simplifying assumptions for the

alpha decay of the factors, their risk and the correlation of their scores. These simplifications

are not critical to the methodology and they could, in principle, be improved in order to model

performance more accurately. However, we find these assumptions to be particularly useful in

helping us to pinpoint the salient features of our method.

6.2.1 The Multi-period IC

In this section we follow Sneddon (2008) and explicitly model the alpha decay of the

factors. Within our framework, an important input that will be instrumental in understanding

14

how the manager allocates weight amongst his/her information sources is the multi-period IC.

This quantity is the appropriate measure of a factor predictive ability in our methodology.

First, we assume that the predicting power of factor j decays as follows:

tp,j

tp,jj

tp,j ICIC 1 (14)

where 10 j and 0][E tp,j . The expected value of the IC in (14) can therefore be

expressed as:

]IC[E]IC[E tj

pj

tp,j (15)

where tjIC is the one-period ahead (p = 0) IC for factor j at time t. In other words, we

assume that the predictive power of factor j follows an exponential decay in p. If j is close

to 0, the factor is said to have a high alpha decay, and its predictive power disappears quickly.

Using equation (15), we then define the multi-period IC (MIC) of factor j as the weighted

average of the expected ICs of past and current factor scores, with )p( as a weighting

function. Namely, we have:

*p

p

pjjj )p(]IC[EMIC

0 (16)

Recognising that the sum in (16) is a geometric series, one can further simplify the

expression to give (see Appendix 4 for more details):

1 factor discount 0

jjj ]IC[EMIC

11 (17)

The MIC of factor j can be understood as the product of the factor�’s expected one-period

ahead IC and a discount factor, which depends upon the turnover penalty (via ) and upon

the alpha decay of the factor ( j ). Within our framework this is the relevant quantity to

consider when assessing the predicting power of a factor, since it measures performance over

the relevant investment horizon.

15

In Exhibit 6, we simulate different values of the discount factor, for varying and j .

Since the predictive power of a factor is assumed to decay over time, higher turnover

constraints will lead to higher discount rates being applied and, therefore, lower expected

performance. Similarly, for every levels of turnover penalty, factors with higher alpha decay

(or equivalently lower j ) will receive a higher discount rate.


6.2.2 The Multi-period Variance of IC

Similar to the multi-period IC, in a multi-period setting, the active manager should

account for the risk of his/her portfolio due to its exposure not only to current but also past

factor scores. To compute the variance of the portfolio return, two more assumptions are

made. First, we take the variance of each factor IC to be constant for all lags p (i.e. 2jp,j )ICvar( ). Second, we assume that the covariances between the ICs of different

factors are negligible, so that 0)IC,ICcov( m,jl,k for all factor indices k and j, such that

jk , and all possible lags. We will see that the volatility of factor ICs is indeed remarkably

stable, even at relatively long lag lengths. To ensure that the assumption of uncorrelated ICs

makes practical sense, one could orthogonalise the factors. Several methodologies have been

advocated for this purpose. We note, again, that these simplifying assumptions are not

necessary to derive a solution to our approach, and one could for example take the correlation

between different factors into account. However, they serve to significantly improve the

tractability of our results as well as the understanding of the mechanics behind our approach.

Given these working assumptions, the only significant components for the risk of the

portfolio are the autocovariances of the ICs, expressed in terms of factor lags. The covariance

between the ICs of factor j at lags l and m is given by 2j

mljm,jl,j )IC,ICcov( .

The information risk of factor j is therefore summarised by its multi-period variance of IC

(MVIC), which can be expressed as follows (see Appendix 5 for more details):

1 factor discount 0

j

jjjMVIC

11

112 (18)

16

We find that the MVIC of factor j is the product of the variance of its IC ( 2j ) and a

discount factor, which depends upon the turnover penalty (via ) and upon the alpha decay

of the factor ( j ).

In Exhibit 7, we simulate different values of the discount factor of MVIC, for varying

and j . Since the autocorrelation of IC decreases with the lag-length, higher turnover

constraints will lead to higher discount rates, and in turn higher diversification benefits.

Similarly, for every level of turnover penalty, factors with steeper alpha decay (lower j )

will experience larger reductions in strategy risk.


6.2.3 The Correlation Matrix of Factor Scores

We use a similar set of assumptions to derive the correlation matrix of factor scores. The

IC decay of a factor typically mirrors the autocorrelation of its scores, so that similar values

for these two parameters could potentially be used. We denote the autocorrelation of the

scores of factor j by j , and note that )�ˆ,�ˆ(corr tt 1 , in the turnover approximation in (9),

can be simplified as follows (see Appendix 6 for more details):

m

j j

jj

m

j j

jj

tt

w

w)�ˆ,�ˆ(corr

1

2

1

2

1

111

(19)

6.3 Assembling the Portfolio Manager�’s Objective Function

Using equations (17) and (18), it is easy to show from equation (13) that the portfolio

return and its variance can be written as follows:

m

jjjp MICw]r[E

1 (20)

and

17

m

jjjp MVICw]rvar[

1

2 (21)

so that the maximisation of the active manager�’s objective function in (11) can be written as:

z%)�ˆ,�ˆ(corr and 10 ,'w .t.s

MVICw

MICwmax

tt

m

jjj

m

jjj

,w

1

1

2

1

11

(22)

The factor-weighting scheme will be chosen to maximise the IR of the portfolio for a

given level of alpha autocorrelation, which will match a desired level of turnover. The

relation between alpha autocorrelation and turnover will be based on the approximation in (9).

The optimisation problem can be easily solved using numerical methods, and we argue

that the resulting set of factor weights forms an appropriate basis to model alpha. Equation

(22) offers some significant insights since it formalises the relation between different building

blocks of the active management process. In particular, instead of considering the alpha-

modelling process and the portfolio-construction process separately, equation (22) points out

the importance of considering these tasks jointly. For instance, considering strategy risk, via

MVIC, could alleviate issues related to the mismatching between a proprietary alpha model

and a third-party risk model. Moreover, in the presence of trading costs, portfolio holdings

will only partially adjust as new information arrives. Maximising the active manager�’s

objective function becomes a multi-period problem where one needs to consider factor

performance over several stages.

Finally, it is easy to see from equation (22) how factor characteristics, such as a

managers�’ forecasts of future performance, will affect factor weights. If a manager�’s view on

a particular style improves, or if the risk of that style is perceived to be lower, the weight on

the factor should increase. The average trading cost of a fund should also guide the selection

of the weighting scheme, via the choice of an appropriate constraint on portfolio turnover. For

instance, a fund in infancy may find it beneficial to be exposed to high turnover signals. As

assets under management increase, however, factors with slower decay will be given more

weight. Similarly, given the set of factors at the manager�’s disposal, different levels of

18

turnover should be prescribed. A manager with very effective short-term signals may be better

off running a high turnover strategy, despite its higher trading cost.

7. Factor-Weighting in Practice

7.1 The Salient Factor Characteristics

The previous section presented a number of important quantities that an active manager

might want to take into account when deciding on his/her factor-weighting scheme. These

include the ICs of the factors as well as their variances, and autocovariances. Because of its

effect on portfolio turnover, we have seen that it is also important to consider the stability of

the factor scores over time. We note, however, that the economic significance of the impact of

these characteristics on the choice of the preferred weighting scheme will ultimately depend

upon the degree by which they differ across factors. For this reason, we present in this section

values for these quantities across a set of typical factors used in alpha modelling. We also

investigate the extent by which our assumptions for their dynamics are validated by the data.

The analysis of the salient features of a typical set of factors is conducted for the

constituents of the FTSE All World Developed Europe Index over a 21-year period,

extending from 1990 to 2010. The five factors that we study are namely:

Value: a composite of three factors (cash flow to price, forward earnings to price, and

sales to price) standardised within sectors;

Long-term momentum: a 12-month moving average of past returns, lagged by one

month;

Short-term momentum (reversal): the performance of a stock over the past month;

Analyst revisions: the monthly difference in the number of upward and downward

revisions in analyst forecasts, scaled by the total of number of revisions each month;

and

Profitability: the sector standardised return on equity of the firm.

Raw values are adjusted to ensure that the correlations between the ICs and the scores of

the different factors remain low, in line with our assumptions.

In Exhibit 8, we investigate the predictive power of our signals up to 16 months ahead.

The performance of the signals is measured in terms of their ICs. The IC of factor j at time

19

t+p is equal to the coefficient of correlation between the scores of the factor at time t and the

returns of the stocks in our universe at month t+p. In Exhibit 9, we plot the variance of these

ICs. Exhibit 10 shows the autocorrelation functions (expressed in terms of p) of factor scores.

From these graphs, it is immediately obvious that factors display markedly different

characteristics of alpha decay, IC volatility and autocorrelation. Given the importance of these

characteristics in determining the adequacy of a particular factor-weighting scheme, their

heterogeneity across factors clearly suggests that they should be taken into account.

For instance, the Momentum factor tends to display high predictive power and low alpha

decay, but its IC volatility is well above that of the average factor. Value has only a modest

predictive power, but its alpha decay and its IC volatility are also low. We find that Analyst

Revisions has a low IC volatility and a relatively high predictive power, but it decays

relatively quickly. The Reversal factor tends to have a high predictive power and an average

level of IC volatility. The IC decay of the factor is, however, extremely large. Finally, the

predictive power of Profitability is relatively low, but it also decays relatively slowly.

[EXHIBITS 8-10 ABOUT HERE]

Moreover, it is interesting to note that IC volatility is remarkably stable, even after

considering relatively long forecasting horizons. In contrast, ICs appear to be significantly

less persistent. In Exhibit 10, we can see how the autocorrelation of factor scores decreases as

the lag length is increased. Not surprisingly, this reduction in autocorrelation is related to the

alpha decay in Exhibit 8. All in all, this empirical analysis of the factor characteristics seems

to validate several of our previous assumptions. In particular, after considering Exhibit 9, it

does not seem totally unrealistic to assume a constant IC volatility at all lag lengths. This

result is hardly surprising since the volatility of stock returns tends to be relatively persistent,

and this persistence carry through at the signal level. In fact, a similar observation is made in

Qian et al (2007) using a universe of US securities. Moreover, visual inspection of Exhibits 8

and 10 suggests that modelling alpha and the autocorrelation function of factor scores as

exponentially decaying functions should broadly capture the salient features of the data.

7.2 A Simulated Experiment of Factor Weighting

In this section we illustrate the derivation of the factor weights in our framework. To

avoid having some estimation errors blurring our results, this analysis is based on a controlled

20

experiment with perfect foresight. In this simulation exercise, the number of assets is set at

500, the target tracking error is equal to 5%, and stock-specific volatility is assumed to be

constant at 30%. Moreover, the values for the relevant traits of each one of our European

factors are presented in Exhibit 11. These are the one-month ahead ICs of the factors, their

variances, and the autocorrelations of ICs and factor scores. We showed in earlier sections

how these characteristics enter the construction of three critical components of the active

manager�’s objective function, namely: the MICs of the factors, their MVICs and the

autocorrelation of the alpha.


Our simulation exercise proceeds as follows, for different levels of targeted portfolio

turnover, we find the factor-weighting scheme that maximises the objective function of the

active manager in (22). Moreover, to gauge the added value of our approach, we compare its

performance against two alternative weighting schemes. The first competing scheme

maximises the objective function in (22) with no consideration of turnover, namely we have:

m

jjj

m

jjj

ww

]IC[Ewmax

1

22

1 (23)

where ]IC[E j and 2j are the one-month ahead expected IC of factor j and its variance,

respectively. The resulting weighting scheme is that presented in Sorensen et al (2004), when

factors are orthogonal. The authors show that the solution to this problem is given by:

2jjj ]IC[Ew (24)

This weighting scheme is a function of the one-month ahead performance of the factors as

well as their risk. However, although it takes strategy risk into account, this methodology fails

to control for the impact of portfolio turnover. We refer to this scheme as Max IR to highlight

the fact that the objective is to maximise an IR without considerations of transaction costs.

The second competing scheme imposes an additional simplification to the selection of the

factor weights. Here, the active manager diversifies his/her bets across factors by investing in

each one of them according to his/her expectations for their future performance. The scheme

21

therefore fails to account for both transaction costs and strategy risk. The selection rule can be

written as follows:

]IC[Ew jj (25)

We refer to this weighting scheme as Max IC since only considerations of future

performance guide its construction.

In Exhibit 12, we show the factor-weighting schemes obtained with each methodology. It

is striking to see how the factor weights, derived with our approach (Max IR with Turnover),

vary according to the level of targeted portfolio turnover. These weighting schemes are also

markedly different from the ones obtained with Max IC and Max IR. For instance, we find

that Max IC puts large weights onto Momentum, Reversal and Analyst Revisions, those

factors with the highest one-month ahead IC. As strategy risk is taken into account, with Max

IR, the weight on Momentum is significantly reduced. This factor has indeed the largest

strategy risk out of all our European factors. Finally, the imposition of increasing degrees of

turnover penalty leads, with our approach, to a heavier weight being assigned on Value, a

factor with low alpha decay, at the expense of Reversal and Analyst Revisions, the two

factors with the steepest alpha decay.


In Exhibit 13, we present the expected performance of each scheme at different levels of

portfolio turnover. These results must be viewed with some caution because they are based on

an assumption of perfect foresight. In practice, however, one will face both model and

estimation error. Nevertheless, the analysis offers some interesting insights. In particular, it

highlights the importance of taking strategy risk into account in the factor-weighting scheme.

Although, Max IC generates the highest expected portfolio performance when turnover is

barely constrained, the risk of the strategy is more than twice the target tracking error. In turn,

the return-to-risk ratio of this scheme is consistently lower than those obtained with Max IR

and our proposed methodology. These results help illustrate the point that maximising the

predictive power of a model is different from maximising a portfolio IR. With Max IC a

manager leaves strategy risk uncontrolled so that he/she will have to resort to a higher level of

risk aversion to achieve his/her tracking error target. In turn, this will lead to smaller active

holdings and a weaker active performance.

22


Finally, and at the risk of stating the obvious, we also show that there is little point to

control for transaction costs in the factor-weighting scheme, if running a high level of

portfolio turnover is appropriate. For example, when turnover is barely constrained at 800%,

we find that taking strategy risk into account is sufficient to bring about most of the

improvements to the naïve Max IC scheme. However, when transaction costs matter and

turnover is constrained at 100%, we find that the IR based on our methodology is 34% higher

than that of the Max IR scheme, and 73% higher than that of the Max IC scheme.

8. Conclusion

The active management of a portfolio is a complex process where several inputs need to

be considered, often in a dynamic fashion. For instance, when deciding which stocks to buy

or sell, an active manager will typically have to balance considerations of risk and transaction

costs with views on future performance.

In this context, building upon some recent work by Sneddon (2008) and Qian et al (2007),

we have shown that an integrated approach to alpha modelling, which takes into account

strategy risk and alpha decay, can be particularly beneficial. Our methodology to combine

factors recognises the interdependencies that exist between the different building blocks of

the portfolio-construction process. It rests on a number of simplifying assumptions, which

help bring about insights that should prove valuable for systematic and fundamental managers

alike. For instance, as the turnover penalty increases, our approach to combine factors

allocates more weight to those signals with lower alpha decay. Factors with higher return-to-

risk ratios are also given a higher prominence since they help allocate strategy risk more

efficiently.

To conclude, although we do not address some of the more intricate constraints faced by

active managers, we believe, given the state of relative infancy of this line of research, that

the methodology to combine factors that we have introduced in this report constitutes an

appropriate basis for a more efficient alpha modelling.

23

Appendices

Appendix 1: The Solution to the Portfolio-Construction Problem

To solve the value-added maximisation in (3) we use the method of Lagrange multipliers,

where the Lagrangian function is:

211

2 21

21 X'h )hh()'hh(h'h'h),h(L tttttttttt (A1)

It is trivial to show that the solution to this problem satisfies the second order conditions for a

maximum, so that we concentrate below on the first order conditions only. The first order

conditions for L are as follows:

021 X)hh(h

hL tttt

t (A2)

02

th'XL (A3)

Solving for th in (A2) we get:

)Xh()(h ttt2

111 (A4)

Substituting th in (A3) we have:

02111 )Xh('X)( tt (A5)

Solving for 2 we obtain:

)h('X)X'X( tt 11112 (A6)

Finally, substituting 2 in (A4) we find the following solution for the portfolio holdings:

)h()'X)X'X(X()(h ttt 111111 (A7)

In this equation )'X)X'X(X( 111 is an idempotent matrix and

tt )'X)X'X(X(�ˆ 111 can be interpreted as the residual of a cross-sectional

regression of t on X, weighted by the inverse of stock specific volatility ( i/1 ).

In equation (A7), given our assumption of constant X, we also have:

0111111111 tttt hh'X)X'X(Xhh)'X)X'X(X(

As a result, the expression for the portfolio holding of stock i can be simplified as follows:

24

12 tii

tit

ih/�ˆ

h (A8)

For clarity we further introduce 2i

ti

ti /�ˆ~ , and write:

1ttt h~

h (A9)

Appendix 2: Simplification of the Solution to the Portfolio-Construction Problem

The solution to the value-added maximisation in (3) is given in (6) by:

pti

*p

pp

pti

~)(

h0

1 (A10)

We define 1p

p

)()p(f , and note that the sum of )p(f is a geometric series with

common ratio :

p*p

p

*p

p)p(f

00

1 (A11)

Equivalently:

1

0

11*p*p

p)p(f (A12)

So that when )p(f * 1 is negligible we have:

10

*p

p)p(f (A13)

For ease of interpretation, we introduce )p( , a scaled version of )p(f , such that

10

*p

p)p( . The expression for the portfolio holdings in (A1) may be written in terms of

)p( as follows:

pti

*p

p

ti

~)p(h0

1 (A14)

Appendix 3: The Impact of Factor Performance on Portfolio Return

25

The derivations in this section build on several insights from Sorensen et al (2004) and Qian

et al (2007).

The return of the portfolio at time t+1 is given by: N

i

ti

ti

tp rhr

1

11 (A15)

Using equation (A14) to replace h, we get: N

i

ti

pti

*p

p

tp r~)p(r

1

1

0

1 1 (A16)

Recall that the alpha in (A16) is a transformed version of the original raw alpha in (1) that is

orthogonal to X in (3), and scaled by stock-specific variance. Therefore, the alpha in equation

(A16) is a weighted average of m factors, as follows: m

j

ti,jj

ti fw~

1 (A17)

where the factors jf are also transformed versions of the raw factors jF in (1). The self-

financing and market-neutrality conditions in X ensure that jf has a mean value of zero and

is orthogonal to the market beta of the securities. Instead of scaling the raw factors jF so that

jf has unit standard deviation, we will see that it is convenient to standardise jF so that

1)f(stdev iti,j .

Replacing alpha in (A16) by expression (A17), we have: m

j

N

i

ti

pti,jj

*p

p

tp rfw)p(r

1 1

1

0

1 1 (A18)

Given that the factors, jf , are orthogonal to the market beta of the securities, the

performance of the portfolio can be equally expressed in terms of stock-specific return ( r~ ).

Moreover, the self-financing condition implies that a convenient, if somewhat arbitrary,

normalisation for r~ can be used, namely: 01

iti

tir~ . As a result, we have that the portfolio

return may be written as:

m

jr~

i

ti

ipt

i,jj

*p

p

tp N)

r~,f(corrw)p(r

1

1

0

1 1 (A19)

In (A19), r~ is the cross-sectional volatility of risk-adjusted stock-specific return

(i

ir~

r~stdev ), which we assume constant over the test period. This assumption is not

unrealistic since we have standardised the stock-specific return by specific risk. Moreover, as

pointed out in the text, for a meaningful range of portfolio turnover penalties the impact of

26

historical data should die out relatively rapidly. Therefore, in practice, the assumption of

constant volatility of stock returns should not prove overly constraining.

Furthermore, we define )r~

,f(corrICi

ti

ipt

i,jt

p,j

11 , where 1t

p,jIC is the information

coefficient of factor j, lagged by p, at time t+1. These ICs are a multi-period extension of the

risk-adjusted ICs of Sorensen et al (2004). These risk-adjusted ICs strip out un-wanted

systematic risk exposures and accommodate for stock-specific risk.

Finally, we introduce N

c r~ and write:

m

j

tp,j

*p

pj

tp IC)p(wcr

1

1

0

1 (A20)

Appendix 4: The Multi-Period IC

The multi-period IC (MIC) of factor j is defined in (15) as follows: *p

p

pjjj )p(]IC[EMIC

0 (A21)

We recall from Appendix 2 that: p

RA

)p( 1 (A22)

So that: *p

p

p

R~j

A~j

*p

p

pjjj RA]IC[E)p(]IC[EMIC

00 (A23)

Given that the above is a geometric series, we have:

RA]IC[E

)R~(A~

)R~(R~A~A~MIC

jj

*p

j 111

1 (A24)

We define , with and being the turnover penalty and the risk-aversion

parameter of equation (3), and write:

jjj ]IC[EMIC

11 (A25)

Finally, using (A20), the expected return of the portfolio can be written as follows: m

jjjp MICwc]r[E

1 (A26)

27

Appendix 5: The Multi-Period Variance of IC

To compute the variance of the portfolio return, we assume that the variance of each factor IC

is constant for all lags p (i.e. 2jp,j )ICvar( ); and that the covariances between the ICs of

different factors are negligible, so that 0)IC,ICcov( m,jl,k for all factor indices k and j, such

that jk , and all possible lags.

Given these working assumptions, the only significant components for the risk of the

portfolio are the autocovariances of the ICs, expressed in terms of factor lags. This

information for factor j can be summarised as follows:

2222

222

2222

jjjjj

jjjjj

jjjjj

jV (A27)

Where l,nV j is the autocovariance of jIC with lag ln . We define the multi-period

variance of IC (MVIC) for factor j as:

)()()(

)()()(

MVICjjjjj

jjjjj

jjjjj'

j 210

210

2222

222

2222

(A28)

Using equation (A20) and the definition of in Appendix 4, we can write the variance of the

portfolio return as follows:

22

2

21

2222

1

11

11

1jj

jj

jj'

F

jjjp wc]rvar[ (A29)

We look at the different diagonal matrix components related to the power of j to calculate

this sum.

There is one diagonal with ones:

2642

1111 ...: (A30)

There are two identical diagonals with powers > 0:

253

1

22 j

jj ...)(: (A31)

28

2

2264222

1

22 j

jj ...)(: (A32)

2

333

1

2 jj : (A33)

The right-hand side elements of equations (A31) to (A33) constitute themselves a geometric

series in j . Taking all the sums together we can write:

...jjj2

33

2

22

22 12

12

12

11

...jjjj 3322

221

12

11

)()()()( j

j

j

jj

j

j

11

1

11

21

11

12

11

2222

Therefore, the expression for the portfolio variance becomes:

)()()(wc]rvar[

j

jm

jjjp

11

11

21

2222 (A34)

or

j

jj

m

jjp wc]rvar[

1

111 2

1

22 (A35)

One can use a similar approach to compute )�ˆ,�ˆ(corr tt 1 in (19).

29

References

Brandt M.W., Santa-Clara P., and Valkanov R. (2009) �‘Parametric Portfolio Policies:

Exploiting Characteristics in the Cross-Section of Equity Returns�’, The Review of Financial

Studies, Vol. 22, p. 3411-3447.

Garleanu N.G. and Pedersen L.H. (2009) �‘Dynamic Trading with Predictable Returns and

Transaction Costs�’, NBER Working Papers 15205, National Bureau of Economic Research,

Inc.

Garleanu N.G., Pedersen L.H., and Poteshman A.M. (2009) �‘Demand-Based Option Pricing�’,

The Review of Financial Studies, Vol. 22, p. 4259-4299.

Grinold R. (2010) �‘Signal Weighting�’, The Journal of Portfolio Management, Vol. 36, p. 24-

34.

Lee J.H. and Stefek D. (2008) �‘Do Risk Factors Eat Alphas?�’, The Journal of Portfolio

Management, Vol. 34, p. 12-25.

Qian E. and Hua R. (2004) �‘Active Risk and Information Ratio�’, Journal of Investment

Management, Vol. 2, p. 20-34.

Qian E., Sorensen E.H., and Hua R. (2007) �‘Information Horizon, Portfolio Turnover, and

Optimal Alpha Models�’, The Journal of Portfolio Management, Vol. 34, p. 27-40.

Sentana E. (2005) �‘Least Squares Predictions and Mean-Variance Analysis�’, Journal of

Financial Econometrics, Vol. 3, p. 56-78.

Sorenson E., Qian E., Hua R., and Schoen R. (2004) �‘Multiple Alpha Sources and Active

Management�’, The Journal of Portfolio Management, Vol. 31, p. 39-45.

Sneddon L. (2008) �‘The Tortoise and the Hare: Portfolio Dynamics for Active Managers�’,

Journal of Investing, Vol. 17, p. 106-111.

30

Exhibit 1: The building blocks of the portfolio-construction process

Exhibit 2: An integrated alpha-modelling approach

Return Forecast(Alpha)

Constraints/Penalties(incl. Turnover)Portfolio

Construction

Portfolio Holdings

Risk Estimates(Covariance Matrix of Return)

Portfolio Construction

Portfolio Holdings

Risk Estimates Constraints/Penalties(incl. Turnover)

Alpha Modelling

31

Exhibit 3: Turnover and autocorrelation in alpha Exhibit 3 shows how the expected level of portfolio turnover (one-way) varies with the autocorrelation of the alpha, and the target tracking error (TE) of the strategy. In these examples, the number of assets is equal to 500, and the average stock-specific volatility is set at 30%.

0%

100%

200%

300%

400%

500%

600%

0.950 0.953 0.956 0.959 0.962 0.965 0.968 0.971 0.974 0.977 0.980 0.983 0.986 0.989 0.992 0.995 0.998

Alpha autocorrelation

Turn

over

(on

e-w

ay)

TE (2%) TE (5%)

Exhibit 4: Decay function and portfolio turnover Exhibit 4 shows how the weighting of current and past alphas varies as a function of the targeted level of portfolio turnover. The experiment assumes a tracking error of 5%, and an alpha autocorrelation of 0.8. Moreover, the number of assets is fixed to 500, and the average stock-specific volatility is set at 30%.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Lag 0 Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 7 Lag 8 Lag 9

Lagged alphas

Wei

ghts

800 600 400 200 100

32

Exhibit 5: Calibrating the decay function Exhibit 5 illustrates how the weighting function of current and past alphas varies as a function of the turnover penalty (eta) and the risk aversion parameter (lambda). The information in these two variables is captured by the parameter kappa.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lag 0 Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 7 Lag 8 Lag 9

Lagged alphas

Wei

ghts

kappa = 0.1 kappa = 0.3 kappa = 0.5 kappa = 0.7 kappa = 0.9

Exhibit 6: Simulated values for the discount factor of the multi-period IC Exhibit 6 reports simulated values for the discount factor of the multi-period IC (MIC). These are computed for different levels of alpha decay (rho) and turnover penalty (kappa).

0

0.2

0.4

0.6

0.8

1

1.2

rho = 0.95 rho = 0.5 rho = 0.05

Disc

ount

fact

or o

f MIC

Kappa = 0.99 Kappa = 0.60 Kappa = 0.15 Kappa = 0.00

33

Exhibit 7: Simulated values for the discount factor of the multi-period variance of IC

Exhibit 7 reports simulated values for the discount factor of the multi-period variance of IC (MVIC). These are computed for different levels of alpha decay (rho) and turnover penalty (kappa).

0

0.2

0.4

0.6

0.8

1

1.2

rho = 0.95 rho = 0.5 rho = 0.05

Disc

ount

fact

or o

f MVI

C

Kappa = 0.99 Kappa = 0.60 Kappa = 0.15 Kappa = 0.00

Exhibit 8: Alpha decay

Exhibit 8 shows the ICs of the different European factors. These ICs are computed up to 16 months ahead.

Exhibit 9: Variance of ICs Exhibit 9 reports the variance of the ICs of the different European factors. These ICs are computed up to 16 months ahead.

-2.0%

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

t+1 t+3 t+5 t+7 t+9 t+11 t +13 t+15

Value Momentum Reversal Analyst Rev Profit

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

1.6%

t+1 t+3 t+5 t+7 t+9 t+11 t+13 t+15


Exhibit 10: Autocorrelation of factor scores Exhibit 10 reports the autocorrelation function of the scores of the different European factors.

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

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


34

Exhibit 11: The salient factor characteristics

Exhibit 11 reports the one-month ahead ICs of the factors in our European model, their variances, and the autocorrelations of ICs and factor scores (rho IC and rho Scores, respectively).

IC Var IC rho IC rho ScoresValue 1.79% 0.36% 0.94 0.98Momentum 3.73% 1.35% 0.92 0.87Reversal 3.35% 0.48% 0.05 0.22Analyst Revision 2.97% 0.34% 0.57 0.79Profitability 0.42% 0.55% 0.95 0.95

Exhibit 12: Factor weighting schemes Exhibit 12 reports the factor weights that are obtained with three different methodologies. Max IC only takes future performance into account, while Max IR also controls for strategy risk. The third methodology further incorporates considerations of turnover and alpha decay. For this approach, the target portfolio turnover is varied from 100% to 800% (one-way).

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

MaxIC

MaxIR

800 750 700 650 600 550 500 450 400 350 300 250 200 150 100


Max IR with Turnover

Exhibit 13: Simulated portfolio performance Exhibit 13 presents the simulated performance of three portfolios constructed with different factor-weighting methodologies. Max IC only takes future performance into account, while Max IR also controls for strategy risk. The third methodology further incorporates considerations of turnover and alpha decay. The target portfolio turnover is varied from 100% to 800% (one-way).

800% 500% 200% 100%Expected return Max IR with Turnover 17.73% 15.22% 11.49% 8.41%

Max IR 17.53% 14.39% 9.08% 5.65%Max IC 20.97% 18.35% 12.88% 8.26%

Expected st dev Max IR with Turnover 6.83% 6.77% 7.02% 6.68%Max IR 6.88% 6.68% 6.41% 6.02%Max IC 10.17% 10.58% 11.31% 11.24%

Expected IR Max IR with Turnover 2.6 2.25 1.64 1.26Max IR 2.55 2.15 1.42 0.94Max IC 2.06 1.74 1.14 0.73

integrated alpha modelling (2011)

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