Signal Weighting by Richard Grinold

24 SIGNAL WEIGHTING SUMMER 2010

Active strategies face the challengeof combining information from dif-ferent sources in order to maximizethat information’s after-cost effec-

tiveness.1 This challenge is evident for investorswho follow a systematic, structured investmentprocess. It is also an issue,although often unrec-ognized, for more traditional managers whomust balance top-down views and the viewsof in-house and sell-side analysts. In any case,a decision is made either through default,analysis, or whimsy. We favor analysis and tothat end this article presents a simple and struc-tured approach to the task. This task is com-monly called signal weighting, although riskbudgeting seems to be a better description.The approach is based on standard methodsof portfolio analysis and their extension todynamic portfolio management.

We can consider an active investmentproduct on three levels: strategic, tactical, andoperational.At the strategic level are the targetclients, fee structure, level of risk, amount ofgearing, level of turnover, marketing channel,decision-making structure, and so forth.At theoperational level, we run the product: main-taining and replenishing data, executing trades,tracking positions and performance, informingclients, and so on.At the tactical level,we allo-cate resources to improve the product and itsoperations, and do some fine-tuning on suchparameters as risk level and turnover level.Thesignal-weighting decision belongs at the tactical

level.Every strategy has an internal risk budgetthat shows how the aggregate risk is being allo-cated among the various sources/themes thatwe anticipate can add value. We should peri-odically, say, every six months, re-examine andpossibly change that allocation.A similar recon-sideration of the internal risk budget shouldtake place if there is a significant change in thesignal mix (e.g., adding a new theme) or if thereis a major change in the market environment.

The method we present in this articlegives substance to the signal-weighting processby isolating the important aspects of the deci-sion and providing a structure that links thosefeatures to a decision.The procedure we out-line mimics the investment process. Eachsignal is viewed as a sub, or source, portfoliothat is a potential place to allocate risk. Weallocate risk to the source portfolios in amanner that makes the most sense at theaggregate level. In this effort we are trying tobe comprehensive enough to depict the essen-tials and simple enough to easily capture thelink between inputs and outputs. Thus, wecompromise between the desire to be rea-sonable and robust, on the one hand, and toprovide a clear box rather than a black box,on the other.

To preserve transparency we should erron the side of simplicity rather than on exces-sive elaboration.There is a temptation to moreand more closely mimic the actual investmentprocess (e.g., backtesting). Our unadorned

Signal WeightingRICHARD GRINOLD

RICHARD GRINOLD

was a global director ofresearch at Barclays GlobalInvestor (1994–2009),director of research atBARRA, (1979–1993), anda professor at the Universityof California in Berkeley,CA (1969–1989)[email protected]

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approach is to assume the portfolio is run with the straight-forward and easy-to-analyze method A,although we realizethat we actually use the complex and impossible-to-ana-lyze method B.We have to trade off the benefits of get-ting sensible, comprehensible answers against the costs ofdeparting from the operational reality.Any effect we wantto consider has to argue for a place at the table.The watch-word is simplicity and the burden of proof is against makingit complicated.This focus on simplicity also argues againsta profusion of sources. In our view, the number of themesshould be more than two and certainly less than, say, seven.If you believe there are a larger number of significant alphadrivers of a strategy you should 1) look up “significant”and 2) aggregate the signals into broader themes.

One benefit of the proposed approach is that it isbased on a theory. There is a logical consistency that letsus track cause and effect. The assumptions are few andtransparent. It is not a sequence of ad hoc decisions andjury-rigged improvisations where the often hiddenassumptions are many and their implications are obscure.This clarity makes it easy to link cause and effect. In par-ticular, it facilitates the use of sensitivity analysis to studythe links between important inputs and outputs.

Signal weighting is a forward-looking exercise.Weare planning for the future not over-fitting the past. Pastperformance of signals can, of course, inform us abouteach signal’s presumed future strength,how fast the infor-mation dissipates, and how the signals relate to each other.However, we will usually be in a situation with old sig-nals that are part of the implemented strategy and new sig-nals that have only seen hypothetical implementation. Inaddition, we may feel strongly that some signals will losestrength because they are being arbitraged away.What wesee in a backtest is one sample from a nonstationary prob-ability distribution. By all means consider past perfor-mance, but add a large pinch of salt.

The analysis is designed for an unconstrained imple-mentation. There may be some idiosyncratic features ofa signal that will make it unsuitable for say a long onlyimplementation. These are difficult to predict, so theanalysis here should still provide a starting point of anystudy of the interactions of signals and constraints.

PREVIEW

This article has the following structure.We initiallyconsider a cost-free world with one signal, and then a

cost-free world with multiple signals.This exercise allowsus to introduce many of the terms and concepts we willuse in the more difficult setting with transaction costs.

The next step is to introduce transaction costs.Costsare difficult to handle in an analytic way.Our approach istraditional; we make an assumption.We assume that theportfolio manager acts as if he is following a simple rebal-ance rule. The rebalance rule has three parts:

• For each signal we construct a source portfolio thatcaptures that signal in an efficient (i.e., most signalper unit of risk) manner.

• We construct a model portfolio that is a mixture ofthe source portfolios.

• We stipulate a rebalance rule to fix the rate at whichthe portfolio manager tries to close the gap betweenthe portfolio the manager actually holds and themodel portfolio.

In Appendix A,we show that this type of rebalancerule is optimal under certain conditions.2 We feel that arule, that is optimal under one set of conditions, will bereasonable in a much wider group of situations. InAppendix B, we study the consequences of using such adecision rule.The signal-weighting model and results areillustrated using a three-signal example.

OTHERWORK

This article stems from the author’s previous workon dynamic portfolio analysis,Grinold [2006,2007].Themost directly related paper is Sneddon [2008]. Similartechniques to those described in Appendix A are used byGarleanu and Pedersen [2009].

ONE SIGNAL AND NOTRANSACTION COSTS

We start with a quick description of the frameworkwith only one signal and then expand the analysis to covermultiple signals. There are N assets whose return covari-ance is described by the N by N covariance matrix V.The basic material in the weighting exercise is an N ele-ment vector a, called a raw alpha. It is scaled to have, onaverage, an information ratio of one,

a′ ⋅ V–1 ⋅ a = 1 (1)

SUMMER 2010 THE JOURNAL OF PORTFOLIO MANAGEMENT 25

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To obtain a weight, we need an assessment of thestrength of the signal. Let IR be the information ratio ofthe signal, then

α = IR ⋅ a (2)

We are done.With one signal we simply weight theraw alpha to have the desired information ratio.

We can get to the same place by a longer route thatapproaches the problem from a portfolio perspective.Theportfolio route is worth the extra effort because it gen-eralizes in a natural way when there are multiple signalsand transaction costs.

We associate a source portfolio s with our signal,

(3)

The source portfolio is an efficient implementationof the signal, that is, among all positions x, it maximizesthe ratio

As noted in Equation (3), the source portfolio has anexcessive risk level of 100%. Suppose we choose a morerealistic risk level ω.This is akin to scaling the source port-folio. If ω is the risk level, then the portfolio is q = ω ⋅ s.This portfolio has an alpha of IR ⋅ ω and a variance of ω 2.If we have a risk penalty,λ, and we do the standard mean-variance optimization, we choose the risk level ω tomaximize,

(4)

The solution is . For example, if IR = 0.75,and we have a risk penalty of λ = 18.75, then ω = 4%.

The alpha resulting from this is

α = λ ⋅ V ⋅ q = λ ⋅ V ⋅ s ⋅ ω = {λ ⋅ ω} ⋅ a (5)

which gets us to the same place as the straightforwardapproach in Equation (2) since λ ⋅ ω = IR.

The reasons for the portfolio optimization approachused in Equation (5) are that

• it generalizes nicely when we have multiple signals,and

ω λ= IR

IR ⋅ − ⋅ω λ ω2

2

a x x V x⋅ ⋅ ⋅′

s V a s V s= ⋅ ⇒ = ⋅ ⋅ =−1 2 1ωS ′

• it makes things much simpler when we introducetransaction costs.

First,we will consider the multiple-signal case withno transaction costs.

J SIGNALS AND NO TRANSACTION COSTS

We have J signals (sub-strategies), each with a rawalpha aj j = 1,…, J. Each aj is a vector with elements foreach of the N assets. In addition, we start with a levelplaying field; each aj is standardized so that its informa-tion ratio is one,3

(6)

We can, as in Equation (3), associate a position witheach raw alpha.These positions have,on average,100% riskand will be the building blocks of our investment strategy,

(7)

Because we have multiple signals, we have to takeinto consideration any correlation between the informa-tion sources. We define that correlation in terms of theexpected return covariance between any pair of positions,4

(8)

For example, if i is a value signal and j is a momentumsignal, they will tend to be negatively correlated. Recallthat signal weighting is a forward-looking exercise; the cor-relations ρi,j are predictions.Nonetheless, a historical analysisof past correlations should be informative;5 often, a fairlystable relationship exists. In other cases, the relationshipappears to depend on the caprice of the market.As a prac-tical matter, if any of these correlations are very large (say,above 0.5), it might be wise to combine the signals. If acorrelation is large and negative, then either you have founda very good thing or, more likely, you have uncoveredsomething that is too good to be true.

We build a strategy by assigning risk levels, ωj, toeach of J sub-strategies. The vector representation is

. The result will be portfolio

(9)q s( ),

ω = ⋅=∑ j jj J

ω1

= ={ } ,ω j j J1ω

ρ i j i j, ≡ ′ ⋅ ⋅s V s

a V s s V a

s V s

j j j j

j j

= ⋅ = ⋅

′ ⋅ ⋅ =

−, 1

1

′ ⋅ ⋅ =−a V aj j1 1



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The information ratios of the J sub-strategies aregiven by the J element vector IR = {IRj}j=1, J. The J byJ correlation matrix with elements ρi,j is denoted as R.

With this notation the alpha expected by using theweights ω is

(10)

The variance of the portfolio,q(ω),with weights ω is

(11)

We choose ω to maximize,

(12)

The solution is obtained by solving the first-orderequations,6

or

(13)

The maximizing vector of weights is given by

(14)

We will refer to the optimal portfolio (a.k.a., theideal portfolio) as Q with positions

(15)

The information ratio and risk of the ideal port-folio Q are

(16)

Note that IRQ is independent of the risk penalty λ,and thus we can choose λ to get the desired level of activerisk,ωQ.

IIRR RR= ⋅ ⋅λ ω

IR IRQ Q Q Q≡ ′ ⋅ ⋅ = ⋅ ⋅ = ⋅−IIRR RR IIRR RR1 , ,ω λ ωω ω′

q s( ),

ω = ⋅=∑ j jj J

ω1

ω = ⋅ ⋅−1 1

λRR IIRR

IRi i j jj J

= ⋅ ⋅=∑λ ρ ω,

,1

α λ σ( ) ( )ω ω− ⋅2

2

σ ω ρ ω2

11

( ) ,,,

= ⋅ ⋅ = ⋅ ⋅==∑∑ i i j jj Ji J

′ RR ωωω

α ω( ),

ω ω= ⋅ = ′ ⋅=∑ IR j jj J1

IIRR

EXAMPLE

The following is an example with three sub-strategies. In anticipation of the next section, we haveclassified the signal by the rate of information turnover.The sources are SLOW, INTERMEDIATE, and FAST.The goal is to mix these three sources to get an ideal port-folio with 5% risk.

The data in Exhibit 1 represent our best estimates fora reasonably long planning horizon, say, a year or more.Referring to Equation (16), the net effect of these data is anaggregate information ratio of IRQ = 1.11. If we want anoverall risk of ωQ = 5.00%, we can set the risk penalty asλ = IRQ/ωQ = 1.11/0.05 = 22.17 and achieve the desiredresult. The final column is the optimal risk level derivedfrom Equation (14). We can interpret this as running theINTERMEDIATE sub-strategy at 3.78% risk, the SLOWsub-strategy at 1.96% risk, and the FAST sub-strategy at4.10% risk.

RISK BUDGETING

Signal weighting, like portfolio management, is anexercise in risk budgeting. We can express the result interms of the resulting risk budget. The variance of theideal portfolio Q is

or

(17)

If we allocate a fraction of therisk to alpha source j, we are in effect taking the twocovariance terms for sources j and k,ωj ⋅ ρj,k ⋅ ωk + ωk ⋅ρk,j ⋅ ωj, and splitting the baby, thus attributing ωj ⋅ ρj,k ⋅ωk to signal j and ωk ⋅ ρk,j ⋅ ωj to signal k.

{ } { }, ,

ωω

ωωρj

Q

k

Qk J j k⋅ ⋅=Σ 1

ii j

Qj Jj k

k

Q

.,

,11

=

⋅ ⋅

=∑

ωω

ρωω =

∑k J1,

i Q jj J

j k kk J

.,

,,

ω ω ρ ω2

1 1

= ⋅ ⋅= =∑ ∑


Note: The input data are the information ratio, IR, and the correlation.The optimal result is in the column labeled “Risk.”

E X H I B I T 1Three-Signal Example


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In our example,over 52% of the risk comes from theFAST signal, 34% from the INTERMEDIATE signal,and only 14% from the SLOW signal.

THE ALPHA PERSPECTIVE

We can view the result in terms of alphas as well asof portfolios. According to Equation (15), our idealportfolio is

The alphas that lead to this ideal are7

(18)

WEIGHTS

Equation (18) can be interpreted in terms of weights(i.e., numbers that sum to one), as follows:

(19)

The scale factor is to ensure that the resulting alphashave the correct information ratio, for example,

(20)

The weights in our examples are 38% for INTER-MEDIATE,20% for SLOW,and 42% for FAST.Note thatthese are similar,but not equal, to the risk budgeting num-bers mentioned previously.

The following three ways of presenting the resultsare illustrated in Exhibit 2:

• risk of each source• risk budget• weights that sum to one

Now we turn to the more challenging and realisticsituation where we consider transaction costs.

q s( ),

ω = ⋅=∑ j jj J

ω1

αQ j jj J

= ⋅ ⋅ = ⋅ ⋅=∑λ λ ωV q a ( )

,1

α αQ Q QIR′ ⋅ ⋅ = =−V 1 1 11.

i scale weight

ii s

Q j jj J

. ,

.

,

α = ⋅ ⋅

=

∑ a1

where

ccale

iii weight

jj J

j

j

ii J

= ⋅

=

=

=

∑

∑

λ ω

ωω

1

1

,

,

.

TRANSACTION COSTS

The previous analysis showed how to successfullyapproach the source/signal-weighting problem when thereare no transaction costs. But the presence of transactioncosts introduces three new obstacles. These obstacles are

• Transaction costs are paid either directly throughcommissions, spread, taxes, and so forth,or indirectlyby demanding liquidity and shelling out too muchfor purchases and getting too little for sales (a.k.a.,market impact).

• Transaction costs are intimidating. They keep youfrom fully exploiting your information,which cre-ates an opportunity loss.

• Transaction costs are levied on changes in positions,so the initial position is a crucial part of the analysis.

The last point implies a multi-period perspective isrequired. Our approach will be to look for strategies ordecision rules that can, in some sense, be consideredoptimal in a multi-period setting. To do this, we makeassumptions that allow us to abstract from the reality ofthe day-to-day portfolio management environment andstill capture something of its essence.This is, after all, boththe power and the Achilles’heel of any economic analysis.As Solow [1956] said,“All theory depends on assumptionsthat are not quite true.”

THE PORTFOLIO’S LAWOFMOTION

An active portfolio is an object in motion. Call itportfolio P. If we look at portfolio P periodically, say,every∆t years,8 then we see a sequence of positions.The changesin these positions, defined as

∆p(t) ≡ p(t) – p(t – ∆t) (21)

determine the cost.


E X H I B I T 2Three Views of the Solution to the Signal/Source-Weighting Problem with No Transaction Costs


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The challenge is to capture these changes in a usefulway. To do this we will specify a rule, called the law ofmotion, to show how portfolio P changes in response tochanges in the source portfolios. We will do this in twosteps, as follows:

1. Combine the source portfolios into a model port-folio M.

2. Show how the portfolio P attempts to track themodel portfolio.

THEMODEL PORTFOLIO

The model portfolio is a mixture of the source port-folios. Its holdings at time t are

(22)

The risk levels ωj are the same levels we used todefine the ideal portfolio; see Equations (13) and (14).The parameters ψj, where 0 < ψj < 1, are explained indetail in the next section.

THE TRACKING RULE

The changes in portfolio P are governed by a lineardecision rule of the form

or

(23)

The parameter delta, 0 < δ < 1, is discussed later.In Appendix A, we demonstrate that the rule describedin Equations (22) and (23) is optimal under special cir-cumstances. The leap of faith is to assume that the rule isat least “reasonable” under more general conditions.

The parameters δ,ψj mentioned in Equations (22)and (23) depend on the rate of change of the signals asrepresented by either the raw alphas aj(t) or the sourceportfolios sj(t) and on the change in the portfolio p(t)itself. We now turn our attention to measuring thosechanges and determining values for these parameters.

iitt t

t t t.( ) ( )

{ ( ) ( )}∆

∆=

−∆

⋅ − − ∆p

m p1 δ

m s( ) ( ),

t tj j jj J

≡ ⋅ ⋅=∑ ω ψ

1

i t t t t. ( ) ( ) ( ) ( )p p m= ⋅ − ∆ + − ⋅δ δ1

AMEASURE OF SIGNAL SPEED

We can gauge the rate of change in the signal jbetween times t and t – ∆t by measuring the correlationof the positions sj(t) and sj(t – ∆t) using the asset covariance,V(t), at time t. To do the calculation in a sensible way,weshould tailor the time interval, calling it ∆tj, on a signal-by-signal basis so that a reasonable amount of change takesplace resulting in a value for γj(t) in Equation (24) in therange of 0.8 to 0.975. This will avoid spurious results if∆t is either too long or too short. Even if we keep dailydata we might want to calculate the correlation over oneweek, two weeks, or one month so that we can see sig-nificant changes.

This past correlation is given by

(24)

We can eliminate the effects of the tailored timeinterval selection,∆tj, by calculating a rate of change,

(25)

The data will show how the signal has moved in thepast; for example, seasonality (e.g., quarterly or semi-annualrelease of earnings) or a trend (e.g., getting faster or slower)may be present. The challenge is to use this analysis tochoose a point estimate that we will call gj, that will deter-mine how fast, on average,we believe the signal will movein the future.One way to consider this question is in terms

g tt

te tj

j

j

g t t

jj j( )

ln( ( ))( )

( )= −∆

=− ⋅∆γγor�

γ jj j j

j j

tt t t t

t t t( )

( ) ( ) ( )

( ) ( ) (=

′ ⋅ ⋅ − ∆

′ ⋅ ⋅

s V s

s V s )) ( ) ( ) ( )⋅ ′ − ∆ ⋅ ⋅ − ∆s V sj j j jt t t t t


E X H I B I T 3Values of gj for Various Half-Lives


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of a half-life, that is, the time it will take for the correlation,as calculated by Equation (24), to drop to 0.5. Exhibit 3shows how gj depends on the half-life.

In our example,we will use nine months as the half-life for the SLOW signal,one month for the FAST signal,and six months for the INTERMEDIATE signal.

THE PACE OF PORTFOLIO CHANGE

The parameter delta in Equation (23) governs therate of change of portfolio P. We will express delta asδ = e–d⋅∆t, where d measures how fast we attempt to closethe gap between where we start,p(t – ∆t), and where wewould like to be,m(t).The choice of d,whether it is madeexplicitly or implicitly, is an attempt to balance two costs:

1. Transaction costs: A larger d (thus lower δ ) leads tomore trading and higher costs.

2. Opportunity loss:A smaller d (thus higher δ ) impliesa less efficient implementation; the fraction of port-folio P’s risk budget directed toward alpha declinesand the amount of uncompensated risk increases.

In Appendix A and in Grinold [2007], we showhow, under special circumstances, the optimal choice ofd is possible. In what follows, we first discuss the inter-pretation of d, which should isolate a reasonable range ofvalues, and then we consider procedures that will help usdiscover an implied value of d by examining the pastbehavior of the portfolio.

INTERPRETATION OF d

Portfolio P chases the model M. It is always behindthe curve. Indeed, 1/d is a measure of just how much P

lags M. Both P and M are based on information that hasflowed in over previous periods.The age-weighted expo-sure of P to this information is about 1/d years longerthan the age-weighted exposure of M to the same infor-mation. Slow trading leaves P under exposed to recentinformation and relatively over exposed to older staleinformation. Exhibit 4 shows lags of one to six monthsand the corresponding values for d.

ESTIMATION OF d; REVEALED PREFERENCE

It is possible to get a rough estimate of d by exam-ining the history of the portfolio and the source portfo-lios.As we indicate in Appendix B,we can use a regression,or portfolio optimization, approach to estimate parame-ters and , so that

(26)

and the unexplained variance, , is mini-mized. If things are going along relatively smoothly (i.e.,no radical changes of policy) and the time interval ∆t0 isselected in a reasonable manner,9 then estimates willtend to be positive and less than one. In such cases,we getan estimate of d as

(27)

The analysis will also produce implied risk levels,, and a measure of the fitting error,

.From these estimates of implied past policy in

addition to the intuition that might be gained fromExhibit 4 and future policy plans, we settle on values forthe parameters d and gj, j = 1,…, J. Then, for any rebal-ance interval ∆t, we can calculate the risk adjustments ψjpromised in Equation (22),10

(28)

Now let’s examine how this would work with theexample introduced earlier. The data in Exhibit 5 are

i e e

ii

d tj

g t

jj

j. ,

.

δ γ

ψ δδ γ

= =

≡ −− ⋅

− ⋅∆ − ⋅∆

11

{ˆ ( ) ˆ( )}/{ ( ) ( ) ( )}ε ε′( )t t t t t t⋅ ⋅ ′ ⋅ ⋅V p V p

ˆ ( )/( ˆ( ))β δj t t1−

ˆ( )ln( ˆ( ))

, ˆ( )ˆ( )d t

tt

t e d t t= −∆

= − ⋅∆δ δ0

0

ˆ( )δ t

ˆ ( ) ˆ( )ε ε′( )t t t⋅ ⋅V

p p s( ) ( ) ˆ( ) ( ) ˆ ( ) ˆ(,

t t t t t t tj jj J

= − ∆ ⋅ + ⋅ +=∑0

1

δ β ε ))

ˆ( )δ t ˆ ( )β j t


E X H I B I T 4Relationship between Trading Rate d and Lagbetween Average Age of Information in ModelPortfolioM and Tracking Portfolio P


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based on a value of d = 4 and a rebalance interval of oneweek,∆t = 1/52, thus δ = 0.926.

In Exhibit 5, the gj are based on the assumed half-lives. If HLj is the half-life in months, then gj = 12 ⋅ {ln(2)/HLj}.The γj are calculated using item (i ) of Equation(28) and the ψj are derived from item (ii) of Equation (28).The column “Risk Q” repeats the optimal risk levels,ωj,of the ideal portfolio previously reported in Exhibits 1and 2.The final column,“RiskM,”contains the risk levelsψj ⋅ ωj associated with the model portfolio, as in Equa-tion (22).The risk levels of all the signals fall, although thedecrease,evident in the “psi”column, is more dramatic forthe FAST signal.The risk11 of model portfolioM is 3.06%compared to the 5% risk level for the ideal portfolio Q.To raise the risk level of the model portfolio to, say, 5%,merely decreases the risk penalty lambda from 22.17 to{3.06/5} ⋅ 22.17 = 13.57 and repeats the calculation.Thiswill scale up the risk, but keep the risk budget allocationand weights the same.

THE ALPHA PERSPECTIVE REVISITED

The adjustment for transaction costs works for alphasas well as for portfolios. According to Equation (22), ourmodel portfolio is

The alphas that would lead to this portfolio are

(29)

WEIGHTS

Equation (29) can, as before,be interpreted in termsof weights (i.e., numbers that sum to one), as follows:

α λ λ ψ ωM j j jj J

= ⋅ ⋅ = ⋅ ⋅ ⋅=∑V m a ( )

,1

mm = ⋅ ⋅=∑ s j j jj J

{ },

ψ ω1

(30)

Exhibit 6 compares the results pre-adjustment (idealQ) and post-adjustment (model M) for transaction costs.

OPPORTUNITY LOSS

The adjustment from ideal portfolio Q to modelportfolioM effectively lowers our sights from one unob-tainable goal to another slightly less unobtainable goal. Inthe process, we have left some potential value-added onthe table.

If we measure the value-added from any portfolio,say F with positions f, as

(31)

then the loss we incur by lowering our sights from Qto M is

(32)U UQ M Q M− = ⋅ −

λ ω2

2

UF Q F= ′ ⋅ − ⋅fλ2

2α ω

i scale weight

ii s

M j jj J

. ,

.

,

= ⋅ ⋅

=

∑1

where

ccale

iii weight

j jj J

j

j j

i

= ⋅ ⋅

=⋅

=∑λ ψ ω

ψ ωψ

.{ }

{ }

{

,1

⋅⋅=∑ ωii J

},1

αα


E X H I B I T 5Assumed Half-lives of the Three Signals and Corresponding Values of gj, γj, and ψj based on ∆t = 1/52and δ = 0.926

E X H I B I T 6Risk Budgets and Weights before (Q) and after (M)Adjustment for Trading


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where ωQ–M is the risk of a position that is long the idealportfolio Q and short the model portfolio M.12 In ourexample that risk is ωQ–M = 2.58% and the correspondingloss in value-added is UQ – UM = 0.74%.

THE END RESULT

In Appendix B,we examine the results of followingthe rebalancing strategy portrayed in Equation (23). If welet be the covariance of sourceportfolio iwith the model portfolioM, then we can easilydetermine the alpha, risk, and implementation efficiencyof the implemented strategy P, as follows:

(33)

ADDITIONAL OPPORTUNITY LOST

The opportunity loss just described was due to drop-ping our sights from portfolio Q to portfolio M. Becauseof trading costs, the portfolio we actually hold, P, will notcapture all the potential value added to the model portfolioM. It is possible to take the analysis one step further (seeAppendix B) and look at the efficiency of portfolio P, theportfolio that results from using the updating rule describedin Equations (22) and (23). In our example, portfolio P has2.58% risk. Its correlation (transfer coefficient) withQ andM is 0.727 and 0.857, respectively. The loss in potentialvalue-added moving from the ideal portfolioQ to the actualportfolio P is 1.43%.Given that we lost 0.74% by loweringour sights from the idealQ to the modelM,we lose another0.69% because we cannot exactly trackM.That means a cer-tain fraction of portfolio P’s risk budget will be residual(i.e., not correlated with any source of alpha) and thus thatrisk is taken with no expectation of return. Exhibit 7 con-tains the risk budgets for the three portfolios Q,M, and P.

Exhibit 7 shows the magnitude of the compromisesforced upon us by trading frictions.With no trading costs,we would allocate 51.74% of our risk budget to the FASTsignal.When costs are taken into consideration,however,we allocate only 2.93% to the FAST signal.This is a hugeshift.

ω ρ ψ ωi M j J i j j j, , ,≡ ⋅ ⋅=Σ 1

i

ii

P M i i i Mi J

P i i

.

.

,,

α λ ω λ ψ ω ω

ω ψ ω

= ⋅ = ⋅ ⋅ ⋅

= ⋅ ⋅=∑2

1

2 2 ωω

ω ψ ψ ω ω

i Mi J

M P i i i i Mi J

iii

,,

,,

. ( )

=

−=

∑= ⋅ − ⋅ ⋅1

2

1

1∑∑

REVERSE ENGINEERING

If the target risk level for portfolio P is, say, 4.00%,then we can alter the analysis to get that outcome. If wechange the risk penalty lambda from 22.17 to 22.17 ⋅{2.58/4.00} = 14.30,we will scale up the risks. The riskbudget numbers in Exhibit 7 will not change.

ESTIMATE OF COSTS

As previously discussed, opportunity losses areincurred by lowering our sights from portfolioQ to port-folioM, 0.74%, and by, in fact, lagging behindM, 0.69%.We also will incur direct costs in the form of fees andcommissions or indirect costs through market impact.We can estimate these costs as well (see Appendix A).Let’s call the difference between portfolios M and P thebacklog. This is the trade we would make if we had oneperiod free of transaction costs.We can measure the sizeof this backlog by its variance, .The estimated annualtransaction costs, cP, for portfolio P are a function of thatbacklog variance

(34)

Based on the data, this cost is 0.30%. In addition tothe 1.43% opportunity loss, transaction costs bring thetotal bill to 1.73% and the annual value added after coststo 1.04%.

SENSITIVITY ANALYSIS

One of the attractive features of our approach tosignal weighting is the ability to do sensitivity analysis.Theequations used in this article can be coded in a spreadsheet

cP M P= ⋅ −

λ ω2

2

ωM P−2


E X H I B I T 7Risk Budgets for the Portfolios: IdealQ, ModelM,and Actual P


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in a matter of minutes. As an example, we calculated theweights of the model portfolioM for various values of thetrading rate parameter d, as shown in Equation (30).

The cases in Exhibit 8 bracket the base case withd = 4 that is summarized in Exhibit 6.We can also con-sider the ideal portfolio Q as the case with very large d.The weight for the ideal portfolio is shown in Exhibit 6as 19.95%, 38.42%, and 41.64% for SLOW, INTERME-DIATE, and FAST, respectively.

CONCLUSION

We have presented a relatively straightforwardapproach to the signal-weighting problem in the face oftransaction costs.Our approach has several attractive attrib-utes. First, it is portfolio based. It looks at the signal-weighting problem as an investment problem by linkingeach signal to an investment strategy and then lookingfor the optimal mix of strategies. Second, the approach isan extension of the solution of the signal-weightingproblem in the absence of costs. And third, it is forwardlooking and depends on three traits of signals: predictedinformation ratio, IRj; predicted return correlation, ρi,j;and rate of change of each signal, gj. In addition, ourapproach depends on two investment strategy parame-ters: 1) trading rate of the portfolio, d, a surrogate forturnover, and 2) a risk penalty,λ, that controls the strate-gy’s level of risk.

Additional attributes include the fact that it is easyto perform sensitivity analysis with our approach, a capa-bility that is vitally important in establishing a firm under-standing of the model and its results and in testing therobustness of the answers to reasonable adjustments in theinputs. Our approach also lends itself to reverse engi-neering in that it is relatively easy to adjust the risk penaltyand trading rate to attain the desired level of turnover andactive variance.Furthermore, the approach predicts some

of the characteristics of the resultingstrategy, including level of risk, risk budget,opportunity loss, and annual trading cost.

Finally,our approach can change per-spective by viewing each signal as a sub-strategy and the weights as active risk levelsfor the sub-strategy. These risk levels, inturn, allow the calculation of risk budgetsthat are a better measure of the importanceof each signal. It changes perspective byfocusing on the dynamics of the signalsand the portfolio. The exercise of trying

to measure the rate of change of the signals, gj, and thetrading rate, d, of the portfolio requires a novel, dynamiclook at the signals and the portfolio.

ENDNOTES

1In this article, the following terms—sources, themes, andsignals—are used as synonyms.

2The technical appendices, A (Optimization) and B(Policy Analysis), are available by writing to the author [email protected].

3It is best practice to have this standardization work in atime-average fashion.This allows a natural emphasis on a themewhen more information is available and, in particular, avoidstrying to make something out of nothing when less informa-tion is on hand.

4Note this also implies that5If there is a history of the past source portfolios sj (t), and

dated covariance matrices, V(t), then

gives a measure of the

correlation at time t.6If one or more of the ωj turn out to be negative, it just

means that we are willing to short that particular sub-strategyor replace sj with –sj. This is not of any technical concernalthough it definitely is a danger sign.

7Another way to say this is . Inthis case we have scaled each signal j to have the correct informationratio, IRj. If the signals are not correlated, ρi,j = 0 for i ≠ j, thenthe expression λ ⋅ ωj/IRj equals one for all signals. In the moregeneral case, we see as an adjust-ment for the correlations among the signals.

8In other words, monthly ∆t = 1/12, weekly ∆t = 1/52,and so forth.

9As mentioned earlier,even if the portfolio were to changeon a daily basis,we would likely want ∆t to represent a longer,say, one- or two-week timeframe, so that more substantialchanges take place.

λ ω ω ρ ω⋅ = ⋅=j j j k J j k kIR/ / , ,Σ 1

α λ ωQ j J j j IRIR j

j= ⋅ ⋅=

⋅Σ 1, { } { }a

{ ( ) ( ) ( ) ( ) ( ) ( )}′ ⋅ ⋅ ⋅ ′ ⋅ ⋅s V s s V si i j jt t t t t t

ρi j

t t t ti j,( ) ( ) ( ) ( )/≡ ′ ⋅ ⋅s V s

ρi j i j i j j i, .= ′ ⋅ ⋅ = ′ ⋅ = ′ ⋅−a V a a s a s1


E X H I B I T 8Alpha Weights for the Model PortfolioM as Trading Rate d Variesand Other Parameters Are Held Constant


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10As ∆t goes to zero ψj → d/{d + gj}.11If , then12The portfolio Q – M has positions

REFERENCES

Garleanu, N., and L. Pedersen. “Dynamic Trading with Pre-dictable Returns and Transactions Costs.”Working Paper,HaasSchool of Business at the University of California, Berkeley,February 24, 2009.

Grinold, R. “A Dynamic Model of Portfolio Management.”Journal of Investment Management, Vol. 4,No. 2 (2006), pp. 5-22.

——. “Dynamic Portfolio Analysis.” Journal of Portfolio Man-agement, 34 (2007), pp. 12-26.

Sneddon, L. “The Tortoise and the Hare: Portfolio Dynamicsfor Active Managers.” Journal of Investing, Vol. 17,No. 4 (2008),pp. 106-111.

Solow, R. “A Contribution to the Theory of EconomicGrowth.”Quarterly Journal of Economics, Vol. 70, No. 1 (1956),pp. 65-94.

To order reprints of this article, please contact Dewey Palmieri [email protected] or 212-224-3675.

ω ω ρ ωM i i j i j j2 = ⋅Σ Σˆ { ˆ }.,ω̂ ψ ωj j j= ⋅

q m− = =Σ j J1, ,s j j j⋅ ⋅ −ω ψ( ).1



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Signal Weighting by Richard Grinold

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