a signal weighting

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SignalWeightingRICHARD GRINOLD

RICHARD GRINOLDwas 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).rcgrinold@hotmail,com

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

tiveness.' This challenge is evident for investorswho foUow 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 s truc-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 target

clients, 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 soon. 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 buthat shows how the aggregate riskis being ;icated among the various sources/themes we anticipate can add value. We should podically, say, every six months, re-examine possibly change thatallocation.A similar recsideration of the internal risk budget shotake place if there is a significant change insignal mix (e,g,, adding a new theme) o r if this a major change in the market environm

The method we present in this art

gives substance to the signal-weighting proby isolating the imp ortant aspects ofth e dsion and providing a structure that links tfeatures to a decision. Th e proced ure w e line mimics the investment process. Esignal is viewed as a sub, or source, portfthat is a potential place to allocate risk.allocate risk to the source portfolios imanner that makes the most sense at aggregate level. In this effort we are tryinbe comprehensive enough to depict the es

tials and simple enough to easily capturelink between inputs and outputs. Thus,compromise between the desire to be sonable and robust, on the one hand, anprovide a clear box rather than a black on the other.

To preserve transparency we shouldon the side of simplicity rather than on exsive elaboration. There is a temptation to mand more closely mimic the actual investmprocess (e.g., backtesting). Our unado

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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 w e wantto consider has to argue for a place at the table. The w atch-word is simplicity and the burden of proofis against makingit complicated. This focus on simplicity also argues againsta profusion of sources. In our view, the num ber of themesshould be more than two and certainlyless than, say, seven.If you believe there area 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 is

based on a theory. There is a logical consistency that letsus track cause and effect. The assumptions are few andtransparent. It is not a sequence ofad hoc decisions andjury-rigged improvisations where the often hiddenassumptions are many and their impHcations 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 implem ented 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 arbitragedaway. What wesee in a backtest is one sample froma nonstationary prob-ability distribution. By all means consider past perfor-mance, but add a large pinch ofsalt.

Th e 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 ofsignals 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 allowus to introduce many ofthe terms and concepts we willuse in the more difficult setting with transaction costs.

Th e next step is to introduce transaction costs. Costare difficult to handle in an analyticway. Our approach istraditional; we make an assumption. We assume that thportfolio manager acts as if he is followinga simple rebal-ance rule. The rebalance rule has three parts:

For each signal we construc t a source portfolio thacaptures that signal in an efficient (i.e., most signaper unit of risk) manner.

We construct a mod el portfoho that is a mixture othe source portfolios.

We stipulate a rebalance rule to fix the rate at whicthe portfolio manager tries to close the gap betweethe portfolio the manager actually holds and thmodel portfoho.

In Appen dix A, w e show that this type of rebalancrule is optimal under certain conditions.^ We feel that rule, that is optimal under one set of conditions, will breasonable in a much wider group of situations. IAppendix B, we study the consequences of using such decision rule. The signal-weighting model and results arillustrated using a three-signal example.

OTHER WORK

This article stems from the author's previous woron dynamic po rtfoho analysis, Grinold [2006 ,2007 ]. Thmost directly related paper is Sneddon [2008 ]. Similtechniques to those described in AppendixA are used byGarleanu and Pedersen [2009].

O N E SI G N A L A N D N O

T R A N S A C T I O N C O S T S

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

a = (1)

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o ( i l n , m ia w e i g h t ,w o n e e d a n a \ N C s M n c i ud l t h e

N t i v n i t ho t i h r M i i ; n a l .I c l iii h e t h e H I I D Ii n a t i o n t a t t oo t

t h e s i j ^ n a L l l i e n

a

W e a r e i l o n i . ' -W i t h o i i i 'M L I H Iw i ' M i i i p K ' \ \ \ ' i n h tt h e

r a w a l p h at o h a \ e t h e J e s i i e d t i i t o i a n a t i o n l a t i o .

W e e a n ;j , elt o t h e s a n K ' p l a e eh \ a l o n L : ; e r r o u t e t h a i

a p p r o a e h e x t h e p m b l e i n h " o n ia p e n t i o l i o p e t s p e e t n e .I h i '

p o i l t o h o m u t e I N w o i t ht h e e x t r a( . t l o i tb e e . u i s et t L ; e n

e r a h / e sm a n a t t u a l \ \ a \ 'w h t ' i i t l i e r . ' a ix " m t i l t t p l e M ^ n a K

a i k l t t M i i s a e t i o n e o s t s .

W e , i \ s ( ) t l a t ea s o u r c e p o r t t o h os w i t h o t i r s i ^ n a h

s - - :V a - ' ^ f / ) , = s ' ' V - s - | i.V)

1 h e s o t i i ie p o t t t o h oi s , i i ) e t f k i i ' i i t i n i p l e i u e t i t a t i o n

o t t h e s i L ^ n a h t h a t i s . a n i o i i i ial l p o s i i i o n sx . i t n i . L \ i i i H / e s

t h e r a t i o

I t m a k e s t h i n ; j , s n u u h s u n p k ' i w h e nw e m t r e >

I rans .u ' t ion TO S I N .

a x / \ ' X ' V X

A s n o t e dIII I A j u a l i o n ( 3 ) . t h e s o t i i xe p o i t t o h o h . i sa n

e x i e s s i w r i s k k ' \ e lo t | n ( i " , i . S u p p o s ew \ ' e h o o s c -a m o r e

r e a h s t i e r i s kl e v e l( 0 . I h i s i s a k i i it o s e a h i i u t h e s o u r e e p o i i

t o h o .It (I) IS t h e r i s k l e \ e k t h e n d i e p o r l t o l i oi s q = ( 0 s .

I h i s p o m t o h o l i a s .1 1) . l i p h a ( ) l 7 / < ' 0 ) m d a w i r i a n e eo \ ( 0 .I1 w e h . i \e .1 i i s k p e n , i l t \ ; .. i n dw e d i> t h e s t . m d a r i l m e a n

w i n . n u e o p t n m / . i t i o n .w e e l i o o s et h e r i s k l e \ e l(0 t o

i n a \ i i n i / e .

/ / \ ' (I) - - - 0)1

\ h e sohlt Kill IS (I) - "V . Io r ex . i i npk ' , it iR = 11.7

.m d w e have a r i sk p e na l t y o r A = 1


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I l u ' m o r i n . i t i o n M tK ) ^ o t t l u - / M i h s r r. i t c j i ; i c \ . u c

g i v e n hv the / c l c n u - n t w c t o r I R - I / \ \ l , ; y I lie / bv

/ t o i r c L i r i o n n i . i t r ix w i t h c l c n i c n t s p^ ^ is d e n o t e d .i^ II

W i t h t h i s n o t a t i o n t h e a l pl u i e x p e e t e d h\ U M I I L " ,t h e

CO IS

( C O )

I he \ a n . u i e e o l t l i e p o r t i o l i o . q (C O),w ith w e i u h t s 0) is

(T ( CO) = V 0 = 0 ) ' K C O ( 1 1 )

We ehoose (0 lo maximi / e .

(X (CO) -CT " (CO) (1 - )

bhe soltition is obtained bv solving the first-order

equations. '

(0

= A - R - ( O (13)

1 li e n i a \ i n i i / i n L ! ; v e c t o r ot w e i g h t s is g i w n b y

CO = ^i-- R ' IR (14)A

W'e will refer to the optimal portfolio (a.k..),. the

portfolio) as Q with positions

q(CO) = 0) ( 1 5)

1 h e H i f o r n i a t u i i i r a t i o a n d r i s k ot the i d e a l ] i o r t -

f o l i o ( ) a re

= 7 l R R ^ 11^ (0,, = T o o ' - R CO. I R , , = A

E X H 1 B r r 1

Three-Signal Example

Source IR SLOW INTERMEDIATE FAST Risk

SLOW 0.4 1.0 -0.15 0.1 , 1.96%

INTERMEDIATE : 0.5 -0.15 I 1.0 | -0.3 i 3.78%FAST 0.7 0.1 -0.3 1.0 i 4.10%

. V ; . < c : 7 / rI l l l ' l l ' , 1.1 1 ,1, ! l , i h , l , l ! c n , , , l l l , ' i l < M h \H i . . l l h li l l - , . < ih 7 , / ; f


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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

q ( ( D ) = 0 ).

The alphas that lead to this ideal

(18)

WEIGHTS

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

H i. weight. =

/'. C L Q= s c a l e ' j ^ a , u / e i g h t . > , w h e r e

i i . s ca l e= X-^

(19)

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

VQ (20)

The weights in our examples are 38% for INTER-MEDIATE, 2 0% 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.

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

Ideal Portfolio

itRisk

^ mQ Risk Budget Q

mofm

Weight Q

BUS

TRANSACTION COSTS

The previous analysis showed how to successfullyapproach the source/signal-weighting problem when thereare no transaction costs. But the presence of transaction

costs introduces three new obstacles. These obstacles are

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

Transaction costs are intimidating. They keep youfrom fuUy 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, botthe power and the Achilles' heel of any economic analysis.As Solow [1956] said, "All theory depends on assumptions

that are not quite true."

THE PORTFOLIO'S LAW OF MOTION

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

Ap(i) =

determine the cost.

- p(t - Ai) ( 2

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Tbe challenge is to capture these changes in a usefulway. Todo this we will specifya rule, called the lawofmotion, to show bow portfolioP changesin responsetochangesin the source portfolios. We willdo this in tw o

steps,as follows:1. Combine the source portfolios intoa model port-

folio M,2, Show how the portfolioP attemptsto trackthe

model portfolio,

T H E M O D E L P O RT E O L IO

Th e m odel portfolio is a mixture oft he source port-folios.Its ho ldingsat time are

A MEASURE OF SIGNAL SPEED

We can gaugethe rate of changein the signal/between timest and Af by measuring the correlofthe positions s.(i) and s.((-A


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ofa half-life, thatis,the time it will take for the correlation,as calculated by Equation (24), to drop to 0.5. Exhibit 3shows how^. depends on the half-life.

In our example,wewill use nine monthsasthehalf-

life for the SLOW signal, one month for the FAST signal,and six months for the INTERMEDIATE signal.

THE PACE OF PORTFOLIO CHA NGE

The parameterdeltain Equation (23) governs therate of change of portfolioP.We will expressdeltaasS= e''''", whered measures how fast we attempt to closethe gap between where we start, p(t Ai),and where wewould like tobe,tn(i).The choice of, whether it is madeexplicitly or implicitly, is an attempt to balance two costs:

1. Transactioncosts:A largerd (thus lowerS) leads tomore trading and higher costs.

2. Opportunityloss:A smallerd (thus higher) impliesa less efficient implementation; the fraction of por t-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 ofd, which should isolate a reasonable range ofvalues, and then we consider procedures that will help usdiscover an implied value ofd by examining the pastbehavior of the portfolio.

INTERPRETATION OF d

PortfolioP chases the model M. It is always behindthe curve. Indeed,1/dis a measure of just how muchP

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

Months LagOne monthTwo monthsThree monthsFour monthsFive monthsSix months

d

12.06.04.03.02.42.0

lags M. BothP and M are based on informatioflowed in over previous periods. The age-wsure of P to this information is aboutl/d years lothan the age-weighted exposure of M to the

mation. Slow trading leaves P under exposinformation and relatively over exposed toinformation. Exhibit 4 shows lags of one tand the corresponding values ford.

ESTIMATION OF d; REVEALED PREFERENCE

It is possible to get a rough estimate ofdby exining the history of the portfolio and the solios.Aswe indicate in AppendixB,wecan use a regror portfolio optimization, approach to estim

ters{t)and j{t),so that

p(0 = p ( i - {t) 0 (2

and the unexplained variance, '(0 V(i) (i), is mmized. If things are going along relatively sno radical changes of policy) and the time iAselected in a reasonable manner,' then estim{t)wtend to be positive andlessthan one. In suchcases,we an estimate ofd as

(2

The ^analysis will also produce implied.{t)/{\o{t)),and a measure of the fittin

( ) ( ) } { P ( ; ) ( ) P ( ) }From these estimates of implied pas

addition to the intuition that might be gExhibit 4 and future policy plans, we settle the parametersd andg.,j = 1,...,]. Then, for any ance intervalAi,we can calculate the risk adjuspromised in Equation (22),'

1-5 (2I I .

1-5-7,

Now let's examine how this would woexample introduced earlier. The data in E

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E X H I B I T 5As sum ed H alf-live s of the Three Sig nals and Co rresp ond ing Va lues of g., / . , and ^. bas ed on Ai = 1/52and S= 0.926

Source HL Months g.SLOW Hl^^ltP '^^INTERMEDIATEH P M | I | M ^h AST IH iy S i i P l^ ijiiio)y{ii

Risk Q Risk M

based on a valueo d 4 and a rebalance interval of oneweek, Ai = 1/52, thus= 0.926.

In Exhibi t 5, theg. are based on the assumedhalf-lives. If i-/L. is the half-hfe in months, theng.= 12 {In( 2 ) / H L . } .T h e y.are calculated using item (/) of Eq uatio n(28) and the yA. are derived from i tem (/i) of Eq uat ion (28).Th e colum n " Ri sk Q " repeats the optimal risk levels,CO,of the ideal portfolio previously reported in Exhibits 1and 2. The final column, "Risk M," contains the risk levelsl/- CO.associated with the model portfolio, as in Equa-tion (22).T he risk levels of all the signals fall, altho ugh thedecrease, evident in the"psi"colu mn , is mo re dram atic forthe FAST signal. Th e risk" of mod el portfoho M is 3.06%com pared to the 5% risk level for the ideal portfolio Q.To raise the risk level of the m od el portfo lio to, say, 5%,merely decreases the risk penalty lambda from 22.17 to{3 .06 /5} 22.17 = 13.57 and repeats the calculation. This

will scale up the risk, but keep the risk budget allocationand weights the same.

T H E A L P H A P E R S P E C T I V E R E V I S I T E D

Th e adjustm ent for transaction costs works for alphasas well as for portfolios. Accord ing to Equa tion (22), ou rmodel portfolio is

The alphas that would lead to this portfolio are

A, = A - V - m = A- //,. CO.) (29)

W E I G H T S

Equa tion (29) can, as before, be interp reted in term sof weights (i .e., num bers that sum to on e), as follows:

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

Source | Risk Budget Q Risk Budget MSLOWINTERiVIEDIATEFAST

m > m > f m s m , ''Weight Q I Weight M

i. (X^ , = scale -i ^ Ot. weight. >, w h e r e

(Y. scale = - ^{y/ CO.} (30)

m.

Exh ibit 6 compares the results pre-adjustment (ideaQ) and post-adjustment (modelM) for transaction costs.

O P P O RT U N I T Y L O S S

The adjustment from ideal portfol io Q to modeportfolio M effectively lowers our sights from one unobtainable goal to a no ther slightly less unobta inable goal. Ithe process, we have left som e poten tial valu e-adde d othe table.

If we measure the value -added from any portfoliosay F with positionsf, as

2(31)

then the loss we inc ur by low erin g our s ights f rom Qto .M is

(32)

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where (OQ_^, is the risk of a position that is long the idealportfolio Q and short the model portfolio M.'^ In ourexample that risk is(OQ_H^= 2.58% and the correspondingloss in value-added isU Q - U ^ = 0.74%.

THE END RESULT

In AppendixB, we exam ine the results of followingthe rebalancing strategy portrayed in Equation(23).If welet C.^i =Z.^| .p. y/.-CO. be the covariance of sourceportfolio / with the model portfolio M , then we can easilydete rmine the alpha, risk, and implem entation efficiencyof the implemented strategy P, as follows:

n . c o ; = (33)

E X H I B I T 7Risk B udgets for the Portfolios: Ideal Q, Model M,and Actual P

RiskBudget

SLOW^ -'^U-ITERMEp:i|r

Residual ' -:

PortfolioQ (ideal)

mofm

PortfolioM (model)


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

Months Lagd

SLOW

INTERMEDIATE

FAST

0.524

'4um

1

12

2

6

' V f

4

I12

1

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

The cases in Exhibit 8 bracket the base case with

rf = 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, IR.; predicted return correlation, p..;and rate of change of each signal,^.. 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. A, 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, g., and thetrading rate, d, of the portfolio requires a novel, dynamiclook at the signals and the portfolio.

ENDNOTES

'In this article, the following termssources, themes, andsignalsare used as synonyms .

technical appendices, A (Optimization) and B(Policy Analysis), are available by writ ing t o the a uth or atr cg r i no ld @ ho tm a i l . co m.

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

tion is on hand.Note this also implies tliat p. . = a ' - V " ' -a . = a'-s . = a ' -s ..

'If there is a history of the past source portfolios s. (i), and

dated covariance matric es, V(f), then P,.(0 = s'(f)'V(f)-s^ .(i)/

a measure of the.^s'(i) V(/) s.(r) Js'j (i) V(i) s .(i)} gives

correlation at time f.""If one or more of the CO .turn out to be negative, it just

means that we are willing to short that particular sub-strategyor replace s. with - s. . This is not of any technical co ncer nalthough it definitely is a danger sign.

'Another way to say this is Q = ^j=, j{"^,- ^j} ' {-JT^}. In

this case we have scaled each signal j to have the correct informationratio, IR.. If the signals are not correlated, p. . = 0 for / ^j, thenthe expression , CO./IR.equals one for all signals. In the moregeneral case, we see X CO./IR. CO.fL^^^^^p.^, CO^,as an adjustment for the correlations among the signals.

"In other words, monthly Ai = 1/12, weekly Ai = 1 /52 ,and so forth.

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

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'"As Ai goes to zero \f/. -^ d/{d-\-g).

"If ft)^. = \f. CO., then )j, = 'L..{L.p..-)^).

'^The portfolio Q- M has positions q - m = Z,^|

REFERENCES

Garleanu, N., and L. Pedersen. "Dynamic Trading with Pre-dictable R etu rns and Transactions C osts." Wo rking 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, N o. 2 (2006),pp. 5-22.

. "Dynamic Portfolio Analysis."JoMmij/of Portfolio Man-

agement, 34 (2007),pp. 12-26,

Sneddon, L. "The Tortoise and the H are: Portfolio Dynamicsfor Active Managers."7oHrafl/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, N o. 1 (1956),pp. 65-94.

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a signal weighting

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