Information Horizon, Portfolio Turnover, and Optimal Alpha...

FALL 2007 THE JOURNAL OF PORTFOLIO MANAGEMENT 27

The bottom-line value of an active investmentprocess has two parts: the theoretical value ofthe alpha skill (the gross paper profit), minusthe cost of implementation. The higher the

former and the lower the latter, the happier the investoris. Clearly total assets under management (AUM) influ-ence the latter. A strategy might be profitable with a low level of assets under management but unprofitablewith more assets—as asset amounts grow, so do transac-tion costs.

Kahn and Shaffer [2005] note that one approach toremedy the size problem is to reduce portfolio turnover.While this is a sensible suggestion, Kahn and Shaffer’swork is based on a hypothetical relation between turnoverand expected alpha that might be too general to be applicable. In reality, any relation between turnover andexpected alpha is not exogenous. It depends on alphafactors, their weights in an alpha model, and the rebal-ance horizon.

We propose an analytic framework for integratingalpha models with portfolio turnover. In practice, manyalpha models are not constructed in such an integratedframework. Typically, managers first develop an alphamodel (giving little consideration to turnover), and thenthrow the alpha model into an optimizer, setting turnoverconstraints to handle the transaction costs. There are twodrawbacks to this two-step process: 1) It makes it hard toknow the true effectiveness of the alpha model; and 2) itdoes not let managers adjust the alpha model along theway as AUM grow.

To integrate an alpha model with portfolio turnover,we extend work by Qian and Hua [2004] and Sorensenet al. [2004] that provides an optimal solution for

Information Horizon, Portfolio Turnover, and Optimal Alpha ModelsOptimizing multiple factors for the best IR.

Edward Qian, Eric H. Sorensen, and Ronald Hua

EDWARD QIAN

is the chief investmentofficer and head ofresearch, MacroStrategies, at PanAgoraAsset Management in Boston, [email protected]

ERIC H. SORENSEN

is president and chief executive officer ofPanAgora Asset [email protected]

RONALD HUA

is the chief investmentofficer and head ofresearch, Equity Group,at PanAgora Asset [email protected]

IT IS

ILLEGAL T

O REPRODUCE T

HIS A

RTICLE IN

ANY F

ORMAT

Copyright © 2007

28 INFORMATION HORIZON, PORTFOLIO TURNOVER, AND OPTIMAL ALPHA MODELS FALL 2007

multifactor models with the objective of the highest infor-mation ratio. This work relies heavily on the time seriescorrelation of information coefficients (IC) as well as con-temporaneous correlations between factor signals.

We explore these critical correlation analyses toinclude serial correlations of factors and the concept of afactor information horizon. This allows us to evaluate imple-mentation costs in finding the factor weights that opti-mize the information ratio (IR) with net returns.1

We first discuss the information horizon generally,and then derive an analytic formula for portfolio turnoverconditioned on changes in forecasts. This solution allowsus to estimate portfolio turnover for different quantitativealpha factors and related investment strategies. We findthat portfolio turnover can be endogenous in a completesystem, and that factor autocorrelation is the key exoge-nous ingredient.

Finally, we build a multifactor model by maximizingthe IR under portfolio turnover constraints. A numericalexample including a value and a momentum factor is usedto illustrate the framework.

INFORMATION HORIZON

Information horizon is crucial to portfolio turnover.If a factor has a relatively short information horizon, it pre-dicts security returns only for the very near term. Thesignal decays quickly. Momentum factors, especially theone-month reversal factor, behave this way. Such factorscause high turnover because their exposures must be con-stantly adjusted. If a factor has a relatively long informa-tion horizon, it predicts returns long after its informationbecomes available. The signal decays slowly. Valuationfactors tend to behave this way. These factors will promptlow turnover—portfolios constructed on the basis oflagged factors can still generate excess returns.

In reality, no two factors (models) have the sameprofile in terms of information horizon. Typically, mostfactors lose all power by nine months; a few can keepgoing for as long as several years. Depending on the pre-dictive power of different factors, it seems reasonable thatturnover frequency will vary with the particular alphamodel. The two should be in balance.2

Information Coefficient Terms

We study the general concept of informationhorizon through two specific expressions related to the

information coefficient (IC): lagged IC and horizon IC.We denote IC as the cross-sectional correlation coeffi-cient between the factor’s value at the start of time t andthe security returns over time period t, i.e., ICt,t = corr(Ft, Rt). Consider this the typical one-period IC mea-sure, for a month or a quarter. An example is the first-quarter return IC. The factor values are observedDecember 31, and the return period is January throughMarch.

The lagged IC is the correlation coefficient betweentime t factor values and a later period (by one, two, or morequarters) return vector, i.e., ICt,t + l = corr (Ft, Rt + l), withlag l. The factor is lagged by l periods compared to the return. For example, we can correlate factor readingson December 31 with returns for later periods (secondquarter [l = 1], third quarter [l = 2]), and so on. The ICwill typically decay in power as the lag increases, but therate of decline differs across different factors such asmomentum and value. For simplicity, we assume the ICsare generated by stationary processes implying that ICt,t+l =ICt + s,t + l + s.

Exhibit 1 shows the average quarterly ICs and stan-dard deviation of ICs of a specific price momentum (PM)factor based on nine-month returns. The average IC ishigh with no lag, but it declines steadily with greater lags.With a lag of three, the average IC is close to zero, indi-cating no predictability for the return.

Exhibit 2 shows the average quarterly ICs and stan-dard deviation of ICs for a specific value factor based ontrailing earnings yield (E2P). The average IC is lowercompared to that of momentum, but shows less decay inpower as the lag increases.

In both cases, the IC standard deviations are stablewith respect to the change in lags. The standard devia-tion of IC is much higher for the momentum factor thanthe value factor. As a result, the IR of the value factor(annualized by multiplying by two) is higher than the IR

IR is the annualized ratio of the average IC to the standard deviation of IC.

E X H I B I T 1Momentum Factor Statistics for IC and Lagged ICs

Copyright © 2007

of the momentum factor at all lags. With a lag of threequarters, the IR of the momentum factor is nearly zero,while that of the value factor is still close to one.

Another important variant on the standard IC helpsus understand the cumulative effect of multiple laggedICs. We define horizon IC as an IC of a factor at a giventime, t, for subsequent returns over multiperiod horizons.For example, if we have a factor available at December 31,we are interested in its correlations with cumulativereturns of the next quarter, the next two quarters, or thenext three quarters, and so on. We denote by Rt,t + h therisk-adjusted cumulative returns from period t throughperiod t + h, and denote by

, the horizon IC. For example, ICto is the

standard IC for the return in period t, and is the cor-relation between the factor and the return vectors over thenext six months, periods 1 and 2.

Relation Between Lagged IC and Horizon IC

Although the lagged IC typically decays with thelag, horizon IC often increases with the horizon, at leastinitially. By definition, the cumulative multiperiod returnin the horizon IC is related to the single-period returnby . When theperiod’s returns are low, we can approximate return by

. Using this expression inthe horizon IC yields:

(1)

If we further assume that the risk-adjusted returns in different periods are uncorrelated, the IC sim-plifies to:3

(2)

The horizon IC is an average of lagged ICs timesthe square root of the horizon length (l + 1). Note that the horizon IC covers returns of multiple periods,and the lagged ICs cover forecasts for single intervals for future periods. Suppose there is no information decay in

ICIC IC IC

l

IC l

tl t t t t t t l≈

+ + +

+= ( ) +

+ +, , ,1

1

1

L

avg

IC tl t t t t l

t t t t l

≈+ + +( )

( ) + + +( )+ +

+ +

cov ,F R R R

F R R R1

1

L

Ldis dis

R R R Rt t h t t t h, + + +≈ + + +1 L

R R R Rt t h t t t h, ( )( ) ( )+ + += + + + −1 1 1 11 L

IC tl

h H= 0 1, , ,…IC t

ht t t h= +corr( , ),,F R

the lagged forecasts, i.e., . Thenfrom Equation (2) we have . In this case,the horizon IC is the IC times the square root of thehorizon length, and it therefore increases as the horizonlengthens.

Even when there is information decay, the horizonIC can still initially increase with the horizon length. Itthen declines as the horizon lengthens further or as thelagged ICs decline more rapidly.

Exhibit 3 plots one such case, in which the initialperiod IC is 0.10. The lagged IC is 0.08 with lag 1, 0.06with lag 2, and so on. It reaches zero with lag 5, and turnsnegative thereafter. The horizon IC increases at first. Forexample, the IC is 0.128 for returns over the next twoperiods (1 on the horizontal axis), and 0.139 for returnsover the next three periods (2 on the horizontal axis).Then, the horizon IC is eventually dragged down by thedeclining lagged ICs.

Exhibit 4 shows the horizon IC of the two factorsin Exhibits 1 and 2. The horizon IC for the PM factor

IC IC ltl

t t= +, 1

IC IC ICt t t t t t l, , ,= = =+ +1 L


IR is the annualized ratio of the average IC to the standard deviation of IC.

E X H I B I T 2Value Factor Statistics for IC and Lagged ICs

E X H I B I T 3Lagged IC and Horizon IC of Signal

Copyright © 2007

starts to decline at three quarters while that of E2P keepsincreasing.

Horizon IC and Trading Horizon

The propensity for the horizon IC to increase ini-tially with the horizon does not necessarily mean thatwe can increase the total IC with a longer trading horizon.Longer trading horizons allow fewer opportunities torebalance, or fewer chances along the time dimension.Basically, the longer horizon reduces the breadth of theprocess, which adversely impacts the information ratio.

The reduced breadth leads to a higher variability ofhorizon IC. As a result, when we measure IR in termsof the ratio of average IC to the standard deviation of IC,the longer trading horizon may not confer any advan-tage. Suppose both forecasts and returns are of quarterlyfrequency. The quarterly IC has a mean of 0.1 and a stan-dard deviation of 0.2. Then the quarterly IR is 0.5, andthe annualized IR is .4

Let’s assume that all subsequent one-period laggedICs have the identical distribution, and are uncorrelated.Then, the horizon IC of one year, or four quarters, willhave a mean of , and a standard devia-tion of . Hence, the annual IR is also1—the same as the annualized IR of quarterly trading.There is no difference in terms of the performance, grossof any trading costs.

There could be differences in trading costs that arisefrom the implication of quarterly trading (rebalancing)rather than annual trading. In one case, we trade once a year.In the other case, we trade four times a year. The questionis whether the total trading costs in the latter case exceedsthe costs in the former case. The answer depends on thenature of the turnover induced by changes in alpha factorsover each quarter versus the one-year holding period. Thus,the alpha model profile and trading costs become endoge-nous with respect to each other, and interact in determiningthe maximum deliverable net (after-cost) IR.

4 0 2 4 0 2⋅ =. / .4 0 1 4 0 2⋅ =. / .

0 5 4 1. =

TURNOVER CAUSED BY CHANGE IN FORECASTS

Consider turnover over a single trading period,where the active weights change from to , foreach security i. We define one-way turnover as one-halftimes the sum of the absolute value of weight changes:5

(3)

We assume the active weights for each security resultfrom an unconstrained mean-variance optimization basedon residual return and residual risk at times t and t + 1:

(4)

where and are risk-adjusted forecasts at t and t + 1, and λt and λt + 1 are risk aversion parameters. Forsimplicity, we assume that all stock-specific risks remainunchanged and that the number of stocks remainsunchanged. If we hold the targeted tracking error σmodel

for the portfolio constant, then the risk aversion para-meter is given by:

(5)

Substituting gives:

(6)

where and are now standardized with ,. In other words, they reduce to simple z-

scores. Equation (6) states that the active weight of a stockdis( )F̃t+ =1 1

dis( )F̃t = 1F̃it+1F̃i

t

wN

Fit i

t

i

++

=−

11

1

σσ

model˜

wN

Fit i

t

i

=−

σσ

model

1

˜

λσt

tN+

+

= −1

11dis( )

model

F

λσt

tN= − 1dis( )

model

F

Fit+1Fi

t

wF

it

t

it

i

+

+

+

=1

1

11λ σ

wF

it

t

it

i

= 1λ σ

T w wit

it

i

N

= −+

=∑1

21

1

| |

wit+1wi

t


E X H I B I T 4Horizon ICs for Two Factors

Copyright © 2007

is directly proportional to the portfolio target trackingerror and its z-score, but is inversely proportional to thespecific risk and the square root of the number of stocks.Using Equation (6) in (3) gives:

(7)

The most difficult aspect of analyzing turnover isdealing with the absolute value function. Our solutionto this problem is to approximate the turnover in Equa-tion (3) as the expectation of the absolute difference oftwo continuous variables underlying two sets of forecasts.When we do this, we can rely on standard statistical theoryto evaluate various expectations. In the appendix, weshow that portfolio turnover is given by:

(8)

Equation (8) represents our solution to forecast-induced turnover for an unconstrained portfolio.6 Itdepends on four elements. Turnover is higher:

• The higher the tracking error.• The higher the number of stocks—proportional to

the square root of N.• The lower the forecast autocorrelation-cross-

sectional correlation between consecutive forecasts,.

• The lower the average stock-specific risk.

The forecast autocorrelation becomes the most rel-evant for our analysis of turnover. There is considerableintuition behind this. For example, consider two extremes.If the correlation between the consecutive forecasts isequal to one, then the weights are identical and there isno turnover. When the correlation is less than unity, it willbe advantageous to incur turnover. At the other extreme,with a correlation of –1, turnover will be at the max-imum. Indeed, all weights flip signs and the portfolioreverses itself.

Suppose stock-specific risks are the same for allstocks; in this case, the turnover is reduced to:

(9)T f= −N model

0πσ

σρ1

ρ ft tF F= +corr( )˜ , ˜1

T f= −

NE

1modelπ

σ ρσ

1

TN

F Fit

it

ii

N

=−

−+

=∑σ

σmodel

2 1

1

1

˜ ˜

For an active portfolio (which could be either long-short, market-neutral, or active versus a benchmark)with N = 500, σmodel = 5%, σ0 = 30%, and ρf = 0.9, theone-time turnover would be

Exhibit 5 plots the function . This is thedependence of turnover on the forecast autocorrelation.Turnover is a declining function of forecast autocorrela-tion. The function behaves much like a linear function formost of the range, but it drops more precipitously whenρf is greater than 0.8.

Among the common quantitative factors practi-tioners use, value factors as a category generally have thehighest forecast autocorrelation, and thus the lowestturnover. They also have the slowest information decay,characterized by high lagged ICs. Their forecast auto-correlation can be as high as 0.95. Among value factors,those based on cash flow have slightly lower forecast auto-correlation.

The momentum factors have the lowest forecastautocorrelation, and thus the highest turnover.Momentum factors’ forecast autocorrelations generallylie between 0.6 and 0.7. For price momentum factors,the autocorrelation typically increases as the time window used for return calculation lengthens, up to 12 months. Therefore, one should use a longer timewindow to measure price momentum in order to reduceturnover.

1 − ρ f

T = − =5003 1415

530

1 0 9 66.

%%

. %


E X H I B I T 5Dependence Function of Turnover on Forecast Autocorrelation

Copyright © 2007

TURNOVER OF MULTIFACTOR MODELS

Multifactor models offer diversification among fac-tors and can improve the information ratio of the overallmodel (see Sorensen et al. [2004]). Drawing on the resultsso far, we analyze the turnover of multifactor models bystudying their forecast autocorrelation. Multifactor modelscan depend on both cross-sectional and time series aver-ages of multiple factors. We can reduce portfolio turnoverif the composite model has higher forecast autocorrela-tion than individual factors.

Cross-Sectional Average

In the cross-sectional dimension, the average con-sists of different factors whose performance is evaluatedover the same time interval. To illustrate, consider a two-factor case, where the composite forecasts are linear com-binations Fc=v1F1 + v2F2; both F1 and F2 are standardizedfactors; and v1, v2 are their weights. The autocorrelationof the composite factor is:

(10)

where is the contemporaneous correlation between the two factors, and is the correlation between and

. For i = j, , is the serial autocorrelation of a singlefactor, and for , is the serial cross-correlationbetween two different factors.

The autocorrelation of the composite factor dependson this correlation structure and on factor weights. Allelse equal, the autocorrelation of the composite factorwill be high if the two factors have high serial auto- andcross-correlation, and the composite factor has lowvolatility. The low composite volatility depends, in turn,on a low contemporaneous factor correlation. Generally,low contemporaneous correlation also means better factorscore diversification and therefore may imply a lowervolatility for the IC.

For a numerical example, suppose the serial autocorrelations of two factors—PM and E2P—are

and ; the serial cross-correlationsρ220 1 0 94, .=ρ11

0 1 0 68, .=

ρij0 1,i j≠

ρij0 1,F j

t+1Fi

tρij0 1,

ρ1 20 0,,

ρ

ρ ρ ρ ρ

ρ

f

ct

ct

ctc

v v v v

v v v v

=( )

( )=

+ + +( )+ +

+cov ,

var

, , , ,

,

F F

F

1

12

110 1

22

220 1

1 2 120 1

210 1

12

22

1 2 120 02

are and ; and the contempora-neous correlation . Then:

Exhibit 6 plots the autocorrelation as a function ofv1, and we have v2 = 1 − v1. When v1 = 0, it is equal tothe autocorrelation of factor 2, while when v1 = 1, it isequal to the autocorrelation of factor 1. When v1 is low,there is a slight increase in the composite autocorrelation.

Mathematically, all the correlation coefficients dis-cussed above can be neatly placed into a single symmetricnon-singular matrix—the correlation matrix for thestacked vector :

(11)

The autocorrelation of composite factors can be calcu-lated using this correlation matrix.

Forecast Autocorrelation of Moving Averages

When signals are volatile, we can smooth them using moving averages. In a multifactor model, moving

C

F

F

F

F

=

+

+1

1

21

1

2

120 0

111 0

121 0

120 0

211 0

221 0

111 0

211 0

120 0

121 0

220 0

120 0

1

1

1

1

t

t

t

t

ρ ρ ρρ ρ ρρ ρ ρρ ρ ρ

, , ,

, , ,

, , ,

, , ,

( , , , )F F F F11

21

1 2t t t t+ +

ρ f c

v v v v

v v v v= + −

+ −0 68 0 94 0 09

0 1612

22

1 2

12

22

1 2

. . .

.

ρ12 0 08t t, .= −ρ21

0 1 0 00, .=ρ120 1 0 09, .= −


E X H I B I T 6Autocorrelation of Composite Forecast as Function of Weight in Factor 1

Copyright © 2007

averages are also considered composite factors—a linearcombination of new and past information. A natural ques-tion is why we would use outdated information in fore-casts, as one tends to think that a forecast based on themost recent information is better than a lagged forecast,or has more predictive power for subsequent returns, i.e.,a higher IR. This may be true, but one should verify thisempirically.

The primary reason to use lagged forecasts for manyalpha factors is that moving averages increase the auto-correlation, thus lowering turnover. Despite possible infor-mation decay in the lagged forecasts, a trade-off betweenpotential profit reduction and improved transaction costsmay favor the inclusion of lagged factors in a multifactormodel. Our next step is to provide a mathematical frame-work for analyzing this important trade-off.

Autocorrelations of moving averages are calculatedsimilarly to cross-sectional averages. Given forecast series

, we form a moving average of orderL as . For L = 2 . Theserial autocorrelation is given by:

(12)

where ρf (h) is the serial autocorrelation function of Ft,with ρf (0) = 1.

For given serial autocorrelations ρf (h), h = 1, 2, the cor-relation is a function of the weights, v0 and v1. Since the cor-relation is invariant to a scalar, we can require v0 + v1 = 1.Exhibit 7 plots Equation (12) as a function of v1—the weightof the lagged forecast for the E2P factor, with ρf (1) = 0.94and ρf (2) = 0.84. When v1 = 0, the moving average isidentical to the original factor, so the serial autocorrela-tion is 0.94. As v1 increases, the lagged forecast is added tothe moving average, and the serial autocorrelation of increases; it reaches a maximum close to 0.96 at v1 = 0.5when the two terms are equally weighted. As v1 changesfrom 0.5 to 1, the autocorrelation declines from the max-imum to 0.94.

Exhibit 8 graphs Equation (12) for the PM factor, withρf (1) = 0.68 and ρf (2) = 0.40. As v1 increases, the laggedforecast is added to the moving average, and the serial autocorrelation of increases; it reaches a maximum closeFma

t

Fmat

ρ

ρ ρ

ρ

f

t t t t

t t

f f

f

ma

v v v v

v v

v v v v v v

v v v v

=+ +( )

+( )=

+ +( ) ( ) + ( )+ + ( )

− +

−

cov ,

var

0 11

01

1

0 11

0 1 02

12

0 1

02

12

0 1

1 2

2 1

F F F F

F F

F F Fmat t tv v= + −

0 11F Fma

tlL

lt lv= =

− −Σ 01

( , , , )F F Ft t t− −1 2 …

to 0.82 at v1 = 0.5 when the two terms are equallyweighted. For both factors, the composite forecast is mademore stable by including the lagged factors leading toreduced portfolio turnover. This is generally the case inpractice.7

Cross-Sectional and Time Series Averages

A comprehensive multifactor model should haveboth cross-sectional and times series averages. We writesuch models as

F Fc mat

lj jt l

l

L

j

M

v, = −

=

−

=∑∑

0

1

1


E X H I B I T 7Serial Autocorrelation of Forecast Moving Average

L = 2, ρf (1) = 0.94, ρf (2) = 0.84.

L = 2, ρf (1) = 0.68, ρf (2) = 0.40.

E X H I B I T 8Serial Autocorrelation of Forecast Moving Average

Copyright © 2007

where there are M factors each with a moving average oforder L. For clarity, we consider here the case of two factors and one lag, all standardized with standard devia-tion one:

(13)

Although it is still possible to calculate the serialautocorrelation of (13) algebraically as before, the expres-sion becomes cumbersome and intractable with more factors and more lags. It is much more succinct to derivethe autocorrelation by matrix notation instead. To this end, we denote the weights in (13) as a vector,

. We consider the stacked vectorand denote its correlation

matrix by:

where . The autocorrelation of thecomposite (13) is the ratio of the covariance between thecomposite and its lagged value to its variance. To calcu-late the covariance and variance, we denote the matrix in the upper left-hand corner of C as C4 and the

matrix in the upper right-hand corner of C as D4:

(15)

Then the variance of isFc mat,

D4

111 0

121 0

112 0

122 0

211 0

221 0

212 0

222 0

120 0

111 0

121 0

120 0

211 0

221 0

1

1

=

ρ ρ ρ ρρ ρ ρ ρ

ρ ρ ρρ ρ ρ

, , , ,

, , , ,

, , ,

, , ,

C4

120 0

111 0

121 0

120 0

211 0

221 0

111 0

211 0

120 0

121 0

221 0

120 0

1

1

1

1

=

ρ ρ ρρ ρ ρρ ρ ρρ ρ ρ

, , ,

, , ,

, , ,

, , ,

4 4×

4 4×

ρijl k

it l

jt k, ,= + +corr( )F F

C

F

F

F

F

F

F

=

+

+

−

−

11

21

1

2

11

21

120 0

111 0

121 0

112 0

122 0

120 0

211 0

221 0

212 0

222 0

111 0

211 0

120 0

111 0

121 0

121 0

221 0

1

1

1

t

t

t

t

t

t

ρ ρ ρ ρ ρρ ρ ρ ρ ρρ ρ ρ ρ ρρ ρ

, , , , ,

, , , , ,

, , , , ,

, , ρρ ρ ρρ ρ ρ ρ ρρ ρ ρ ρ ρ

120 0

211 0

221 0

112 0

212 0

111 0

211 0

120 0

122 0

222 0

121 0

221 0

120 0

1

1

1

, , ,

, , , , ,

, , , , ,

( , , , , , )F F F F F F11

21

1 2 11

21t t t t t t+ + − −

v = ′( )v v v v01 02 11 12

F F F F Fc mat t t t tv v v v, = + + +− −

01 1 02 2 11 11

12 21

(16)

And the covariance is

(17)

Combining Equations (16) and (17) yields the serialautocorrelation of :

(18)

For a given factor correlation matrix C, the auto-correlation is an analytic function of the weight vector v.

OPTIMAL ALPHA MODEL WITH INCLUSIVE TURNOVER CONSTRAINTS

We have laid the groundwork for building optimalalpha models that explicitly consider turnover constraints,using multifactor linear models that include both currentand lagged factors. The autocorrelation of the model setsthe constraint on portfolio turnover while the model’s IRis optimized according to the average ICs and covariancesof ICs, based on the framework developed first in Qianand Hua [2004] and extended in Sorensen at al. [2004].

The key insight from this constrained optimization isthe optimal use of lagged forecasts as part of the compositealpha model even if the lagged ICs are sometimes weakerthan the contemporaneous ICs. Including lagged forecastsincreases composite forecast autocorrelation, and thus reducesportfolio turnover, giving rise to savings in transaction costs.The equilibrium trade-off between the lagged ICs and theforecast autocorrelations determines the optimal modelweights in the current as well as the lagged factors.

Constrained Optimization

Continuing with the case of two factors and onelag, Equation (13) describes an alpha model, and we areinterested in the model weights that maximize the IR while controlling the turnover. Theautocorrelation of the composite is given by Equation (18)and the matrices C4 and D4 are submatrices of C, givenby Equation (15). As turnover is a function of , wethus constrain the autocorrelation to be at a targeted value.

ρ f c ma,

v = ′( )v v v v01 02 11 12

ρ f

c mat

c mat

c matc ma,

cov ,

var

, ,

,

=( )

( ) = ′ ⋅ ⋅′ ⋅ ⋅

−F F

F

v D v

v C v

1

4

4

Fc mat,

cov ,, ,F F v D vc mat

c mat−( ) = ′ ⋅ ⋅1

4

var ,F v C vc mat( ) = ′ ⋅ ⋅4


(14)

Copyright © 2007

The IR of the alpha model is approximately the ratioof average IC to the standard deviation of IC (Qian and Hua[2004]). Denote the average IC of by and the IC covariance matrix by ΣIC. The average IC ofthe model is , and the standard deviation of IC is

. The constrained optimization problem is:

(19)

The target autocorrelation is denoted by ρt. Theautocorrelation constraint is quadratic in nature. Thisimplies that (19) is a non-linear optimization with a qua-dratic constraint for which no analytic solution is readilyavailable. It is easy to solve the problem numerically, as inthe example below. We also note that the problem can beeasily extended to include more factors and multiple lags.

Example—Inputs

We describe a numerical exampleusing the two factors, PM and E2P. To makethe example more realistic, we considermodels with three lags.

First, we consider the IC inputs asso-ciated with the factors in Exhibits 1 and 2.These are quarterly data from the Russell3000 universe for 1987-2004, and includethe average ICs and standard deviations of ICfor the two factors and their lagged factors.Thus, with two factors and three lags, wehave eight different sources of alpha. Next,we compute the IR of the composite modelusing factor IC correlations (Exhibit 9). Thesubscripted numbers denote lags.

The most notable feature of Exhibit 9is the negative IC correlation between thetwo factors.8 For instance, the ICs of PM_0and E2P_0 have a correlation of −0.42, indi-cating significant diversification benefit. Thatis, when the PM factor has a higher ICadding more alpha, the E2P factor tends tohave a lower IC adding less alpha.

The diversification carries over to ICsof the lagged forecasts. For example, the ICs

Maximize: IR

subject to:

= ′ ⋅′ ⋅ ⋅

= ′ ⋅ ⋅′ ⋅ ⋅

=

v IC

v v

v D v

v C v

ΣIC

f tc maρ ρ

,

4

4

′ ⋅ ⋅v vΣIC

′ ⋅v IC

IC( , , , )F F F F1 2 11

21t t t t− −

of PM_1 and E2P_1 have a correlation of −0.45, and theICs of PM_0 and E2P_1 have a correlation of −0.37.

We also should note that the IC correlations amongthe same factors but of different lags tend to be high, indi-cating less diversification of information, although thecorrelation drops with an increasing time period betweenforecasts. For instance, for the PM factor, the correlationis 0.86 between PM_0 and PM_1, 0.78 between PM_0and PM_2, and 0.61 between PM_0 and PM_3. For thevalue factor, the correlations are even higher: 0.92 betweenE2P_0 and E2P_1, 0.84 between E2P_0 and E2P_2, and0.78 between E2P_0 and E2P_3.

These IC correlations determine the compositealpha model’s strategy risk. The factor correlations deter-mine the alpha model autocorrelation.

Exhibit 10 presents the average factor correlationsbetween factors of different lags that are needed to deter-mine the forecast autocorrelation. Notice there are fourlags in Exhibit 10, instead of the three lags in Exhibit 9.This is because we need to consider serial autocorrelation


E X H I B I T 1 0Factor Correlation Matrix of Current and Lagged Values for Momentum and Value Factors

E X H I B I T 9IC Correlation Matrix of Current and Lagged Values for Momentum and Value Factors

Copyright © 2007

(with one lag) of forecasts that are made offactors of three lags.

Note that correlations among the samefactors are high, for E2P in particular. Thisis not surprising, as earnings yields or PEmultiples change slowly. Essentially, highserial autocorrelation of value factors is con-sistent with their minimal information decay.

The factor correlations are much lowerfor the PM factor; the lag 1 correlation is0.68; the lag 2 correlation is 0.40; and thelag 3 and lag 4 correlations drop nearly tozero. This implies that winners and losersdefined by price momentum change drasti-cally over time—winners today have littleresemblance to the winners nine monthsearlier. Thus, the construction process willincur more turnover in maintainingmomentum exposure as the PM factor updates frequently.

Finally, note that the correlations between PM andE2P of different lags are low and not quite as negative asthe IC correlations.

Optimal Alpha Models— Results and Insights

Given the inputs, we solve the optimization problemin Equation (19) for a series of forecast autocorrelation tar-gets ρt, ranging from 0.85 to 0.97. The optimal modelweights for each autocorrelation target, together with thecorresponding IR, are presented in Exhibit 11.

First, notice that as ρt goes from 0.85 to 0.97, theoptimal IR increases from 2.30 to 2.39 when ρt is 0.89and then drops to 1.88 when ρt reaches 0.97. The highestIR occurs when the optimal weights are 36% for PM_0and 64% for E2P_0, with no lagged factors. This repre-sents the unconstrained optimal alpha model using thecurrent factors only.

Second, we can see that for targeted autocorrela-tion above 0.89, the optimal model begins to add weightto lagged factors, which will slow turnover—but this isat the expense of the IR. PM_1 is the first to appear, andit is followed by E2P_1, E2P_2, and E2P_3. The othertwo lagged momentum factors, PM_2 and PM_3, neverattain any significant weight in the model. The reason fortheir absence is both PM_2 and PM_3 have very lowaverage ICs. All E2P factors, however, have consistentICs and high autocorrelation. Their optimal weights near10% each as the target autocorrelation gets very high.

Finally, we see that the weight increase in the lagged fac-tors comes first at the expense of the current momentumfactor PM_0 and then the current value factor E2P_0.

To understand the quantitative trade-off betweenIR and turnover, consider a comparison. The maximumIR 2.39 occurs when the forecast autocorrelation ρt is at0.89, and the model IR drops to 2.33 when ρf is at 0.93.Hence, the IR drops by roughly 2.5%. At the same time, the turnover changes according to the function

, dropping from to , or by asmuch as 20%.

In Exhibit 12 we aggregate optimal weights intoweights for PM and E2P, and into weights for factors oflags 0, 1, 2, and 3. As ρt increases from 0.85 to 0.97, thePM weight drops from 45% to 28%, while the E2P

1 0 93− .1 0 89− .1 − ρ f


E X H I B I T 1 1Optimal Weights for Different Levels of Autocorrelation and Optimal IR

E X H I B I T 1 2Aggregated Optimal Weights with Autocorrelation Targets and Associated IRs

Copyright © 2007

weight increases from 55% to 72%. At the same time,the weight with lagged factors increases from 0% to 58%,offset by reductions in the weights for zero-lagged factors.

Transaction Costs and Net Returns

One last piece to the puzzle is an assumption fortrading costs. First, we measure the effect of autocorre-lation on turnover by calculating turnover according toEquation (9), on an annual basis for a long-short portfolio.The inputs are N = 3,000, target risk σmodel = 4%, andstock-specific risk σ0 = 30%. The turnover results and theIR are graphed in Exhibit 13.

First note the extremely high turnover when auto-correlation is low; it is nearly 550% when ρt is 0.89.9 Butthe most important feature of the graph is that the rateof decline is markedly different for the IR and theturnover, as the autocorrelation ρt increases. While theturnover drops consistently, the IR changes rather slowlyexcept when the autocorrelation reaches a very high level.Since the turnover drops more rapidly than the IR overa wide range, it is entirely feasible that net expectedreturn—expected return less transaction costs—is higherfor alpha models more highly autocorrelated with laggedfactors.

We compute net expected return by imposing dif-ferent levels of transaction costs that represent simply alinear proportion of the portfolio turnover. For example,at 50 basis points or 0.5%, a turnover of 100% would cost0.5% and a turnover of 200% would cost 1%, and so on.With increasing assets under management, one can use ahigher multiple of turnover as transaction costs.

Exhibit 14 shows the gross return given by IR timesthe target tracking error, turnover, and net returns withdifferent transaction assumptions. We also include trans-action costs of 1.0% and 1.5%.

As expected, the gross return is maximized at ρf =0.89, where the IR reaches its maximum. But the netreturn attains its maximum at a higher factor model auto-correlation ρf , corresponding to a different alpha modelwith lagged factors. When the cost of 100% turnover is0.5%, the maximum net return is at ρf = 0.93. In thisinstance, the IR is 2.33, but the turnover drops to 436%from 547%. When the transaction cost is higher at 1.0%,the maximum net return is at ρf = 0.95 where the paperIR is 2.21 and the expected net return is higher than themodel with ρf = 0.89, by over 1.0%. At the even highertransaction cost of 1.5% for 100% turnover, the optimal

model for net return would be at ρf = 0.96. Alpha modelswith these autocorrelation targets include significantweights of lagged factors (see Exhibits 11 and 12).

In Exhibit 15 we plot the data in Exhibit 14. Thesquare on each curve denotes the model with the maximumnet return. As the transaction costs increase, the net returngets lower and lower. This is especially true for the leftsides of the return curves, due to higher turnover. Theright side of the curves drops to a lesser extent becausethe turnover is lower (higher model autocorrelation). Oneach cost curve, the point of maximum net return shiftsto the right as the autocorrelation increases. We also note


E X H I B I T 1 3IR and Portfolio Turnover of Optimal Alpha Modelswith Given Forecast Autocorrelation

IR scale is on the left axis, and turnover scale is on the right axis.

E X H I B I T 1 4Gross Excess Return and Net Excess Return under Different Transaction Cost Assumptions

Portfolios with N = 3,000, target risk σmodel = 4%, and stock-specific risk σ0 = 30%.

Copyright © 2007

we should build optimal alpha models that inte-grate transaction costs in the modeling process.We provide an analytic framework to do so bymaximizing the information ratio of alphamodels under portfolio turnover constraints.

The solution we provide is quite gen-eral. We show that portfolio turnover is analgebraic function of forecast autocorrelation.Hence, factor autocorrelation is a key diag-nostic in evaluating single-factor efficacy andin creating optimal composite models. Forlinear composite forecasts, the autocorrelationdepends on auto- and cross-factor correlations.Optimal models by definition must dependon factor correlations for contemporaneousand lagged values as well as average ICs, stan-dard deviation of ICs, and IC correlations.

These results should be useful to prac-titioners in several ways. First, autocorrela-

tion is an important diagnostic for evaluating factors.Second, comprehensive alpha models necessitate the directintegration of transaction costs. Third, the analyses herelead directly to reasonable estimations of strategy capacityassociated with increasing assets under management levels.They suggest adjusting weights in favor of lagged factorsto reduce turnover by increasing the target autocorrela-tion, thus maximizing the net return.

APPENDIX

Portfolio Turnover

In this appendix we derive the theoretical solution of port-folio turnover. We first rewrite Equation (7) as an expectation:

(A-1)

To evaluate the expectation, we further assume that thechange in the risk-adjusted forecast and the stock-specific riskare independent. Hence, Equation (A-1) can be written as:

(A-2)TN

F Ft t= −

+σσ

model E(| |)E2

11˜ ˜

TN

N

F F

N F F

it

it

ii

N

t t

=−

=−

+

=

+

∑σσ

σσ

model

model E

21

2

1

1

1

˜ ˜

˜ ˜


E X H I B I T 1 5Gross Excess Return and Net Excess Return under Different Transaction Cost Assumptions

that when transaction costs are high, there is a more rapidincrement in net return as autocorrelation increases—optimal models with high autocorrelation have more ofa chance to yield positive net returns as trading costs rise.

These results have strong implications for the con-struction of optimal alpha models. First, using the modelwith maximum gross IR can be suboptimal in terms ofnet return when transaction costs are taken into account.For example, with 1% cost, the net return of that modelis lower than that of the optimal model with ρf = 0.95,by more than 1 percentage point.

Second, the optimal model in terms of highest netreturn changes as transaction costs rise. This indicates theneed to adjust the alpha model as assets under managementgrow. Our results reveal that one way to do this is to addlagged factors based on the risk/return/turnover trade-off.

Third, both the net return and the optimal model aresensitive to the IR assumption. If the IRs are lower thanthose in the example, then for a given level of transactioncosts, the maximum net return would be achieved witheven higher ρf models. In other words, when the informa-tion content of the factors is lower, we need to pay even moreattention to constrain portfolio turnover to reduce transac-tion costs. This inevitably leads to more weight for the laggedfactors, especially lagged value factors.10

SUMMARY

Realistic alpha models necessitate the direct endoge-nous analysis of implementation costs. This means that

Portfolios with N = 3,000, target risk σmodel = 4%, and stock-specific risk σ0 = 30%.

Copyright © 2007

Both sets of forecasts have a standard deviation of one. We fur-ther assume they form a bivariate normal distribution withmean zero, and the cross-sectional correlation between the twosets of consecutive forecasts is ρf. This is simply the lag 1 auto-correlation of the risk-adjusted forecasts. If the correlation ishigh, the change in forecasts is minimal, and the turnover islow. Conversely, if the correlation is low, the forecast change issignificant, and turnover will be high.

Since both forecasts are normally distributed, the changeis still a normal distribution with zero mean and

standard deviation . For a random variable x withdistribution , we can show that:

Therefore:

(A-3)

Substituting Equation (A-3) into (A-2) yields:

(A-4)

Since targeted tracking error is linked to the portfolioleverage, we can derive a relation between leverage and forecast-induced turnover, using expectations. We have:

(A-5)

It should be apparent from Equation (A-5) that the leverageof the portfolio should remain the same if the targeted trackingerror and the specific risks remain constant.

Since is a standard normal variable, we have. Using this result in Equation (A-5) yields:E(| |)˜ /F t = 2 π

F̃ t

L wN

F

NF

N F

ii

Nit

ii

N

t

t

= =−

=

= ( )

= =∑ ∑

1 11

1

σσ

σσ

σσ

model

model

model

E

E E

˜

˜

˜

T f= −

NE

1modelπ

σ ρσ

1

E(| |)˜ ˜F Ft t f+ − =−

12 1 ρ

π

E(| |)x d= 2π

x N d⋅ ( , )0 2

d f= −2 1( )ρ˜ ˜F Ft t+ −1

(A-6)

Combining (A-6) and (A-4) yields:

(A-7)

Therefore, turnover is directly proportional to leveragetimes the square root of 1 minus forecast autocorrelation.

ENDNOTES

1We assume that on an aggregated portfolio level theimplementation cost is proportional to the amount of trading,or portfolio turnover. The majority of implementation cost isrelated to trading. These costs could be commission, bid-askspread, and market impact. The trading cost may vary fromstock to stock. For an approximation, we simply assume thattrading cost is a fixed multiple of portfolio turnover.

2Turnover can also be caused by flows in and out of port-folios. These forced turnovers are not due to portfolio rebal-ancing, and they are easy to analyze. We ignore them in ouranalysis.

3If there is short-term reversion between consecutive-period returns, the horizon IC will be higher.

4See Qian and Hua [2004] for a development of the IRexpression: .

5Our definition of turnover measures the percentagechange of the portfolio versus total portfolio capital, which ismost relevant in terms of amount of trading and costs. Thereare other variations that use total portfolio leverage or nota-tional exposures as denominators, which tend to reduce theturnover percentage.

6For constrained portfolios such as long-only portfolios,turnover can be substantially less, since constraints work to sup-press changes in portfolio weights (see Qian, Hua, and Tilney[2004]). Turnover can also be reduced through other meanssuch as cutting weights proportionally or ignoring small trades.For details, see Grinold and Stuckelman [1993] and Qian, Hua,and Tilney [2004].

7Our analysis shows that the inclusion of a lagged fore-cast would increase the serial autocorrelation as long as auto-correlation of lag 2 is above a certain threshold. The value ofthe threshold is given by two times the lag 1 correlation squaredminus one. These values are easily exceeded for most factorsencountered in practice.

8 While factor correlation has some bearing on the cor-responding IC correlation, the two are not the same. In building

IR Expected IC IC)≈ / (σ

TL f=

−1

2

ρ

LN=

2 1π

σσmodelE


Copyright © 2007

an alpha model, it is the IC correlation not the factor correlationthat plays a crucial role in determining the weights of factors.

9In practice, the actual turnover will be lower when thetransactions are incorporated into the portfolio optimization.

10It is not hard to see this situation might apply to marketsegments that are relatively less inefficient, such as U.S. large-cap stocks.

REFERENCES

Grinold, Richard C., and M. Stuckelman. “The Value-Added/Turnover Frontier.” The Journal of Portfolio Management,vol. 19, no. 4 (Summer 1993), pp. 8-17.

Kahn, R. N., and J. S. Shaffer. “The Surprisingly Small Impactof Asset Growth on Expected Alpha.” The Journal of PortfolioManagement, vol. 32, no. 1 (Fall 2005), pp. 49-60.

Qian, Edward, and Ronald Hua. “Active Risk and Informa-tion Ratio.” The Journal of Investment Management, vol. 2, no. 3(2004), pp. 20-34.

Qian, Edward, Ronald Hua, and John Tilney. “Turnover ofQuantitatively Managed Portfolios.” Working paper, PanAgoraAsset Management, 2004.

Sorensen, Eric H., Ronald Hua, Edward Qian, and RobertSchoen. “Multiple Alpha Sources and Active Management.”The Journal of Portfolio Management, vol. 31, no. 2 (Winter 2004),pp. 39-45.

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


Copyright © 2007

Information Horizon, Portfolio Turnover, and Optimal Alpha...

Documents

Transcript of Information Horizon, Portfolio Turnover, and Optimal Alpha...