a structural model of dynamic market timing · a structural model of dynamic market timing jérôme...

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A Structural Model of Dynamic MarketTiming

Jérôme DetempleBoston University School of Management

Marcel RindisbacherBoston University School of Management

This paper derives and analyzes dynamic timing strategies of a fund manager with privateinformation. Endogenous timing strategies generated by various information structures andskills, and associated fund styles, are identified. Endogenous fund returns are characterizedin the public information of an uninformed observer. Timing components are identified.The paper provides foundations for regression analyses of fund returns and tests of markettiming. (JEL G11)

This article develops and analyzes a structural dynamic model of markettiming. Endogenous market-timing strategies of a fund manager endowed withprivate information are derived. The endogenous excess return of the managedfund is characterized and examined. From the point of view of an observerwith public information, (noisy) market timing adds three terms to excessreturns: a time aggregation component, a timing claim component, and an error.Explicit formulas are obtained for the timing-induced components. The analysisprovides foundations for regression models of fund returns and associated testsof market timing.

The evaluation of the performance of managed funds has a long history.Early studies by Treynor and Mazuy (1966) and Henriksson and Merton (1981)postulate that professionally managed funds generate returns with additionalnonlinear components reflecting the timing skills of the manager. Theoreticaljustifications were proposed by Merton (1981). His study shows that market-timing ability—that is, the ability to extract information about future marketreturns—explains the emergence of nonlinear components with option-like

The paper was presented at HEC Montreal, BI Norwegian Business School, Boston University, the EPFL-Swissquote Conference, Université de Genêve, UNC Charlotte, Rutgers University, Georgia State University,Brown University, EDHEC Singapore, the Columbia Conference on Probabity, Control and Finance, the PrincetonConference on Risk Measurement, FIRS 2012, SMU-ESSEC 2012, BFS 2012, EUROFIDAI-AFFI 2012 andAQF2013. We thank Felix Kübler and Stephan Jank for discussing early versions of the paper and Yacine Ait-Sahalia,Wolfgang Buehler, Laurent Calvet, Peter Carr, Harald Hau, Olivier Scaillet, and Jialin Wu for their comments. Wewould also like to thank two referees and the editor, Pietro Veronesi, for insightful and constructive comments.Send correspondence to Marcel Rindisbacher, Boston University School of Management, 595 CommonwealthAve, Boston, MA 02215, USA; telephone: (617) 353-4152. E-mail: [email protected].

© The Author 2013. Published by Oxford University Press on behalf of The Society for Financial Studies.All rights reserved. For Permissions, please e-mail: [email protected]:10.1093/rfs/hht028

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The Review of Financial Studies / v 0 n 0 2013

structures in fund returns. The failure of classical performance measures inthe presence of private information is further discussed in Dybvig and Ross(1985). The distinction between timing and selectivity is examined, from botha theory and an econometric perspective, by Admati et al. (1986). A procedurefor performance evaluation, which relies on the approximation of a fund’spayoff by a series of options, is developed by Glosten and Jagannathan (1994).Scores of empirical studies have examined managed portfolio returns (Lee andRahman 1990; Fung and Hsieh 1997, 2000, 2001;Agarwal and Naik 2004; Chanet al. 2006; Chen 2007; Chen and Liang 2007; Bilio, Getmansky, and Pelizzon2012). Evidence of timing behavior has been found in individual funds aswell as specific categories of funds (fund styles). Over time, the sophisticationof the return models estimated has increased. Time-varying betas have beenaccounted for (Ferson and Schadt 1996). The effects of discrepancies betweentiming and observation windows have also been examined (Ferson and Khang2002; Goetzmann, Ingersoll, and Ivkovic 2002; Ferson, Henry, and Kisgen2006). Corrections for interim trading biases, based on high-frequency data,have been proposed (Patton and Ramadorai 2012). For the most part, theoreticalexplanations of timing and the structure of models tested rely on static analysesof the market-timing problem. Recent exceptions are Koijen (2010), who testsfor selectivity skill in a dynamic model with constant selectivity alpha, andCvitanic et al. (2006), who consider Bayesian learning about alpha.

The typical timing regression model developed and applied in the literaturetakes the form

rp

t+1 =α+βrmt+1 +ft+1 +εt+1, (1)

where rpt+1 is the excess return on a managed portfolio at time t +1, rmt+1 is theexcess return on the market portfolio, ft+1 is a random variable depending onthe market excess return, and εt+1 is an independent error term. The coefficientαrepresents an abnormal excess return component, often interpreted as a rewardfor selection skill. The coefficient β is the exposure to market risk (systematicrisk). The random variable ft+1 is the reward for timing skill, called the timingoption. Various parametrizations of ft+1 have been studied. Popular formsinclude

ft+1

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩γ(rmt+1

)2Treynor and Mazuy (1966)

γmax{rmt+1,0

}Henriksson and Merton (1981)∑K

i=1γimax{rmt+1 −κi ,0

}Glosten and Jagannathan (1994)

γ(∏J

j=1 max{

1+Rmtj ,1+Rftj

}−(

1+rmt+1

))Goetzmann, Ingersoll, and Ivkovic (2002)

(2)

where γ,γi,κi are constant parameters. For these specifications, the timingoption has a nonlinear dependence on the market excess return. Nonlinearitytakes the form of a quadratic function or a piecewise linear function such as a call

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A Structural Model of Dynamic Market Timing

payoff function. In the last specification the timing option is nonlinear and path-dependent. It compounds the maximum of the gross market and risk-free returnsover a partition

{tj :j =1,...,J

}of the time interval [t,t +1]. Regressions such

as (1)–(2) are run. Evidence of timing skill is reported when estimates of thetiming coefficients γ,γi are found to be significantly different from zero.

This paper develops a dynamic theory of market timing that seeks toendogenize the regression model (1)–(2) linking fund excess returns to marketexcess returns. In essence, it seeks to develop the foundations for regressionmodels and tests of market timing.

For this purpose, a theoretical model of dynamic market timing is solved andexamined. The informed manager, the market timer, is endowed with privateinformation about future market returns—that is, anticipative information.Private information is noisy, but it enhances the public information set. Thestructure of the private signal is general, leading to conditional distributions thatcan be outside the Gaussian class. The market timer selects an optimal dynamicinvestment policy reflecting the private signal obtained. The optimal policy isidentified and the endogenous fund return derived. From the perspective ofan outside investor with public information, who observes fund returns ata lower frequency, the fund excess return has three nonlinear componentsrelated to market timing.1 The first component is a time aggregation factorψ due to the computation of returns at the coarser observation frequency. Itconstitutes a part of the intercept. The second component is a timing claim f

generated by the processing and use of anticipative information for dynamictrading purposes. This timing claim is intrinsically path-dependent. The lastcomponent is an error term ε, orthogonal to public information, due to theindependent noise that affects the anticipative signal. Special cases of themodel are studied to understand the return structures associated with particularinformation extraction skills. Managers adept at generating information aboutexcess return levels (noisy return forecast) and about the direction of themarket (directional forecast) are analyzed. Additional issues, such as theability to distinguish between asset selection and market timing, are alsoexamined. Finally, extensions of the model are developed, to deal with variousapplications including equilibrium with asymmetric information and volatilitytiming.

The article is organized as follows. Section 1 presents and solves the market-timing problem of an informed manager. It also derives the endogenousmanaged fund excess return in the private and public information. Specificapplications are studied in Section 2. Section 3 examines the difference betweenselection and market timing. Extensions of the basic model are carried out inSection 4. Conclusions are in Section 5. Appendix A provides an introduction

1 The fund excess return is measured by the log of the gross excess return.

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to forward integration. Appendix B presents an extension to constant relativerisk aversion. Proofs are collected in Appendix C.

1. Optimal Timing Policies with Anticipative Information

This section develops a dynamic model of market timing and derives theendogenous fund return. The private information structure of the fund manageris described in Section 1.1. The optimal portfolio with private informationand the associated fund return are derived in Section 1.2. Section 1.3 studiesthe private information price of risk and the value of information. Section 1.4examines the structure of the fund return in the public information filtration.Section 1.5 highlights the relation between timing skill and second-orderstochastic dominance.

1.1 A model of timing informationConsider a fund manager with private information about future equity returns.Information is represented by the filtration

G(·) =Fm(·)∨

FY(·) (3)

where Fm(·) is the public information generated by market excess returns dRm,

and FY(·) is the filtration generated by a private signal Y . The instantaneous

market excess return (dRmv ) and the gross market excess return (Smτi−1,τi) over

a period [τi−1,τi) are given by

dRmv =σmv(θmv dv+dWm

v

)and Smτi−1,τi

=exp

(∫ τi

τi−1

dRmv − 1

2

∫ τi

τi−1

(σmv

)2dv

)(4)

where σmv θmv =μmv −rv represents the expected instantaneous excess return. The

interest rate r and the volatility coefficient σm are stochastic processes adaptedto public information. The volatility σm is also positive and bounded awayfrom zero. The private signal informs about future market returns. It takes thegeneral anticipative form

Yv≡N∑i=1

Gi1[τi−1,τi) (v), (5)

where τi is a sequence of deterministic times with τ0 =0, and τN ≤T , the process1[τi−1,τi) (v) is the indicator of v∈ [τi−1,τi), and

Gi≡g(Smτi−1,τi

,ζi

)(6)

for some function g and sequence of random variables ζi independent fromFmT .2 The times {τi : i =0,...,N−1} are times of new information arrival. The

2 The noise ζi(Wζ

)is a functional of Wζ , a Brownian motion independent from Wm.

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time elapsed between two information signals, hi≡τi−τi−1, is the interarrival

time. At time τi , the signal Gi+1 ≡g(Smτi ,τi+1

,ζi+1

)is observed, providing

anticipative information about the market gross excess return over the up-coming period [τi,τi+1). Information is noisy due to the presence of theindependent random variable ζi+1. The information in Gi+1 remains valuableat all times s∈ [τi,τi+1), even if s is very close to τi+1. At the next arrival time

τi+1, a new signal Gi+2 ≡g(Smτi+1,τi+2

,ζi+2

)materializes, providing valuable

information about gross excess returns at the future time τi+2. The informationadvantage is therefore renewed. This information arrival process repeats itselfaccording to the information clock {τi : i =0,...,N−1}. Renewal ensures thatthe private information filtration is always more informative than the publicfiltration conveyed by market excess returns, G(·) ⊃Fm

(·).Actively managed funds report realized returns on a periodic basis. Their

reporting schedule is typically coarse and allows fund managers to collectand exploit private information signals in between reporting points. In order tocapture this aspect, it will be assumed that the reporting dates are τ0 and τN . Thepublic information at time v∈ [τ0,τN ) consists of realized market excess returns(Fm

v ) and previously reported managed fund excess returns (Faτ0

). Thus, Fv =Fmv ⊗Fa

τ0. At the reporting date τN , the public information is FτN =Fm

τN⊗Fa

τN.

Modeling private information as a signal about gross excess returns places thefocus on the relative performances of equities and money market instruments.It should also be noted that the information structure described above differsfrom the typical setting considered in the mathematical finance literature. Thepaper by Pikovsky and Karatzas (1996) is the first one to consider the dynamicproblem of an informed investor with anticipating information. They assumelogarithmic utility function. In their model, the investor receives a single initialsignal about the future state, and the information conveyed can be perfector noisy. The literature that follows uses the same assumption (Biagini andOksendal 2005; Kohatsu-Higa and Sulem 2006). The present paper generalizesthis standard setting by considering a sequence of signals collected by theinformed manager at a set of deterministic information arrival times. Eachsignal is valuable only during the period following its arrival. When a newsignal is received, the previous one becomes obsolete. But, even though theinformed fund receives several signals, the public observes the fund returnsonly at the initial and the terminal reporting dates. As a result, outside investors(the public) cannot learn from the fund trades.

1.2 Information premium and optimal timing strategiesAnticipating information changes the price of risk, which captures theinstantaneous reward for risk. A publicly informed investor has a premiumper-unit risk given by the market price of risk θm. Private information givesinsights about upcoming events affecting returns. It changes the reward for risk

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by the private information price of risk (PIPR),

θGv ≡ 1

σmvlimε↓0

1

εE[Rmv,v+ε

∣∣Gv]−θmv , (7)

which represents the premium per-unit risk due to private information. Inthis expression Rmv,v+ε =

∫ v+εv

dRmv is the cumulative excess return over thetime interval [v,v+ε]. The informed investor’s price of risk is the (public)market price of risk augmented by the PIPR, θmv +θG

v . In the absence of privateinformation, when G(·) =Fm

(·), the price of information risk is null because

E[Rmv,v+ε

∣∣Gv]=E[Rmv,v+ε

∣∣Fmv

]and limε↓0

1εE[Rmv,v+ε

∣∣Fmv

]=σmv θ

mv . With per-

fect foresight, when Gτi =Fmτi+1

, it explodes because limε↓01εE[Rmv,v+ε

∣∣Fmτi+1

]=

σmv θmv +limε↓0

1ε

∫ v+εv

σms dWms =±∞ for any v∈ [τi,τi+1).3 There is an arbitrage

opportunity. With noisy private information, when Fmv ⊂Gv⊂Fm

τi+1for all

v∈ [τi,τi+1) and i =0,...,N−1, the PIPR is finite and there are no arbitrageopportunities for the privately informed fund manager.

Consider now a fund manager with private information G(·) and logarithmicutility function, acting as a price taker.4 The optimal portfolio and induced fundexcess return are described next.5

Proposition 1. Let G(·) =Fm(·)∨FY

(·) be the private information filtrationgenerated by market returns and the private observation process Y in (5). Theoptimal informed investment policy and the associated fund excess return are6

πav =1

σmv

(θmv +θG

v

)and (8)

d−Rav =1

σmv

(θmv +θG

v

)d−Rmv , (9)

where θGv is the PIPR (7). An informed timer with positive (negative) PIPR

invests more (less) in the market. The volatility coefficient of the optimallymanaged fund return then increases (decreases) relative to a similar fund undermanagement without timing skill.

3 For ε small,∫ v+εv

(σms

)2ds =

(σmv

)2ε+o(1). The scaling property of Brownian motion then implies

limε↓0

∣∣∣ 1ε

∫ v+εv σms dW

ms

∣∣∣=limε↓0

∣∣∣ 1√εWm

1 σmv

∣∣∣=+∞.

4 Logarithmic utility is not essential. Appendix B studies the case of a manager with constant relative risk aversion.

5 Pikovsky and Karatzas (1996) derive the optimal policy (8) of a logarithmic investor with private information,in a simpler informational setting with a single initial signal. Their result relies on an enlargement of filtration.The approach underlying Proposition 1 is based on the modeling of gains from trade through forward integrals.

6 The notation d−Rmv in (9) stands for the forward differential (see Russo and Vallois 1993). Forward integrationis the natural concept to model gains from trade and stochastic integrals when information anticipates the future.See Appendix A for more details.

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The fund manager with logarithmic utility function displays myopia. Theoptimal portfolio policy (8) has a mean-variance structure, but relative to theinformation possessed. Private information changes the risk premium per-unitrisk from θm to θm+θG , and this becomes the basis for the manager’s portfolioselection. In the absence of true timing ability, G(·) =Fm

(·), leading to a null PIPRand an optimal portfolio equal to that of an investor with public information.Private information can lead to a positive or a negative component θG

v . When thefuture return innovation is expected to be sufficiently large given the privateinformation signal, the PIPR is positive. The privately informed agent withpositive (negative) PIPR will then naturally invest more (less) than an individualrelying solely on public information. A decomposition of the portfolio canbe made to highlight the two informational motives driving investments. Thecomponent motivated by public information corresponds to θmv /σ

mv . The one

motivated by private information is θGv /σ

mv .

The instantaneous excess return process generated by the fund manageris described in (9). The volatility of the fund return reflects the strategypursued. When private information induces a positive PIPR, the volatilitycoefficient of the fund return increases. Volatility reflects the nature of theinformation collected. It does not reveal private information to the publicbecause fund returns are observed only at the reporting dates τ0,τN , and fundtrades are unobserved. As for the portfolio policy, the fund excess return hastwo components. The component generated by public information trades is(θmv /σ

mv

)d−Rmv . The excess return component due to private information is(

θGv /σ

mv

)d−Rmv .

1.3 Private information price of risk and value of informationThe PIPR θG

v determines the optimal active portfolio of the skilled investor. Thenext proposition relates it to the density of the signal generating the anticipativeinformation.

Proposition 2. Let pG(·) (z)≡{pGv (z) :v∈ [τi−1,τi)

}be the conditional density

process of the private signal Gi given public information. The PIPR is theinstantaneous covariance between the conditional density process and theinnovation in the market return

θGv =

(d[logpG (z),Wm

]v

dv

)|z=Gi

=Dmv logpGv (z)|z=Gi (10)

for v∈ [τi−1,τi), where Dmv is the Malliavin derivative relative to Wm. The

expected PIPR with respect to public information is null (Ev[θGv+h

]=0 for all

h≥0).

Formula (10) also shows that the PIPR corresponds to the instantaneousvolatility coefficient of the density of the information signal with respectto market return innovations. This parallels the standard property of the

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market price of risk, which is the negative of the volatility coefficient ofthe risk-neutral density. The private information price of risk vanishes ifand only if the conditional density of the signal does not covary with thepublic information source (d

[logpG (z),Wm

]=0). A signal of this sort has a

conditionally independent density and is noninformative about the fundamentalsource of uncertainty driving excess returns. The lack of informational contentleaves the perceived risk-return tradeoff unaltered.

The martingale property relative to public information is intriguing. It reflectsa no-arbitrage condition satisfied by returns, under public information. Theexcess return at time s =v+h under private information is

dRms =σms((θms +θG

s

)ds+dWm,G

s

),

where E[dWm,G

s

∣∣Gs]=0, by the properties of Brownian motion.7 Taking theconditional expectation relative to public information givesEs

[dWm,G

s

]=0, so

thatEs[dRms

]=σms

(θms +Es

[θGs

])ds. IfEs

[θGs

] �=0, the market price of risk forthe uninformed differs from θmv , in contradiction with the initial assumption.In that case, the uninformed public perceives the active fund as being able togenerate systematic arbitrage opportunities, which cannot be. Thus,Es

[θGs

]=0.

Projecting on public information prior to time s then shows that Ev[θGs

]=0.

It is also useful to examine the relation between the value of information andthe PIPR. The (ex ante) value of private information received at time τi−1 canbe measured by the ratio IG

i−1 of the certainty equivalent achieved with privateinformation to the certainty equivalent without the information. It is describednext.

Proposition 3. The value of the private information signal Gi is

IGi−1 =exp

(1

2

∫ τi

τi−1

Eτi−1

[(θGv

)2]dv

)=exp

(Eτi−1

[DKL

(pGτi (z)

∣∣∣pGτi−1(z))])(11)

where DKL

(pGτi (z)

∣∣∣pGτi−1(z))≡Eτi

[log

pGτi(Gi )

pGτi−1 (Gi )

]=∫

Rlog

pGτi(z)

pGτi−1 (z)dPGτi (z) is

the relative entropy of the signal. A signal has no value if and only if thePIPR is null. The value of the information structure {Gi : i =1,...,N} at the

beginning of the interval [0,τN )=⋃Ni=1[τi−1,τi) is IG

0,τN=E

[∏Ni=1I

Gi−1

].

The value of information (11) is an increasing function of the absolute valueof the PIPR. In the absence of trading constraints, the increased size of thePIPR effectively represents an enhanced opportunity set for the informed.

7 The process Wm,G , defined by dWms ≡θGs ds+dWm,G

s , is a Brownian motion relative to the private filtrationG(·). Note that the process A, which appears in the semimartingale decomposition ofWm provided by Kohatsu-

Higa and Sulem (2006) and Biagini and Oksendal (2005), corresponds to dAs ≡θGs ds. The existence of thesemimartingale decomposition in the enlarged filtration is discussed in Jacod (1979).

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The value function can also be expressed as an increasing function of

the relative entropy DKL

(pGτi (z)

∣∣∣pGτi−1(z))

—that is, the Kullback-Leibler

divergence of pGτi−1(Gi) from pGτi (Gi) (Kullback and Leibler 1951). Relative

entropy measures the information gained by learning from public informationduring the timing interval. If DKL is positive over some event in Fm

τi,

public information leading to that event provides valuable information aboutthe signal. If expected relative entropy increases, the value of informationincreases. When the private signal is unrelated to public information (i.e.,Gi isuninformative), the conditional distributions are the same and relative entropyis null. Neither the private signal nor the public news collected during the timinginterval is a source of private information value.

Certainty equivalents can also be used to measure the losses incurred bypursuing an arbitrary anticipative strategy πv (Gi) instead of the optimal one.The ratio of certainty equivalents in the case of an arbitrary informed policy is,after simplifications,

IG,πi−1 ≡ CEQπ

CEQa=exp

(−1

2

∫ τi

τi−1

Eτi−1

[(θmv +θG

v −πv (Gi)σv)2]dv

).

The gain realized by pursuing the optimal informed policy is related to thesquared difference between the optimal and suboptimal policies. Any deviationfrom optimal behavior results in a loss of value, given by the negative of(πav σv−πv (Gi)σv

)2. The identification of the correct PIPR is therefore critical

for full extraction of the benefits associated with the information collected.

1.4 Endogenous fund return and public informationThe next proposition describes the fund excess return in the public information.

Proposition 4. Let F(·) be the public information filtration generated bymarket returns (Fm

(·)) and by the actively managed fund returns observed at thereporting dates (Fa

τ0for v<τN and Fa

τNfor v=τN ). In the public information

filtration, the excess return of the fund with timing skill is

logSaτ0,τN =−1

2

∫ τN

τ0

δvdv+∫ τN

τ0

βvdRmv − 1

2ψτ0,τN +fτ0,τN +ετ0,τN (12)

where∫ τNτ0δvdv and ψτ0,τN are time aggregation factors, β is the exposure to

market risk associated with public information, fτ0,τN is the cumulative timingpayoff (measured in excess return), and ετ0,τN is an orthogonal noise component.The return components are

δv =(θmv)2, βv =θmv /σ

mv , (13)

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and

ψτ0,τN =∑

i:τi∈(τ0,τN ]

ψi, fτ0,τN =∑

i:τi∈(τ0,τN ]

f mi , ετ0,τN =∑

i:τi∈(τ0,τN ]

(εmi +εai

),

(14)

with

ψi≡∫ τi

τi−1

E[ηGv

∣∣Fτi−]dv (15)

f mi ≡E[∫ τi

τi−1

γ Gv d

−Rmv

∣∣∣∣∣Fτi−

]=∫ τi

τi−1

E[γ Gv

∣∣Fτi−]d−Rmv (16)

εmi ≡∫ τi

τi−1

(γ Gv −E[γ G

v

∣∣Fτi−])d−Rmv , εai ≡−1

2

∫ τi

τi−1

(ηGv −E[ηG

v

∣∣Fτi−])dv

(17)

γ Gv ≡θG

v /σmv , ηG

v =(2θmv +θG

v

)θGv , (18)

where Fτi− ≡ lims↑τi Fs . The temporal aggregation factor has a componentrelated to the market price of equity risk in the public information (δ) anda component due to timing skill (ψτ0,τN ). The timing option (fτ0,τN ) generatedby a skilled market timer is path-dependent and can be positive or negative. Itsvariance increases with the absolute value of the expected PIPR.8

Formula (12), along with (13)–(18), describes the structure of the fund’sexcess return for a publicly informed investor. Trading based on publicinformation gives the optimal portfolio βv =θmv /σ

mv resulting in the first two

components of (12). The first of these terms, −(1/2)∫ τNτ0δvdv, is a quadratic

variation correction due to public information trading. The second term,∫ τNτ0βvdR

mv , is the cumulative gain from trade based on public information.

Private information leads to the next three components. The first, −(1/2)ψτ0,τN ,is a quadratic variation correction due to private information trading. Thesecond, fτ0,τN , is a sum of expectations of returns due to private informationtrading E

[ (θGv /σ

mv

)d−Rmv

∣∣Fτi−]

conditional on the public information priorto time τi . It captures the timing payoff, from the perspective of a publiclyinformed individual, generated by the endogenous timing activity of the fundmanager. Noise in the information signal ensures that private informationtrades are not sure winners and that the timing payoff can take positive aswell as negative values. Variations in the PIPR imply that the timing payoffis path-dependent. The last component, ετ0,τN , is the noise relative to public

8 The decomposition (12)–(18) also holds when the active fund is subject to a no-short-sale constraint. In this case

θGv is replaced by max

{θGv ,−θmv

}.

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information. This part of the excess return is orthogonal to public information.It represents an innovation that cannot be synthesized by trading in the marketportfolio. The innovation component ετ0,τN becomes public at the reportingtime τN , when the latest realization of the fund return is disclosed.

Formula (12) is an endogenous regression model linking the fund excessreturn to the market excess return. It is the analog of (1), except that thedependent variable is measured as the logarithm of the gross excess return.9

This introduces an intercept, − 12

∫ τNτ0δvdv− 1

2ψτ0,τN , that depends on time

aggregation factors related to public and private information.10 The systematicrisk component,

∫ τNτ0βvdR

mv , is a functional of instantaneous market excess

returns. As shown by (16), the timing claim, fτ0,τN , is also a functional ofinstantaneous market excess returns. The error, ετ0,τN , is the difference betweenthe realized fund excess return and its expectation based on public information.It has two parts. The first corresponds to the error in the time aggregationfactor related to private information. The second is the difference betweenthe cumulative excess return generated by private information trades and itsexpectation based on public information. The endogenous regression modelhas important ramifications for empirical studies. It highlights, in particular,the path-dependent structure of the regression components. It also shows theheteroscedastic property of the regression error.

With the particular information structure (6), more precise expressions areavailable for the timing option and other excess return components.

Corollary 1. For v∈ [τi−1,τi) and i =1,...,N , define the conditional PIPRfunction

x �→θGv (x)≡ limε↓0Ev

[Rmv,v+ε

ε

∣∣∣∣Gi=x

]=Dm

v logpGv (x) (19)

and suppose that θGv (x) satisfies the technical conditions in Lemma 1 in theAppendix. The PIPR is θG

v =θGv (Gi). Let P ζi (dζ ) be the probability measureof the independent random variable ζi in (6). The fund excess return is givenby (12)–(18) with

E[ηGv

∣∣Fτi−]

=∫ ∞

−∞

(2θmv +θGv (x)

)θGv (x)|x=g

(Smτi−1,τi ,ζ

)P ζi (dζ ) (20)

E[γ Gv

∣∣Fτi−]

=∫ ∞

−∞

(θGv (x)

σmv

)|x=g

(Smτi−1,τi ,ζ

)P ζi (dζ ). (21)

9 The realized gross excess return over the observation period, Saτ0,τN , depends nonlinearly on systematic risk

and timing factors. A regression based on the cumulative excess return,∫ τNτ0 d−Rav , cannot be implemented

because the local excess returns d−Rav are, by assumption, not observable in the interval [τ0,τN ). Also note thatlogSaτ0,τN can be interpreted as the continuously compounded excess return over the observation interval.

10 The intercept in this regression does not measure selection skill—that is, alpha. It is negative for both skilled andunskilled market timers.

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If ζi =ζ (z) where ζ (·) is a function and z is a standard normal variate, then

P ζi (dζ ) is replaced by φ (z)dz where φ (z)≡(

1/√

2π)

exp(−z2/2

)is the

standard normal density function.

The function θGv (x) in (19) is the private information price of risk at timev conditional on a given realization x of the private signal. At time τi apublicly informed investor knows the realized market excess return and usesthat information to evaluate the fund’s excess return components driven byprivate information. For instance, the conditional expectation of the PIPR basedon public information is E

[θGv

∣∣Fτi−]

=∫∞−∞θ

Gv (x)|x=g

(Smτi−1,τi ,ζ

)P ζi (dζ ). This

rationale underlies the conditional expectations in (20)–(21). The formulasobtained take the form of integrals relative to a density function, which areeasy to calculate in implementations.

1.5 Return distribution and stochastic dominanceTiming skill is the ability to extract private information about future excessreturns and to use it for optimal trading. The gross return achieved by theskilled fund is Xa,sτi−1,τi

. An unskilled manager is unable to extract usefulinformation and generates the gross return Xa,uτi−1,τi

. Explicit expressions for(Xa,sτi−1,τi

,Xa,uτi−1,τi

)are in the proof of Proposition 5 in the Appendix.

Let F si−1 (respectively Fui−1) be the cumulative distribution functionassociated with the skilled (respectively unskilled) return Lsi ≡ logXa,sτi−1,τi(respectively Lui ≡ logXa,uτi−1,τi

). The cumulative spread between the two

distributions is Ti−1 (y)≡∫ y−∞

(F si−1 (z)−Fui−1 (z)

)dz. The next proposition

describes the returns.

Proposition 5. The returns generated by the skilled and unskilled fundssatisfy

Lsi =Lui +μi−1(Lui)+εi, (22)

where μi−1(Lui)≡Eτi−1

[logpGτi−1

(Gi)/pGτi (Gi)∣∣∣Lui ] is a positive (condi-

tional) compensation for risk with Eτi−1

[μi−1

(Lui)]

=logIGi−1>0 and εi≡

logpGτi−1(Gi)/pGτi (Gi)−μi−1

(Lui)

is a conditionally centered noise term with

Eτi−1

[εi |Lui

]=0. The skilled return is more risky, in the sense of second-

order stochastic dominance, than the compensated unskilled return Lui +μi−1

(Lui). Expected compensation is positive and equals the logarithm of

the value of information. Moreover, Eτi−1

[Lsi]>Eτi−1

[Lui]

and Ti−1 (∞)=−(Eτi−1

[Lsi]−Eτi−1

[Lui])

=−logIGi−1<0.

Proposition 5 provides an alternative decomposition of the skilled return. Itshows that the skilled return is a noisy version of the compensated unskilledreturn Lui +μi−1

(Lui). By Rothschild and Stiglitz (1970), it follows that every

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risk averter prefers the compensated unskilled return to the skilled return. Thecompensation μi−1

(Lui)

is the reward for the added risk, which is a by-productof the claim manufactured by the skilled manager. This compensation dependson the unskilled return. It ensures that the skilled manager with logarithmicutility finds it profitable to gather information and trade based on it. It alsoensures that the active fund is sufficiently profitable to attract a clientele ofnoninformed investors.

The result provides perspective about the forces that shape skilledreturns. There are three effects. First, timing skill has value and increasesthe unconditional expected return of the fund (Eτi−1

[Lsi]

=Eτi−1

[Lui]+

Eτi−1

[μi−1

(Lui)]>Eτi−1

[Lui]). Second, private information yields additional

insights about the true value of future excess returns. It induces the skilledmanager to place bets on events revealed by the private signal. The performanceof these bets depends on the realization of the unskilled return and is reflected inthe conditional expected return of the fund (Eτi−1

[Lsi

∣∣Lui ]=Lui +μi−1(Lui)>

Lui ). Third, noise in the information structure implies that these gambles arenot sure winners after compensation and that the skilled return is more riskythan the unskilled return adjusted by the compensation for risk.

The various characterizations of second-order stochastic dominanceestablished by Rothschild and Stiglitz (1970) shed further light on the result.The return distribution generated by a skilled fund lies above (below) thecompensated unskilled return distribution in the lower (upper) tail of theirdomains (odd number of crossings). Alternatively, the expected return isproduced at higher risk than the compensated unskilled return (mean-preservingspread). The most significant aspect is that the compensated unskilled returnis preferred by all risk averters, irrespective of their risk aversion profiles andtheir initial wealths. Of course, the uncompensated unskilled return—that is,the unskilled return—is not preferred by all.

Figure 1 illustrates the shapes of the distributions and densities of theskilled and unskilled fund returns in the directional forecast model describedin Section 2.2. In this example, the skilled and unskilled distributions have thesingle crossing property. Risk aversion implies that the skilled return profile isdominated if the means are the same. The compensation for additional risk isin the expected return. By construction, this compensation is enough for theskilled logarithmic investor to prefer gathering private information. It is alsosufficient to attract a clientele of potential investors in the skilled fund.

In order to identify potential investors in a fund it is useful to expresspreferences in terms of the underlying state price density (SPD). Let Ube the class of utility functions u(·) :R→R that are increasing, concave,and twice continuously differentiable.11 The indirect utility derived frominvesting an initial wealth x with a manager with logarithmic utility function is

11 The differentiability assumption can be relaxed.

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0.6 0.8 1 1.2 1.4 1.60

0.51

1.52

2.53

3.54

skilledunskilled

0.6 0.8 1 1.2 1.4 1.60

0.10.20.30.40.50.60.70.80.9

1skilledunskilled

0.6 0.8 1 1.2 1.4 1.6−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Figure 1Gross excess return distribution in the directional timing modelThe left panel shows the density functions, and the middle panel the cumulative distribution functions for theskilled and unskilled returns. The right panel shows the spread between the skilled and unskilled cumulativedistributions. Plots show function values (y-axis) associated with possible realizations of gross excess returns(x-axis). The market volatility is σm =0.2, the market price of risk is θm =0.1, the timing window is one year,and the level of skill is s =0.1.

J (ξ/x)≡u(Ia (ξ/x)), where Ia (ξ/x)≡ (ξ/x)−1 is the inverse marginal utilityof the manager and ξ is the SPD. The associated value function x, is Ui−1 ≡Eτi−1 [J (ξ/x)]. An unskilled (respectively skilled) fund optimizes using theSPD ξm (respectively ξG) based on public (respectively private) information(see (A8)–(A9) in theAppendix). Investing in a skilled fund dominates investingin an unskilled fund if and only ifUs

i−1>Uui−1. Let U∗ ⊆U be the subset of utili-

ties that find it optimal to invest in the skilled fund instead of the unskilled fund

U∗i−1 =

{u∈U :Us

i−1 ≡Eτi−1

[J(ξGτi−1,τi

/x)]>Eτi−1

[J(ξmτi−1,τi

/x)]

≡Uui−1

},

where ξGτi−1,τi

=ξmτi−1,τipGτi−1

(Gi)/pGτi (Gi). Let T ξi be the cumulative spreadbetween the distributions of skilled and unskilled SPDs.

Proposition 6. The state price density for public information second-orderstochastically dominates the state price density for anticipative information

ξmτi−1,τiSSD ξG

τi−1,τi⇔T

ξ

i−1 (y)≥0 for ally∈R+. (23)

The set U∗i−1 (x)≡{

u∈U :Usi−1>U

ui−1

}of investors who prefer the skilled fund

is

U∗i−1 (x)=

{u∈U :

∫ ∞

0u′(x

y

)1

y3

(2−R

(x

y

))Tξ

i−1 (y)dy>0

}(24)

whereR(·) is the relative risk aversion of u(·). As T ξi−1 (y)≥0 for all y∈ [0,∞),skilled fund returns are preferred by all investors with R(·)<2. Conversely,investors with relative risk aversionR(·)≥2 will never prefer the skilled returnirrespective of the skill level.

The fact that investors with relative risk aversion greater than 2 prefer theunskilled over any skilled return is notable. The benchmark 2 corresponds to

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0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6skilled

unskilled

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

skilled

unskilled

0 0.1 0.2 0.3 0.4 0.50.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6skilled

unskilled

Figure 2Portfolio performance statistics in the directional timing modelThe left panel shows the Sharpe ratios, the middle panel the expected excess returns, and the right panel thevolatilities generated by skilled and unskilled funds. Plots show statistics (y-axis) versus skill (x-axis). Themarket volatility is σm =0.2, the timing window is one year, and the market price of risk is θm =0.1.

the relative prudence coefficient of logarithmic utility. Individuals whose riskaversion lies above the relative prudence of the logarithmic fund manager willnever find it beneficial to delegate portfolio management to the skilled fund,independently of the level of skill.

The condition follows directly from the second-order stochastic dominancerelation between public and private SPDs and the fact that the indirect utilityis strictly concave (respectively convex) if and only if 2<R(·) (respectively2>R(·)). Second-order stochastic dominance follows from the fact that theratio of informed and public SPDs corresponds to the reciprocal of the densityprocess of the information signal. The density process concentrates probabilitymass around the realization of the signal. Its reciprocal spreads probability massto the tails. As the SPDs have the same unconditional mean (see the proof ofProposition 6 in the Appendix), the informed SPD is obtained from the publicSPD by a mean-preserving spread.

The fact that some investors prefer the unskilled return for any level skill isstriking and has important practical ramifications. Figure 2 illustrates propertiesof the fund return in the directional timing model. It shows that the expectedexcess return is an increasing function of skill, whereas the return volatilityreaches a maximum and decreases for sufficiently high levels of skill. Theresulting Sharpe ratio of the active fund is increasing and convex. Sharpe ratiosand expected excess returns are often used in practice to rationalize portfolioallocations to active funds. Proposition 6 implies that some asset allocationdecisions are inappropriate. If risk aversion is sufficiently high, there is nolevel of skill that justifies a full investment in a skilled fund with logarithmicutility function, even if the Sharpe ratio engineered is extremely high.

Proposition 6 can be used to formulate a second-order stochasticdominance test (SSDT) to detect timing skill in fund returns. Given the

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inverse relation between gross returns and SPDs (23) is equivalent to(Xa,sτi−1,τi

)−1SSD(Xa,uτi−1,τi)−1. The SSDT for skill can be implemented using

a Kolmogorov-Smirnov statistic or a von Mises statistic. It tests whetherTξ

i−1(y)≥0 for all y>0, under the null hypothesis of no skill, where thecumulative spread is between the empirical distribution of (Xa,sτi−1,τi

)−1 and

the known parametric distribution of (Xa,uτi−1,τi)−1.12

2. Parametric Timing Models

This section specializes the general results of Section 1. Section 2.1 examinesthe case of private information about future excess return levels. Section 2.2studies directional information. Throughout the section, it is assumed that σm

and θm are constants.

2.1 Return forecast and return-based fund styleSuppose that the market timer observes the future market excess return withnoise. The private signal (5) has the linear-multiplicative form13

Gi≡g(Smτi−1,τi

,ζi

)=Smτi−1,τi

ζi

with ζi≡exp

(σy√hi

∫ τi

τi−1

dWζv − 1

2(σy)2

),

(25)

where hi≡τi−τi−1 is the information interarrival time, logζi∼N(−(σy)2/2,(σy)2), and E

[ζi |Fm

v

]=1 for all v∈ [0,T ]. The logarithm

of the signal is the sum of the cumulative local excess return of the marketand of a normally distributed noise term. Given that the discounted value ofthe market Smτi−1,τi

is positive, the signal is always positive. The volatility σy

measures the informativeness of the signal. The information extraction skillof the manager is related to the inverse volatility s =(σy)−1. A manager withgreater skill generates more precise information. A manager without skillproduces pure noise and has an uninformative signal.

The fund excess return specializes as:

Corollary 2 (Noisy Return Forecast). Suppose that θm and σm are constant,and consider the private information filtration G(·) =Fm

(·)∨FY

(·) generated bymarket returns and the private signal (5), (25). For v∈ [τi−1,τi), the PIPR is

θGv =σm

(logGi−Ev [logGi]

VARv [logGi]

)(26)

12 With an i.i.d. market return process and a constant interest rate, as in the models of Section 2,(Xuτi−1,τi

)−1has

a log-normal distribution with mean −(r+ 1

2(θm

)2)hi and variance

(θm

)2hi . For general market coefficients

the distribution can be obtained by simulation or partial differential equation methods.

13 The information structure in (25) is a continuous-time extension of the Gaussian framework typically found instatic models with private information (e.g., Grossman and Stiglitz 1980). Log-normality ensures that pricescannot be negative.

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where

VARv [logGi]= (σm)2 (τi−v)+(σy)2 ≡�v,τi (27)

Ev [logGi]=∫ v

τi−1

dRms +σmθm (τi−v)− 1

2

((σm)2

hi +(σy)2). (28)

The optimal informed investment policy and the associated fund excessreturn process are given by (8)–(9) evaluated at θG

v . The PIPR is positive(negative) if the innovation in the log signal is positive (negative). Underthese circumstances the informed manager invests more (less) in the market,and the volatility coefficient of the optimally managed fund return is greater(smaller) than the volatility of a similar fund under uninformed management.The optimal fund management style is return-based: it is an affine function ofthe logarithm of the private signal and the observed market return. Let νmiv =

logSmτi−1,τi−Ev

[logSmτi−1,τi

]be the innovation in the continuously compounded

market excess return logSmτi−1,τigiven public information at time v∈ [τi−1,τi).

In the public information filtration, the endogenous fund excess return has thedecomposition (12)–(18) with

ψi≡∫ τi

τi−1

(2θmσm

(νmiv

�v,τi

)+(σm)2

((νmiv)2

+(σy)2(�v,τi

)2

))dv (29)

f mi ≡∫ τi

τi−1

(νmiv

�v,τi

)d−Rmv , εmi ≡

∫ τi

τi−1

(σywi

�v,τi

)d−Rmv (30)

εai ≡−1

2

∫ τi

τi−1

2σm(θm+σm

νmiv

�v,τi

)(σywi

�v,τi

)dv

− 1

2

∫ τi

τi−1

(σm)2 (σywi)2 −(σy)2(

�v,τi)2 dv

(31)

where wi≡(1/

√hi)∫ τiτi−1

dWζv .

As shown by Equations (26)–(28), the PIPR is linear in the innovationlogGi−Ev [logGi] and inversely related to the conditional variance �v,τi .Moreover, as time passes, the informed manager learns from publicinformation, prompting revisions in the assessment of risk and the PIPR. For afixed innovation, the information content about gross excess returns increases,raising the absolute value of the private information price of risk. With perfectforesight/infinite skill (i.e., σy =0), the information signal eventually revealsthe instantaneous excess return, and the PIPR explodes as the next tick ofthe information clock approaches (θG

v →∞ as v→τi). In the limit, localuncertainty is resolved and an arbitrage opportunity emerges. Prior to τiuncertainty about instantaneous excess returns remains, ensuring a finite price

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Gi = 0.5 Gi = 1.5

00.5

11.5

2

0

0.020.04

0.060.08−0.8

−0.6

−0.4

−0.2

0

skillelapsed time0

0.51

1.52

00.02

0.040.06

0.080

0.1

0.2

0.3

0.4

skillelapsed time

Gi = 0.5 Gi = 1.5

00.5

11.5

2

0.90.95

11.05

1.1−0.8

−0.6

−0.4

−0.2

0

skillgross return0

0.51

1.52

0.90.95

11.05

1.10

0.1

0.2

0.3

0.4

0.5

skillgross return

Figure 3PIPR in the return forecast modelUpper panels plot the PIPR (z-axis) as a function of skill s =1/σy (x-axis) and the elapsed time in the timinginterval v−τi (y-axis). Parameter values are σm =0.2, θm =0.3, Smτi−1,v

=1. Lower panels plot the PIPR (z-axis)

as a function of skill s =1/σy (x-axis) and the realized gross excess return Smτi−1,v(y-axis). Parameter values are

σm =0.2, θm =0.3, v−τi =0.5/12. Left-hand panels correspond to signal realization Gi =0.5, right-hand panelsto Gi =1.5. All plots are for signal errors logζi =E [logζi ].

of risk. As long as information is noisy (i.e., σy >0), the price of informationrisk remains finite at all times.

Figure 3 provides further insights about the structure of the PIPR for tworealizations of the signal. The plots illustrate the effects of skill s =1/σ y , elapsedtime v−τi−1 since the beginning of the timing window, and realized grossexcess return Smτi−1,v

, for a one-month timing window (τi−τi−1 =1/12). Whenskill increases, the PIPR increases in absolute value for both signal realizations.When the signal points to a significant increase (respectively decrease) of themarket, it is positive (respectively negative). The impact of time variations,within this short timing window, is small. In contrast, the PIPR exhibits a

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clearer state dependence. Indeed, the lower panels show a negative relation tothe realized gross excess return for highly skilled funds—that is, low levels ofsignal noise.

The optimal informed policy (8) and the fund return volatility (9) inheritthe structure and properties of the PIPR. The fund return decomposition in thepublic information has several interesting features. The most notable aspectis the timing option f mi in (30), that displays a path-dependent, quadratic-likerelation to market returns To see this, note that the first term of f mi , whichequals

logSmτi−1,τi

∫ τi

τi−1

d−Rmv�v,τi

=∫ τi

τi−1

(dRmv − 1

2(σm)2

hidv

)∫ τi

τi−1

dRmv

�v,τi,

is the product of an affine function of the cumulative excess returnRmτi−1,τi

=∫ τiτi−1

dRmv and of a weighted sum of realized local excess returns∫ τiτi−1

(�v,τi

)−1dRmv . The presence of the time-dependent weight �v,τi is due

to the dynamic trading of the fund manager, which is motivated by changesin the private information set. As time passes, the variance of the forecastdecreases, inducing a manager with positive news to increase the share ofequities in the portfolio (for a given information innovation). This phenomenonleads to an increase in the response of the timing option to market excessreturns. In the absence of information accumulation, as in a static model, theweight �v,τi ≡1/a is a constant and the weighted sum becomes proportionalto cumulative excess returns, aRmτi−1,τi

. The timing option inherits a term

a(Rmτi−1,τi

)2proportional to the squared excess return over the period, as in the

reduced-form model postulated by Treynor and Mazuy (1966). Local timingactivity, based on information accumulation, is a source of path-dependence inthe timing payoff. It is also a source of heteroscedasticity in fund excess returns.As revealed by

(εmi ,ε

ai

)in (30)–(31), the error term has a variance related to

past market excess returns. The conditional error distribution is non-centralchi-square (see Detemple and Rindisbacher 2012).

Amanager with greater timing skill is able to extract more precise informationabout future returns. For a given innovation logGi−Ev [logGi], the absolutevalues of the PIPR and the timing component of the portfolio increase. Thetiming component (30) of the endogenous fund excess return becomes moresensitive to the market excess returns.

Corollary 3 gives an explicit formula for the value of information.

Corollary 3. The values of the information signal Sτi−1,τi ζi and structure{Sτi−1,τi ζi : i =1,...,N

}are deterministic and given by IG

i−1 =√

1+(σm/σy)2hi

and IGτ0,τN

=∏Ni=1

√1+(σm/σy)2hi .

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0 0.20.40.60.81 1.21.41.61.82

00.01

0.020.03

0.040.05

0.060.07

0.081

1.001

1.002

1.003

1.004

1.005

1.006

1.007

skilltiming window

00.5

11.5

2

0

0.5

1

1.51

1.05

1.1

1.15

1.2

skillmarket volatility 00.5

11.5

2

0

0.5

1

1.51

1.001

1.002

1.003

1.004

1.005

skillmarket price of risk

Figure 4The value of information in the return forecast modelThe upper panel plots the value of information (z-axis) as a function of skill s =1/σy (x-axis) and the lengthof the timing interval hi (y-axis). Parameter values are σm =0.2, θm =0.3. The lower panels plot the value ofinformation (z-axis) as function of skill s =1/σy (x-axis), and the market volatility σm (y-axis, left panel) or themarket price of risk θm (y-axis, right panel). Parameter values are θm =0.3, hi =0.5/12 (left panel), and σm =0.2,hi =0.5/12 (right panel).

The value of the information signal is increasing in the market volatility, thelength of the timing interval, and skill. It does not depend on the market priceof risk θm. Figure 4 illustrates these relations for different levels of skill.

2.2 Directional forecast and long-short styleThe second application focuses on directional information—that is, informationpertaining to the direction of the market. The private information filtrationG(·) =Fm

(·)∨FY

(·) is now generated by market returns and the private signal (5)with

Gi =g(Smτi−1,τi

,ζi

)= sgn

(logSmτi−1,τi

)sgn(ζi−κi), (32)

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where sgn(x)=1{x>0}−1{x≤0} is the sign of x and ζi is a noise term, independentof the market.

In order to interpret the elements of this information structure, note that thesignal indicates the direction of the market, but is corrupted by the noise ζi .If sgn(ζi−κi)=+1, the signal Gi gives a correct forecast. In this instance, anup signalGi =+1 (respectively down signalGi =−1) gives correct informationabout the direction of the market as Rmτi−1,τi

>0 (respectively Rmτi−1,τi<0). In

contrast, if sgn(ζi−κi)=−1, the signal gives an incorrect forecast. In thiscase, the up signal Gi =+1 (respectively down signal Gi =−1) indicates thatRmτi−1,τi

<0 (respectively Rmτi−1,τi>0). As ζi is not observed, there is noise in

the directional forecast Gi .The constant κi ∈R captures the precision of the directional excess return

forecast. Let Fi (x)=Pτi (ζi >x) be the complementary conditional distributionfunction of ζi at time τi and define the percentile function x =F−1

i (pi) forpi ∈

[12 ,1

]. With the parametrization κi =F

−1i (pi), the conditional probability

of a correct forecast is

Pτi

(sgn

(logSmτi−1,τi

)=Gi

)=Pτi

(ζi−F−1

i (pi)>0)

=pi. (33)

The probability pi measures the quality of the signal and quantifies theforecasting skill of the manager. If pi >1/2, the likelihood of a correctdirectional inference is greater than pure chance and the manager has timingskill. If pi =1/2, the signal is useless for inference about the direction of themarket. A skilled timer has pi >1/2 and timing skill increases with pi . Thedistance si≡pi−1/2 measures the skill level. Timing skill and skill levelalso depend on public information at the time of forecast arrival τi−1 (hencethe subscript i). This permits the modeling of persistence in forecasting andtiming skill across forecasting intervals and could be used to test the hot handsphenomenon (see Jagannathan, Malakhov, and Novikov 2010).

The optimal portfolio and the fund excess return process are now:

Corollary 4 (Directional Forecast). Suppose that θm and σm are constant andconsider the private information filtration G(·) =Fm

(·)∨FY

(·) generated by marketreturns and the private directional signal (32)–(33). Let si≡pi−1/2 be theskill level and define the functions

di (z,v)=−log(z)+

(12 (σm)2 −σmθm

)(τi−v)

σm√τi−v , Ki (z,v)≡ φ (di (z,v))√

τi−v (34)

Hv (y)≡ypGv (y)−1 =y

[1

2+si

(2y(

1−�(di

(Smτi−1,v

,v)))

+1y=−1 −1y=1

)]−1

(35)

where hi≡τi−τi−1 is the interarrival time; φ (·),�(·) are the standard normaldensity and cumulative distribution functions; and y∈{−1,+1}. The PIPR is

θGv =2siKi

(Smτi−1,v

,v)Hv (Gi) for v∈ [τi−1,τi). (36)

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The optimal informed investment policy and the associated fund excess returnprocess are given by (8)–(9) evaluated at θG

v . The PIPR is positive (negative) ifthe signal is up (down) and pi >1/2 Under these circumstances the informedmanager invests more (less) in the market and the volatility coefficient ofthe optimally managed fund return is greater (smaller) than the volatilitycoefficient of a similar fund under uninformed management. The optimal fundmanagement is a directional trading style. When the uninformed premiumis null, θm=0, the optimal management is a long-short style. In the publicinformation filtration, the endogenous fund excess return has the decomposition(12)–(18) with temporal aggregation factor, timing option, and error given by

ψi≡4θmsi

∫ τi

τi−1

Ki

(Smτi−1,v

,v)H (1)v dv+4s2

i

∫ τi

τi−1

Ki

(Smτi−1,v

,v)2H (2)v dv (37)

f mi ≡2si

σm

∫ τi

τi−1

Ki

(Smτi−1,v

,v)H (1)v d

−Rmv (38)

εmi ≡2si

σm

∫ τi

τi−1

Ki

(Smτi−1,v

,v)(Hv (Gi)−H (1)

v

)d−Rmv (39)

εai ≡−2siθm

∫ τi

τi−1

Ki

(Smτi−1,v

,v)(Hv (Gi)−H (1)

v

)dv

+2s2i

∫ τi

τi−1

Ki

(Smτi−1,v

,v)2(Hv (Gi)

2 −H (2)v

)dv

(40)

where H(k)v ≡E[Hv (Gi)

k∣∣Fτi−

]is the kth conditional moment

of Hv (Gi), k=1,2. Moreover, H(k)v =Hv

(sgn

(logSmτi−1,τi

))kpi +

Hv

(−sgn

(logSmτi−1,τi

))k(1−pi) for k=1,2. Alternative expressions for

these moments are in (A10)–(A11) in the Appendix.

The directional information structure (32)–(33) can be viewed as a dynamicextension of the setting in Henriksson and Merton (1981). The signal points tothe direction of the market. But the quality of the information and its usefulnessdepend on the skill of the market timer. In addition, the information transmittedby a given signal changes over time as public information accumulates.

The PIPR generated by such a market timer can be positive or negative. If theprobability of a positive noise indicator is pi =1, the forecast is perfect. In thiscase, the informed has the ability to identify the true direction of the market,and the information price reaches its maximal size (in absolute value)

θGv =Ki

(Smτi−1,v

,v)Gi

2

[1+2Gi

(1−�

(di

(Smτi−1,v

,v)))

+1Gi=−1 −1Gi=1

]−1.

If the probability of a positive noise realization is pi =1/2, the signal isuninformative about the direction of the market. In this instance, there is equal

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Gi = −1 Gi = 1

00.02

0.040.06

0.080.1

0

0.02

0.04

0.06

0.08−2.5

−2

−1.5

−1

−0.5

0

skillelapsed time0

0.020.04

0.060.08

0.1

0

0.02

0.04

0.06

0.080

0.5

1

1.5

2

2.5

skillelapsed time

Gi = −1 Gi = 1

00.02

0.040.06

0.080.1

0.90.95

11.05

1.1

1.15−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

skillgross return0

0.020.04

0.060.08

0.1

0.90.95

11.05

1.11.15

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

skillgross return

Figure 5PIPR in the directional forecast modelUpper panels plot the PIPR (z-axis) as a function of skill s =p−1/2 (x-axis) and the elapsed time in the timinginterval v−τi (y-axis). Parameter values are σm =0.2, θm =0.3, Smτi−1,v

=1. Lower panels plot the PIPR (z-axis)

as a function of skill s =p−1/2 (x-axis), and the realized gross excess return Smτi−1,v(y-axis). Parameter values

are σm =0.2, θm =0.3, v−τi =0.5/12. Left-hand panels correspond to signal realizationGi =−1, and right-hand

panels to Gi =+1. All plots are for signal errors sgn(ζi−F−1

i(p)

)=1.

likelihood of a correct and an incorrect directional forecast, and the PIPRis null, θG

v =0. In other cases, the behavior depends on the likelihood of acorrect forecast and on the signal received. If the signal is positive (respectivelynegative) and the odds of a correct positive forecast exceed 1/2, the PIPRis positive (respectively negative). Ceteris paribus, the absolute value of thePIPR increases with the informativeness of the signal—that is, with the skilllevel si .

Figure 5 illustrates the PIPR’s behavior in the directional forecast model. Thepatterns are markedly different from those in the return forecast model. ThePIPR is now increasing (respectively decreasing) in elapsed time even at low

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levels of skill, for positive (respectively negative) signals. The dependence onthe gross excess return is determined by the Gaussian density in Ki(Smτi−1,v

,v).In that regard, the PIPR is no longer monotone and in fact vanishes forextreme realizations of the gross excess return. Moreover, excess return effectsare nonsymmetric with respect to the signal (it can be shown that θGv (1)=−θGv (−1)�(di)/�(−di)). From an economic point of view, the realized grossexcess return Smτi−1,v

is a public signal about the unknown gross excess returnover the whole timing period, Smτi−1,τi

. An increase in the absolute value ofthis public signal reduces the value of private information, leading to a non-monotone PIPR. An extreme realization of Smτi−1,v

signals the direction of themarket with very high likelihood. In the limit, private information becomesuseless and the PIPR vanishes.

The fund manager invests more aggressively, relative to an uninformedinvestor, when the signal becomes more precise. The pattern of tradesimplemented by a directional market timer is even more interesting. If the signalreceived at an information arrival time points to an increase (decrease) in themarket, the skilled directional market timer increases (decreases) the position inequities by a discrete amount relative to a publicly informed investor. Over time,switches from positive (negative) directional indicators to negative (positive)ones will prompt large decreases (increases) in the fraction invested in equities.Corresponding changes in endogenous fund return volatility accompany thesetrading patterns. If the market price of equity risk were null, the skilledmarket timer would take a long or short position in equities depending on thenature of the signal (buy or sell). Under these circumstances, this directionalmarket timer would switch between long and short equity positions as timepasses.

The endogenous fund excess return reflects the skill and behaviorof the manager. A manager without skill is unable to generate timing-related components (ψi =f mi =εmi =εai =0). Skilled management generates path-dependent excess returns and timing components. Inspection of (38) shows thatthe sensitivity of the timing claim to market excess returns increases with theskill level. The regression errors (39)–(40) can be shown to inherit the binomialstructure of the signal noise (see Detemple and Rindisbacher 2012 for details).

Explicit formulas for the value of information are given next.

Corollary 5. The value of the signal sgn(logSmτi−1,τi)sgn(ζi−F−1

i (pi)) isdeterministic

IGi−1 =exp

⎛⎝pi logpi +(1−pi)log(1−pi)−∑

z∈{−1,1}logpGτi−1

(z)pGτi−1(z)

⎞⎠(41)

where pGτi−1(z)≡2(1−�(di (1,τi−1)))si (1z=1 −1z=−1)+(1−pi)1z=1 +pi1z=−1

for z∈{−1,1}. The value of the information structure is deterministic: IGτ0,τN

=∏i I

Gi−1.

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00.1

0.20.3

0.40.5

0

0.02

0.04

0.06

0.080

0.2

0.4

0.6

0.8

skilltiming window

00.1

0.20.3

0.40.5

0

0.5

1

1.50

0.2

0.4

0.6

0.8

skillmarket volatility 00.1

0.20.3

0.40.5

0

0.5

1

1.50

0.2

0.4

0.6

0.8

skillmarket price of risk

Figure 6The value of information in the directional forecast modelThe upper panel plots the value of information (z-axis) as a function of skill s =p−1/2 (x-axis) and the lengthof the timing interval hi (y-axis). Parameter values are σm =0.2, θm =0.3. The lower panels plot the value ofinformation (z-axis) as a function of skill s =p−1/2 (x-axis), and the market volatility σm (y-axis, lower leftpanel) or the market price of risk θm (y-axis, lower right panel). Parameter values are θm =0.3, hi =0.5/12 (leftpanel), and σm =0.2, hi =0.5/12 (right panel).

The value of information (41) depends on the length of the timing intervalhi , the market price of risk θm, and the market volatility σm through theparameter di(1,τi−1)=−(θm− 1

2σm)

√hi (see (34)). The effect of the timing

interval length depends on sgn(θm− 12σ

m). In contrast to the return forecastmodel, the market price of risk θm has an impact. Figure 6 illustrates theseproperties.

To conclude this section, it is worth stressing the impact of the informationstructure on a fund’s excess return. As documented above, managers trading onlevel forecasts behave differently from those trading on directional information.The characteristics of the information-gathering technology condition theoptimal portfolio strategies, which in turn affect the structure of a fund’sreturn. Being able to discriminate between different return structures and

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understanding the link between investment styles and returns has importantramifications for the design of empirical studies.

3. Selection and Market Timing

Suppose now that there are d individual assets andm factors (indices) that canbe synthesized by trading in factor mimicking portfolios. The d-dimensionalvector of individual asset excess returns RA has a factor structure relative tothe m-dimensional vector of factor excess returns RF ,

dRAv =σv (θvdt +dWv)=βvdRFv +σAv dW

Av (42)

dRFv =σFv(θFv dt +dW

Fv

)(43)

where β is a d×m-dimensional matrix of factor loadings, WF a vector of mBrownian motions affecting factors, and WA a vector of d Brownian motionsindependent ofWF . The coefficient σA is the d×d volatility matrix associatedwith non-factor risk sources, σF is them×m volatility matrix of factor excessreturns, and θF is the m-dimensional vector of market prices of factor risks.The market price of risk vector for the d+m-dimensional Brownian motion

W ′ =((WF

)′,(WA

)′)is θ ′

v =((θFv)′,0′)

, and the local excess return variance

is σv (σv)′ =βvσFv

(βvσ

Fv

)′+σAv

(σAv)′

. Factor risks WF are priced, but noiserisks WA are not. The coefficients

(β,θF ,σF ,σA

)are adapted to the filtration

generated by(WF ,WA

). The matrices σA, σF are assumed to be invertible.

Public information F(·) is conveyed by individual asset excess returns (FA(·)),

factor excess returns (FF(·)), and the active fund excess returns observed at the

reporting dates (Faτ0

for v<τN and FaτN

for v=τN ). Private information G(·) has

the structure outlined in Section 1 with(Smτi−1,τi

)′ ≡[(SAτi−1,τi

)′,(SFτi−1,τi

)′].

Let pGv (z) be the conditional density process of the signalG given Fmv ≡FF

v ⊗FAv —that is, public information generated by sources of factor and noise risk,(WF ,WA

). The next result shows a key decomposition of the PIPR into timing

and selection components.

Proposition 7. Assume that excess returns have the factor structure (42)–(43).The PIPR has timing (θG,F

v ) and selection (θG,Av ) components given by

θGv =

[θG,Fv

θG,Av

]=

[d[logpG(z),WF ]v

dvd[logpG(z),WA]v

dv

]|z=Gi

=

[ (DFv logpGv (z)|z=Gi

)′(DAv logpGv (z)|z=Gi

)′ ] (44)

for v∈ [τi−1,τi), where d[logpG (z),WF

](respectively d

[logpG (z),WA

]) is

the covariation between the conditional log-density process and the BrownianmotionWF (WA) and DF

v (respectively DAv ) is the Malliavin derivative relative

to WF (respectively WA). In (44),(d[logpG (z),WF

]/dv

)′(respectively

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(d[logpG (z),WA

]/dv

)′) represents the volatility of the rate of growth of the

conditional density pGv (z) with respect to WF (respectively WA).

Like (10), the representation (44) of the PIPR is a deep result, whichturns out to be very useful for financial interpretation purposes. The resultfollows because the conditional expectation of a random variable given privateinformation (as in (7)) is the same as a conditional expectation given publicinformation of the random variable times a density process. This density processputs all the weight on events compatible with the private information received.It corresponds, in fact, to the density process of the conditional measure of theprivate signal given public information. Covariation with the density processmodifies the drift of a process by the volatility coefficient of the densityprocess. Here, the conditional density process depends onWF andWA, becausepublic information corresponds to the information generated by WF and WA.Its volatility structure has two types of components, capturing the respectivesensitivities of the density to innovations in WF and WA.

The financial implications of the decomposition are interesting. Theproposition shows that the price of private information (the shift in the driftdue to private information) has two terms, each related to one specific typeof uncertainty source. If public factor information is useless for updates inthe conditional density of the private signal, then the volatility componentd[logpG (z),WF

]v/dv is null.As factor information does not help in describing

the distribution of the signal, this information has no value. Its associatedprice of risk, θG,F

v , is null. Likewise, if noise information does not helpto update the conditional density, then d

[logpG (z),WA

]v/dv and θG,A

v

vanish.Ultimately, the existence of two types of information prices permits a

distinction between selection and timing. When the conditional densityof the signal does not respond to factor innovations, it means that theinformation signal pertains to the noise components. This type of informationreflects the selection ability of the manager. Its value represents a price forselectivity. Symmetrically, failure to respond to noise innovations meansthat information pertains to factors and reflects the timing skill of themanager. The value of information is then derived from its timing value. Ingeneral, a signal can be sensitive to both types of innovations. The value ofinformation can then be decomposed into a price for timing and a price forselection.

The next proposition describes the optimal portfolio for the informed fundmanager.

Proposition 8. Let G(·) =FA(·) ⊗FF

(·)∨FY

(·) be the private information filtrationgenerated by asset and factor returns and by the private observation processY in (5)–(6) with (Smτi−1,τi

)′ ≡ [(SAτi−1,τi)′,(SFτi−1,τi

)′]. The optimal informed

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policy is[πFvπAv

]=

⎡⎢⎣((σFv

)′)−1(θFv +θG,F

v

)−((σFv

)′)−1(βvσ

Fv

)′((σAv)′)−1

θG,Av((

σAv)′)−1

θG,Av

⎤⎥⎦.The demand πA for individual assets is driven by selection information. Thedemand πF for factor-mimicking portfolios has two components. The firstcomponent is a mean-variance demand driven by public-factor informationand by timing information. The second component is a hedging demanddriven by selection information. The associated fund excess return is d−Rav =(πFv

)′d−RFv +

(πAv

)′d−RAv . In the absence of selection skill, the optimal demand

is to invest in factor-mimicking portfolios, generating a fund excess returnd−Rav =

(πFv

)′d−RFv . The demand πA for individual assets is not affected if

factor-mimicking portfolios are not available (incomplete market). Timinginformation is then irrelevant, and the endogenous fund excess return becomesd−Rav =

(πAv

)′d−RAv .

The proposition shows that the demands for different securities are drivenby different motives. Individual assets are held solely for selection purposes.Factor-mimicking portfolios are held for diversification, timing, and hedgingreasons. More specifically, the demand for the latter has two parts. The first isa mean-variance demand with a component based on public information and acomponent reflecting the timing skill of the manager. The second is a hedgingdemand induced by the positions taken in individual assets. These positionscreate exposures to factor risks, hedged by entering offsetting positions in thefactor-mimicking portfolios. The net demand for factor portfolios is driven bydiversification and timing considerations. Indeed, the endogenous fund return

d−Rav =(πFv

)′d−RFv +

(πAv

)′d−RAv

=(θFv +θG,F

v

)′(σFv

)−1d−RFv +

(θG,Av

)′(σAv)−1(

σAv dWAv

)reveals a diversification and timing demand for factor portfolios and a pureselection demand for individual asset risks. In the absence of factor-mimickingportfolios (non-investable factors), the informed manager optimally ignorestiming information. This follows from the myopic behavior of logarithmicutility, which is known to ignore missing markets. The proposition shows thatthis result holds even in the presence of private information pertaining to factors.

The next proposition describes the structure of the fund excess return in thepublic information.

Proposition 9. Assume that returns have the factor structure (42)–(43). Inthe public information filtration, the excess return of the fund with timing andselection skill is

logSaτ0,τN =−1

2

∫ τN

τ0

δvdv+∫ τN

τ0

(βFv)′dRFv − 1

2ψτ0,τN +fτ0,τN +ετ0,τN (45)

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where δv≡∥∥θFv ∥∥2

,(βFv)′ ≡(

θFv)′(σFv

)−1and

ψτ0,τN =∑

i:τi∈(τ0,τN ]

ψi, fτ0,τN =∑

i:τi∈(τ0,τN ]

fi, ετ0,τN =∑

i:τi∈(τ0,τN ]

εi (46)

with

ψi =ψFi +ψA

i , fi =fFi +f Ai , εi =ε

Fi +εAi +εai (47)

ψFi ≡

∫ τi

τi−1

E[

2(θFv)′θG,Fv +

∥∥θG,Fv

∥∥2∣∣∣Fτi−

]dv,

ψAi ≡

∫ τi

τi−1

E[∥∥θG,A

v

∥∥2∣∣∣Fτi−

]dv

(48)

f Fi ≡E[∫ τi

τi−1

(γ G,Fv

)′d−RFv

∣∣∣∣∣Fτi−

], f Ai ≡E

[∫ τi

τi−1

(θG,Av

)′d−WA

v

∣∣∣∣∣Fτi−

](49)

εFi ≡∫ τi

τi−1

(γ G,Fv

)′d−RFv −f Fi , εAi ≡

∫ τi

τi−1

(θG,Av

)′d−WA

v −f Ai (50)

and εai ≡−∫ τiτi−1

(ηv−E

[ηv|Fτi−

])dv/2. The remaining coefficients are(

γ G,Fv

)′ ≡(θG,Fv

)′(σFv

)−1and ηv≡

∥∥θFv +θG,Fv

∥∥2 −∥∥θFv ∥∥2+∥∥θG,Av

∥∥2.

With timing and selection skills, the excess return of the managed fundrelative to public information has additional components reflecting each of thetwo skills. In (47)–(50), the terms (ψF

i ,fFi ,ε

Fi ) are due to timing information;

(ψAi ,f

Ai ,ε

Ai ) relate to selection information. If the selection price of risk,

θG,Av , is null, reflecting the lack of selection ability, all the corresponding

components (ψAi ,f

Ai ,ε

Ai ) vanish. If the timing price of risk, θG,F

v , is null, theterms (ψF

i ,fFi ,ε

Fi ) disappear.

Equations (45)–(50) describe the endogenous regression in the presenceof timing and selection information. Selection introduces an additional timeaggregation component—namely, ψA

i —in the intercept of the regression.It also introduces a path-dependent selection factor, f Ai . From a structuralpoint of view, this factor is similar to the path-dependent factor f Fi , exceptthat it stems from selection skill (θG,A

v �=0).14 Finally, selection generates anadditional regression error term, εAi , and modifies the time aggregation error,εai . Proposition 9 provides a foundation for empirical studies seeking to examinethe timing and selection activities of managed funds. It shows how to formulatethe regression model (1) when both skills are present.

14 In classic regression models with selection information, the alpha of the fund includes fAi

. In addition, if the

observation and trading frequencies coincide, the temporal aggregation factors (ψFi

, ψFi

) vanish.

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4. Extensions of the Model

This section extends the results to a more general class of private informationsignals and uncertainty structures. Section 4.1 presents the general setting.Section 4.2 describes the link between the PIPR and the fundamentals in adiffusion setting. An application to volatility timing is presented in Section 4.3.Applications to equilibrium models with asymmetric information and strategictrading are carried out in Section 4.4.

4.1 General information signalsConsider the bivariate Brownian motion (Wm,Wz), and let Fm

(·) ⊗F z(·) be the

associated filtration. Excess returns satisfy

dRmv =σmv(θmv dv+dWm

v

)(51)

dZv =Azvdv+Bzv

(ρvdW

mv +

√1−ρ2

vdWzv

), (52)

where the coefficients are adapted to Fm(·) ⊗F z

(·). The process Z representsa fundamental factor affecting the local moments of market excess returns.Dividend yields or macroeconomic variables such as inflation or gross nationalproduct (GNP) are examples of relevant fundamentals. The public is assumedto observe (Wm,Wz).As before it observes the managed fund returns at discretereporting dates. The public information set is Fm

(·) ⊗F z(·) augmented by the

filtration generated by the periodic reports Fa(·).

The active fund manager receives a sequence of private information signals{Gi : i =1,...,N} at the arrival dates {τi : i =0,...,N−1}. SignalGi is an Fm

τi⊗

F zτi

⊗σ (ζi)-measurable random variable, where ζi is an independent randomvariable and σ (ζi) is the associated sigma-algebra. The corresponding signalprocess is Yv =

∑Ni=1Gi1[τi−1,τi) (v). The private information filtration is G(·) =

Fm(·) ⊗F z

(·)∨FY

(·). Portfolio policies πa of the active fund are adapted to G(·)and generate the local gains from trade d−Rav =πav d

−Rmv described by forwardstochastic integrals.

As opposed to the setting in Section 1, the private signal need not have

the particular form g(Smτi−1,τi

,ζi

)where information relates only to future

market excess returns. The general structure introduced here allows forarbitrary messages pertaining to the future trajectories of the returns and/orthe underlying fundamentals. Particular configurations emerge by specializingthis structure. The model presented earlier is a special case, as further discussedin the next section.

By definition, the PIPR is the increment in the perceived price of risk due toprivate information,

θGv ≡

[θG,mv

θG,zv

]≡ lim

ε↓0

1

ε

[ 1σmvE[Rmv,v+ε

∣∣Gv]E[∫ v+ε

v

(θzs ds+dWz

s

)∣∣∣Gv]]

−[θmvθzv

]

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where(θmv ,θ

zv

)are the prices of risk associated with (Wm,Wz), based on public

information. As there are two underlying sources of uncertainty, the PIPR isbivariate. The next result generalizes Proposition 4 to the structure at hand.

Proposition 10. Let pG(·) (z)≡{pGv (z) :v∈ [τi−1,τi)

}be the conditional den-

sity process of the private signal Gi given public information. The PIPR isthe instantaneous covariance between the conditional density process and theinnovation in the underlying Brownian motion

θGv ≡

[θG,mv

θG,zv

]=

[d[logpG(z),Wm]v

dvd[logpG(z),Wz]v

dv

]|z=Gi

=

[ Dmv logpGv (z)|z=GiDzv logpGv (z)|z=Gi

](53)

for v∈ [τi−1,τi), where Dmv (respectively Dz

v) is the Malliavin derivativerelative to Wm (respectively Wz). If the signal is independent of Fm

(·), then

Dmv logpGv (z)=0 and θG,m

v =0.

Proposition 10 shows that the previous formulas and intuitions apply inthis extended framework. It shows, in particular, that the bivariate PIPRcorresponds to the vector of instantaneous volatilities of the density of theinformation signal with respect to market return and fundamental innovations.With these expressions, the representation of the active fund return as aregression involving public information variables can be derived followingthe same steps as before.

4.2 Private information price of risk and fundamentals in a diffusionsetting

Suppose that the state variable Z evolves according to the diffusion

dZv =A(Zv)dv+B (Zv)

(ρ (Zv)dW

mv +

√1−ρ (Zv)

2dWzv

)(54)

and that Gi = l(Zτi ,ζi;τi−1,Zτi−1

)≡ li (Zτi ,ζi) for some one-to-one mappingli in Zτi . Private information thus pertains to the fundamental factor Z. Theconditional density of the signal is

pGv (x)≡fG|Zv (x|Zv)=∫

R

fZτi |Zv(l−1i (x,ζ )

∣∣Zv)∣∣∇xl−1i (x,ζ )

∣∣P ζi (dζ ) (55)

for v∈ [τi−1,τi), where fG|Zv (x|z) is the conditional density ofGi at the point xgivenZv =z, and fZτi |Zv is the transition density ofZ. The quantity ∇xl

−1i (x,ζ )

is the gradient of l−1i (x,ζ ). The next proposition describes the PIPR.

Proposition 11. Consider the diffusion model (54) with signal Gi =li(Zτi ,ζi

), and assume that the transition density fZτi |Zv of the process (54)

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has a closed form. Under these conditions, the PIPR θGv ≡(

θG,mv ,θG,z

v

)′has a

closed form as well and is given by

θGv =

(∫RfZτi |Zv

(l−1i (x,ζ )

∣∣Zv)hi (x,ζ,Zv)∣∣∇xl−1i (x,ζ )

∣∣P ζi (dζ )

pGv (x)

)|x=Gi

×B (Zv)

[ρ (Zv)√

1−ρ (Zv)2

](56)

for v∈ [τi−1,τi), where hi (x,ζ,Zv)≡∂Zv logfZτi |Zv(l−1i (x,ζ )

∣∣Zv) equals

hi (x,ζ,Zv)=∂Zv logfZv |Zτi(Zv|l−1

i (x,ζ ))−∂Zv logfZv (Zv). (57)

Note that (57) uses Bayes’ rule, fZτi |Zv (x|z)=fZv |Zτi (z|x)(fZτi (x)/fZv (z)

).

Formula (56) relates the PIPR to the transition density of the fundamental factor.When ρ =0, the fundamental does not covary with market returns, implying aPIPR component θG,m

v equal to zero. In this case, private information pertainssolely to the fundamental factor and affects the perceived reward associatedwith the fundamental riskWz. This risk (and the information pertaining to thisrisk) is relevant for the optimal investment policy of a non-myopic managerwho seeks to hedge fluctuations in the coefficients of the excess return process.It does not affect the optimal policy of a manager with logarithmic utility whois solely concerned with the instantaneous risk-reward tradeoff.

Another case of interest is when the excess return has a direct relation tofundamentals. Suppose, for instance, that Smτi−1,τi

=V(τi,Zτi

)/V

(τi−1,Zτi−1

),

where V (·,·) is invertible in the second argument. The private informationsignal Gi = li

(Zτi ,ζi

)can then be rewritten in terms of the excess return as

Gi = li(V −1

(τi,S

mτi−1,τi

V(τi−1,Zτi−1

)),ζi

)≡gi

(Smτi−1,τi

,ζi

). Assuming ρ =1

leads to a special (diffusion) case of the model in Section 1. The link betweenthe excess return and the fundamental can be made explicit in the formula for thePIPR. Specializing (56) to this structure and using (57) gives θG

v ≡(θG,mv ,0

)′,

where for v∈ [τi−1,τi)

θG,mv =

(1

pGv (x)

∫R

fSmτi−1,τi |Zv(g−1i (x,ζ )

∣∣Zv)ki (x,ζ,Zv)P ζi (dζ )

)|z=Gi

B (Zv)

(58)

with g−1i (x,ζ )=V

(τi,l

−1i (x,ζ )

)/V

(τi−1,Zτi−1

)and

ki (x,ζ,Zv)

=(∂Zv logfZv |Smτi−1,τi

(Zv|g−1

i (x,ζ ))−∂Zv logfZv (Zv)

)∣∣∇xg−1i (x,ζ )

∣∣.(59)

The dependence of the excess return on the fundamentals is embedded in thefunction g−1

i (x,ζ ) (in the term V (τi,x)). This dependence affects the PIPR

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through multiple channels, as described in (58)–(59). Moreover, it is clear thatany signal about the fundamental can be written as a signal about the excessreturn (and conversely) if the map V (τi,x) is a bijection.

A special case arises when the fundamental factor is the aggregatedividend.

Corollary 6. Suppose that the discounted value of the market is Smt =V (t,Dt )whereD is the aggregate dividend that satisfies dDv =A(Dv)dv+B(Dv)dWm

v .Assume that private information pertains to future dividends, Gi = li(Dτi ,ζi),and that the conditional density exists. In this environment, the PIPR is givenby (58)–(59) where D replaces Z.

The major difficulty in equilibrium settings with heterogeneous informationis to identify the endogenous equilibrium pricing function V (t,·) and theendogenous information variables generating it. This issue is addressed inSection 4.4.

4.3 Volatility timingVarious empirical studies document the importance of volatility timing (Busse1999; Fleming, Kirby, and Ostdiek 2001, 2003). This is perhaps not surprising,given the evidence of predictability in volatility (Bollerslev, Chou, and Kroner1992). This section develops a model where the fund manager has volatilitytiming skill and discusses some of the implications for the fund return.

The basic setting is a Heston volatility model

dRmv =√Zv(θmv dv+dWm

v

)(60)

dZv =κ(Z−Zv

)dv+σ z

√Zv

(ρdWm

v +√

1−ρ2dWzv

)(61)

where (Wm,Wz) is a bivariate Brownian motion and ρ is the constantcorrelation between market returns and variance changes.Z is the instantaneousvariance of the market excess return process (60). The volatility coefficient ofthe volatility process

√Z is σ z/2.

At time τi−1, the skilled manager collects a private signal Gi = li(Zτi ,ζi

)where ζi is independent noise and li (·,ζi) is a one-to-one transformation. Thesignal provides information about the local variance of the market excess returnat τi . The conditional density of the signal is given by (55) evaluated at thetransition density of the instantaneous variance

fZv+�|Zv (y|Zv)=eκ�

2c(�)

(Zve

κ�

y

)δ/2

×exp

(−y+Zveκ�

2c(�)

)Iδ

(√Zvyeκ�

c(�)

)1Zv≥0

(62)

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where c(�)=((σ z)2/4κ

)(eκ�−1

)and δ=

(2κZ/(σ z)2)−1 and where

Iδ (·) is the Bessel function Iδ (y)≡(y

2

)α∑∞j=0

(−1)j (y2/4)jj !�(δ+j+1) with �(x)≡∫∞

0 exp(−t)tx−1dt as the Gamma function.An application of Proposition 11 then gives:

Proposition 12. Consider the Heston volatility model (60)–(61) along withthe private variance signalGi = li

(Zτi ,ζi

). The PIPR θG

v ≡(θG,mv ,θG,z

v

)′is given

by formulas (56)–(57), with fZτi |Zv as defined in (62). Consider a fund tradingthe market. The optimal informed investment policy and the associated fundexcess return are given by the formulas in Proposition 1 with θG,m

v in place ofθGv . The fund excess return has the regression decomposition of Proposition 4

based on θG,mv .

One insight derived from this proposition is that variance information isuseful to a logarithmic market timer only if the variance process covaries withthe market excess return process—that is, if there is a leverage effect. Indeed,if correlation ρ =0, the PIPR for market risk is null, θG,m

v =0, implying that theinformed logarithmic manager behaves as an uninformed investor. In this case,the resulting timing option in the fund excess return is null. If correlation ρ �=0,the informed logarithmic manager exploits the information and trades so as togenerate a timing option component.

While variance information may help to time the market, it can also be usedto time other assets that depend on volatility. In particular, volatility-dependentderivatives, such as VIX futures, options, or exchange-traded funds (ETFs), canbe timed using variance information. Formulas for excess returns generated bya skilled fund trading volatility derivatives—that is, a volatility timer—can bederived using the methodology developed in this paper.

4.4 Equilibrium with asymmetric informationThis section embeds the settings of Sections 4.1 and 4.2 in an equilibriummodel with asymmetric information. Classic references on this subject includeGrossman (1976, 1980), Grossman and Stiglitz (1980), and Wang (1993). Themodel outlined below builds on this literature. It adds a dynamic elementrelative to static models such as Grossman and Stiglitz (1980). It complementsWang (1993) by considering information about a future dividend payment,assuming logarithmic utility function and a finite horizon. The main result isa detailed characterization of a noisy rational expectations equilibrium. Theregression representation of returns is shown to hold in this equilibrium.

To simplify the presentation, assume that there is a single timing window.Thus, N =1 and τN =τ1 =T . There are two assets, a risky and a risklessasset. The riskless asset is a money market account paying instantaneousinterest rate r . Without loss of generality, set r =0. The risky asset dividendDT , paid at the terminal date, is the final value of the fundamental Dv =

D0exp(∫ v

0

(μDs − 1

2

(σDs

)2)ds+

∫ v0 σ

Ds dW

Ds

). The process WD is a Brownian

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motion generating the filtration FD(·), and the coefficients

(μD,σD

)are adapted

to FD(·). The filtration generated by D is assumed to be the same as FD

(·). Thefundamental process is observed, so FD

(·) is part of the public information set.The risky asset supply of shares is normalized to one. There are three types ofagents, a skilled active fund manager, a (representative) uninformed investor,and a (representative) noise trader.

At the initial date τ0, the fund manager receives a private signal G thatconveys noisy information about the terminal dividend. The noise structure

is ζ =(

1/√h)∫ T +h

Tσ ζv dW

ζv , where Wζ is an independent Brownian motion

and σ ζ is a function of time. The signal is assumed to have a generalstructure G=g(DT ,ζ ). The informed filtration is G(·) =Fm

(·)∨σ (G), where

Fm(·) is the endogenous public filtration. The public filtration includes the

information generated by the equilibrium risky asset price as well as otherobservable equilibrium quantities. It also includes the information conveyedby the fundamental process (i.e., FD

(·) ⊆Fm(·)). The manager’s preferences are

Ua =Ea[

log(XaT

)∣∣G0], where XaT is terminal wealth (the terminal value of the

active fund).The manager is assumed to act as a price taker. He or she maximizesutility by choosing the best portfolio in the class of admissible portfolios.

The uninformed investor observes public information Fm(·).

15 Uninformedpreferences have logarithmic utility function over terminal wealth XuT . Theuninformed agent acts as a price-taker.

The noise trader is a mimicking agent who emulates the demand of the activefund, but based on pure noise. His or her demand for the risky asset equalsXnvπ

nv ≡Xnv (θmv +φv)/σmv , where φ is an independent nonpersistent stochastic

process, and Xn is noise-trading wealth. Unless explicitly stated, noise takesthe form φv =k((1/

√h)∫ T +hT

σ φv (s)dWns +μφv ), where k(·) is a differentiable

function, Wn is an independent Brownian motion, and μφs ,σφs (·) are functions

of time. The noise-trading demand has a mean-variance structure perturbedby φ.

The next proposition provides a characterization of equilibrium.

Proposition 13. Consider the model with heterogeneous information andprice-taking behavior described above and assume that an equilibriumexists. The PIPR of the informed manager is θGv (x)=DD

v logpGv(x|Fm

v

). The

endogenous information revealed in equilibrium includes

Zv =δav θGv +δnvφv (63)

15 The endogenous filtration Fm(·) includes, in particular, aggregate risky asset holdings. From the market-clearing

condition, the uninformed investor can infer the combined equilibrium holdings of other agents in the market.

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where θGv ≡θGv (G) and the wealth shares δav ,δ

nv of the active fund and the noise

traders satisfy

dδav =δav(θGv −Zv

)(−Zvdv+dWmv

), dδnv =δnv (φv−Zv)

(−Zvdv+dWmv

)(64)

with(δa0 ,δ

n0

)=(na,nn) and δuv =1−(

δav +δnv). In equilibrium, the optimal

amounts invested by the active fundXaπa and the uninformed investorXuπu,and the amount invested by the mimicking noise trader Xnπn, are

Xavπav =Xav

θmv +θGv

σmv, Xuvπ

uv =Xuv

θmv

σmv, Xnvπ

nv =Xnv

θmv +φvσmv

. (65)

The equilibrium market price of risk is θmv =σmv −Zv . The equilibrium assetprice Sm, its volatility σm, and the Brownian motion Wm satisfy the forward-backward stochastic differential equation (FBSDE)

dSmv =Smv σmv

((σmv −Zv

)dv+dWm

v

), SmT =DT . (66)

The active fund equilibrium excess return is d−Rav =πav d−Rmv , where

d−Rmv =dSmv /Smv . The regression of Proposition 4 holds with public

information Fm(·) ⊇FD

(·)∨FZ

(·) ≡FD,Z(·) and βmv =1−Zv/σmv . If Fm

(·) �=G(·), theequilibrium is a noisy rational expectations equilibrium (NREE). If Fm

(·) =

FD,Z(·) and d

[Wm,WD

]v/dv=1 then dWm

v =dWDv −θD,mv dv and θD,mv =θZ

v =

E[dWD

v

∣∣FZv

]=Dv logpZ

(z|FD

v

)|z=Z is the price of fundamental risk in the

public filtration.

Proposition 13 describes the structure of equilibrium in this economywith asymmetric information. The endogenous signal (63) shows that theinformation extracted from the price includes the noisy translation Zv =δav θ

Gv +δnvφv of the information of the skilled manager. The components of

this signal are weighted by the wealth shares of the active fund and the noisetrader. Information revelation need not occur because φ is not observed. Theendogenous public information set is such that Fm

(·) ⊇FD,Z(·) . In equilibrium, the

PIPR of the skilled fund is calculated relative to Fm(·).

The regression representation of the active fund return still holds, but isnow calculated using the endogenous information structure Fm

(·). The mostsignificant modification is the endogenous structure of the market beta, whichbecomes stochastic and reflects the information extracted from the equilibriumprice and allocations. Another aspect concerns the market-timing factor andthe regression errors, which are obtained by integrating over the signal noise ζusing the conditional distribution P

(ζ ∈dz|Fm

T

)rather than the unconditional

distribution P (ζ ∈dz). Further discussion of the regression is postponed to theend of the section, pending a review of special cases and variations of the model.

In this equilibrium, the filtration Fm(·) generated by the equilibrium price and

quantities and by the fundamental includes the filtration FD,Z(·) generated by

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the fundamentalD and the noisy signal Z. This shows the link between earliersections and this one, and highlights the relevance of the previous derivationsfor the construction and analysis of an equilibrium with endogenous asymmetricinformation.

As indicated above, the proposition characterizes the equilibrium, but doesnot prove existence. Demonstrating existence boils down to showing theexistence of a triplet (Sm,σm,Wm) solving to forward-backward system (63)–(64), (66). This complex issue as well as the design of numerical algorithms forcomputing the solution of this equation are topics of interest for future research.

Special cases and variations of the model are discussed next. Remark 1illustrates the importance of noise trading for the existence of an equilibriumwhere private information has value.

Remark 1 (Fully revealing equilibrium). Under appropriate regularity con-ditions, equilibrium is fully revealing in the absence of noise trading (φ=0). Theendogenous public filtration is the informed filtration,Fm

(·) =G(·). The conditional

distribution pGv(x|Fm

v

)is then degenerate, leading to the PIPR θGv (x)=

DDv logpGv

(x|Fm

v

)=0. Endogenous incentives to collect private information are

effectively null. Thus Z=0 and θmv =σmv . The distribution of wealth becomes(δav ,δ

uv ,δ

nv

)=(na,nu,nn), and the combined holdingsXavπ

av +Xnvπ

nv =Smv (1−nu)

reflect the price information. As dSmv =Smv σmv

(σmv dv+

(dWD

v −θD,mv dv))

,revelation occurs through prices. Moreover, the price of fundamental risk inthe public information is θD,mv =E

[dWD

v

∣∣Fmv

]/dv=DD

v logpGv(x|FD

v

)|x=G.

In the equilibrium described in Remark 1, the market excess return revealsdWD

v −θD,mv dv. Given that innovations dWDv in the fundamental are observed,

it follows that θD,mv is observed. The relation θD,mv dv=E[dWD

v

∣∣Fmv

]=

E[dWD

v

∣∣Gv] shows that θD,mv depends on the private information of the activefund. The price of fundamental risk in the market filtration becomes the vehiclefor information revelation.

When information is costly, full revelation leads to the well-knownparadox concerning the impossibility of informationally efficient markets (seeGrossman and Stiglitz 1980). Noise trading and other unobservable demandfactors foster incentives to collect private information.

Extreme forms of mimicking “noise” trading can lead to completelyuninformative market filtrations. Remark 2 elaborates on this point.

Remark 2 (Uninformative equilibrium). If mimicking “noise” trading emu-lates the negative of the informed demand (φv =−(δav /δnv )θG

v ), an uninformativeequilibrium exists. In this case, information revealed is nullZ=0 and Fm

(·) =FD(·).

The PIPR becomes θGv (x)=DDv logpGv

(x|FD

v

) �=0. The equilibrium asset pricesatisfies dSmv =Smv σ

mv

(σmv dv+dWD

v

)with SmT =DT , and the equilibrium market

price of risk is θmv =σmv =σD . In this equilibrium, incentives to collect privateinformation are unaffected by the observation of prices or quantities.

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The mimicking noise trader in Remark 2 can be interpreted as an agentwho regularly meets with the active fund manager and is able to extractperfect information about the absolute size of the fund’s informational demand,but is systematically incorrect or misled on the direction. Alternatively, themimicking agent is an investor blindingly seeking to emulate the active fundinvestment policy based on informal reports and rumors. Investments taken onthe basis of these beliefs offset the informational demand of the active fundand result in an uninformative price. An active fund manager in this positioncan strategically use the mimicking noise trader to protect his or her privateinformation. Protection is achieved by leaking false information.

Incentives for strategic behavior stem from the fact that an informed agenthas monopoly power over the information possessed (Hellwig 1980). In theabsence of mimicking noise trading, this monopoly power can be exercised bychoosing portfolios taking their market impact into account. The next remarkprovides basic insights about the resulting equilibrium configuration.

Remark 3 (Strategic behavior). Suppose that the active fund behavesstrategically. Its demand function has the decomposition πav =πa,mv +πa,Gv ,where πa,mv =θmv /σ

mv is based on public information and πa,Gv reflects all

other trading motives, including private information trades. Market clearingimplies θmv =σmv −(

δav πa,Gv +δnvφv

)where π a,Gv ≡πa,Gv σmv . Note that δav ,δ

nv are

functionals of past values of π a,G and, consequently, define Zv(πa,G(·)

)≡

δav πa,Gv +δnvφv . The strategic fund with logarithmic utility maximizes Ua subject

to θmv =σmv −Zv(πa,G(·)

), with respect to π a,G. Suppose that the solution to

this optimization problem exists, and denote it by π a,G,∗(·) . The endogenous

information conveyed in equilibrium includes Z∗v ≡Zv

(πa,G,∗(·)

)=δav π

a,G,∗v +

δnvφv . The equilibrium market price of risk is θmv =σmv −Z∗v . The equilibrium

asset price Sm and its volatility σm satisfy the FBSDE (66), where Z∗ replacesZ and the wealth shares (δa,δn) solve the system (64) evaluated at Z∗. Theregression of Proposition 4 holds with public information Fm

(·) ⊇FD,Z∗(·) and

βmv =1−Z∗v/σ

mv .

Strategic behavior adds the market-clearing constraint to the active fundoptimization problem. Given the path-dependent nature of the distribution ofwealth and of the endogenous market price of risk, the logarithmic investorno longer behaves myopically. The portfolio selection problem becomesnontrivial. Its resolution, which is likely to require perturbation methods andnumerical schemes, is beyond the scope of this paper. Taking the solution asgiven nevertheless leads to the characterization of the equilibrium stock priceand market price of risk stated. This structure differs from the nonstrategic casethrough the informational content of endogenous variables.

In this strategic equilibrium, the news gathered by the uninformed traderinclude (Z∗,D). The public filtration Fm

(·) ⊇FD,Z∗(·) is now used to assess the

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performance of the active fund. The regression representation still applies, butrelative to that information structure and with the corresponding endogenousmarket beta.

The final remark describes equilibrium in a microstructure setting, inspiredby Kyle (1985) and Back (1992), in which the informed agent interacts with amarket marker and noise traders. The market maker is risk neutral and sets theequilibrium asset price.

Remark 4 (Equilibrium with a market maker). Consider a financial mar-ket where the price of the asset is set by a risk-neutral market makerwho observes the aggregate demand schedule of the informed fund and a(representative) noise trader. The equilibrium price is Smv =E

[DT |Fm

v

]where

the endogenous filtration Fm(·) is generated by the aggregate order flow. It

follows that prices are martingales under P and that the market price of riskis θmv =0. Systematic risk is null as well, so that βm=0. The equilibrium pricesolves the backward stochastic differential equation (BSDE) dSmv =Smv σ

mv dW

mv

with SmT =DT . The informed trader has a pure informational demand Xavπav =

XavθGv /σ

mv . The aggregate order flow is Xv

(δav θ

Gv +δnvφv

)/σmv . Information

revelation occurs through Zv≡δav θGv +δnvφv . The market maker’s filtration is

Fm(·) =FZ

(·).

In this market-making framework, the excess return of the active fundsatisfies the regression representation of Proposition 4 with Fm

(·) =FZ(·) and

βm=0. The general structure of the market-timing factor and the error termremain the same.

To conclude the section, let us briefly return to the regression structure of theactive fund excess return in the NREE. From Remarks 1–3, it is clear that theequilibrium measure of systematic risk is one, βm=1, if the equilibrium is fullyrevealing (Fm

v =Gv) or fully nonrevealing (Fmv =FD

v ). In a NREE, as describedin Proposition 13, beta is time varying and deviates from its homogeneousinformation value by −Zv/σmv , where Zv/σmv =δav

(θGv /σ

mv

)+δnv

(φv/σ

mv

)is the

wealth-weighted sum of the informational demand θGv /σ

mv of the fund manager

and of the mimicking demand φv/σmv of the noise trader. This has importantramifications for performance measurement. Indeed, if beta is assumed tobe constant, an unskilled fund can mistakenly be identified as an over- orunderperforming market timer. Moreover, deviations from the symmetricequilibrium beta are seen to depend on the process governing informationrevelation and the market volatility. Ceteris paribus, the absolute value ofthe deviation increases if the market volatility decreases. The deviation alsoincreases if the informational fund demand decreases or if the mimicking noisetrading demand decreases. Likewise, the deviation increases if the wealthshare of the fund (respectively of the noise trader) decreases as long as theinformational fund demand (respectively the mimicking noise trading demand)is positive. Similar qualitative properties hold in the strategic equilibrium ofRemark 3.

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5. Conclusion

This paper lays theoretical foundations for regression analyses of active fundreturns. It develops structural models of market timing when the trading intervaldiffers from the observation interval. The theory shows that the endogenousexcess return generated by an informed timing activity, when projected on thepublic information set, satisfies a nonlinear regression with two parts. The firstpart, generated by trades based on public information, has a time aggregationcomponent and an exposure relative to the market excess return. The secondpart, associated with trades based on private information, consists of a timeaggregation component, a timing factor, and a regression error. The endogenoustiming option is inherently path-dependent. The endogenous regression errordisplays heteroscedasticity. A detailed analysis of specialized timing skillmodels is carried out. The ability to distinguish between selection and timingability is also analyzed. Extensions of the model deal with volatility informationand with endogenous information structures. The regression structure is shownto apply across models with suitable specializations of its coefficients.

The endogenous regression model provides a basis for the development ofnovel econometric inference tools tailored to the analysis of active fund returns.It shows that excess returns can be represented in the form of a nonlinear statespace system where the measurement equation contains both time aggregationand market-timing factors. The model can be used to derive exact distributionalproperties of the noise in the measurement equation. Specific distributionalforms are associated with specific market-timing strategies pursued. Theexplicit regression structure of the measurement equation also permits thederivation of the filtered likelihood of the full system using high-frequencyreturns. The combination of low-frequency information about active fundreturns with high-frequency information about market returns raises issuesthat are similar to those in the realized volatility literature. Questions tobe addressed pertain to the optimal sampling scheme and to the asymptoticconvergence of timing factors. The study of these questions, and more generallythe development of econometric estimation techniques for the class of structuraltiming models derived here, is an important and promising avenue for futureresearch.16

Appendix

Appendix A: An introduction to forward integrationThis appendix provides an economically motivated introduction to forward integrals. Amathematically rigorous exposition can be found in Russo and Vallois (1993).

Ito integrals are important in finance because they can be used to represent the gains from tradewhen continuous trading is admissible. More specifically, suppose that the price process Sm is an

16 Detemple and Rindisbacher (2012) develop a Lagrange multiplier (LM) test of market timing based on theendogenous regression structure. A comparison study documents the respective performances of this and otherstandard tests of market timing.

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Fm(·)-semimartingale and that the trading policy ϕ implemented is adapted to Fm

(·). The cumulativegain from trade realized at the terminal date T is then given by

P− limN→∞G

NT ≡P− lim

N→∞

⎛⎝N−1∑j=0

ϕj�

(Sm(j+1)�−Smj�

)⎞⎠=∫ T

0ϕv−dSmv ≡GT (A1)

where �=T/N and the limit is in probability. Under the conditions described and additionalintegrability conditions, the limit on the left-hand side exists. This limit defines the Ito integral onthe right-hand side, which represents the gains from trade in this setting with continuous trading.

Ito integrals can also be considered as limits in the topology of uniform convergence on compacts(limits in the ucp sense), for policies that are adapted.17 Indeed, consider an adapted ϕ for which

the Ito integral exists. Define the approximating sequence{ϕηN (v) :v∈ [0,T ]

}where

ηN (v)=j� for v∈ [j�,(j +1)�) (A2)

The policy ϕηN (·) is a piecewise constant approximation of ϕ. It samples ϕ at the discrete dates

{j� :j =0,...,N−1}. Gains from trade for this approximation are GNT =∫ T

0 ϕηN (v)dSv . Because{ϕηN (v) :v∈ [0,T ]

}is adapted and in the limit converges pointwise to {ϕv− :v∈ [0,T ]}, the ucp-

limit of GNT corresponds to the Ito integral GT defined above (i.e., GNTucp→GT ).

This reinterpretation of the Ito integral, as a limit in the ucp sense, is the key to its extensionto anticipative integrands. Consider an anticipating (i.e., non-adapted) process ϕ. The associated

summation GNT ≡∑N−1j=0 ϕj�


)does not converge in the sense of (A1). The limit

(the Ito integral) does not exist.Forward integration can be used to circumvent this difficulty. Forward integrals were introduced

by Russo and Vallois (1993) as ucp-limits of approximating sums of possibly non-adaptedintegrands ϕ

ucp− lim�↓0

1

�

∫ T

0ϕv−

(∫ v+�

v

dSms

)dv≡

∫ T

0ϕv−d−Sv. (A3)

The limit on the left-hand side of (A3) exists. It defines the random variable∫ T

0 ϕv−d−Sv . The

notation d− indicates that the differential is taken in the sense of forward integration instead of theusual sense of Ito integration (as in (A1)).

For an application to finance, consider an anticipating policy ϕ and use the sampling scheme

(A2) to construct its approximation,{ϕηN (v) :v∈ [0,T ]

}. The gains from trade with the discretized

policy are

GNT =N

T

∫ T

0ϕηN (v)

(∫ ηN (v)+T/N

ηN (v)dSm

ηN (u)

)dv=

N−1∑j=0

ϕj�


).

The ucp-limit ucp−limN→∞GNT ≡∫ T0 ϕv−d

−Smv exists and defines a forward integral. This limitrepresents the gains from trade associated with the anticipative policy considered.

Given that Ito integrals for predictable integrands can also be considered as ucp-limits, theforward integral is a natural representation of gains from trade in a continuous-time setting withanticipative policies.

Biagini and Oksendal (2005) and Kohatsu-Higa and Sulem (2006), among others, use forwardintegrals to represent gains from trade associated with inside information. They show how torecover the results in Pikovsky and Karatzas (1996) that were derived using techniques from thetheory of enlargement of filtration.

17 GNucp→ G if limN→∞P

(sups<t

∣∣∣GNs −Gs∣∣∣>ε)=0 for all ε,t >0.

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Appendix B: Constant relative risk aversionThe excess market return in the private flow of information G(·) has the representation dRmv =σv((θmv +θG

v

)dv+dWG

v

)whereWG is the G(·)-progressively measurable Brownian motion dWG

v =

dWmv −θG

v dv. The state price density with private information is ξGt =exp

(−∫ t0 rvdv)ZG

t where

ZGt ≡exp

(− 1

2

∫ t0

(θmv +θG

v

)2dv−∫ t

0

(θmv +θG

v

)dWG

v

). A price-taking fund manager with constant

relative risk aversion R and initial capital Xa0 ≡Sa0 solves

V G (Sa0 )= infy>0,y∈G0

⎧⎨⎩ supXaT

∈GTE

[ (XaT

)1−R

1−R −yξGT X

aT

∣∣∣∣∣G0

]+ySa0

⎫⎬⎭.The optimal gross return of the fund isXaT /X

a0 ≡exp(

∫ T0 rvdv)Sa0,T =(ξG

T )−1/R/E[ (ξGT )1−1/R

∣∣∣G0].

The excess return in the public information is given next.

Proposition 14. Let F(·) be the public information filtration generated by market returns (Fm(·))

and by the actively managed fund returns observed at the reporting dates (Faτ0

for v<τN and FaτN

for v=τN ). The optimal informed investment policy has the decomposition πa =πFm+πG , where

πFm(respectively πG ) is a portfolio based on public (respectively private) information. In the

public information filtration, the excess return of the fund with timing skill is given by (12) withcomponents

(δ,β,γG ,ηG) described in (A16).

Proposition 14 extends Proposition 4 to constant relative risk aversion R. With R �=1, each ofthe optimal portfolio components can be split into a mean-variance term and a dynamic hedgingterm. Thus, πFm

=πm,Fm+πh,Fm

and πG =πm,G +πh,G , where πm,Fm(respectively πm,G ) is

the mean-variance demand associated with θm (respectively θG ), and πh,Fm(respectively πh,G )

is the dynamic hedge motivated by stochastic fluctuations in θm (respectively θG ). Relative tothe case of logarithmic utility function, all regression components have additional terms, inducedby πh,Fm

and/or πh,G , reflecting the hedging behavior of the fund manager. The mean-variancerelated terms are the same, but scaled by the relative risk aversion coefficient.

Appendix C: ProofsIn order to prove the main results, the following regularity conditions are assumed.

Assumption. Let θv≡θmv +θGv . For i =1,...,N , the PIPR and the informed trading strategy satisfy

the conditions

(i) (Finite quadratic variation)∫ τiτi−1

θ2v dv<∞ P -a.s.

(ii) (Gains from trade bounded below) Let Hεi ≡ 1

ε

∫ τiτi−1

πv(Rmv,v+ε−E

[Rmv,v+ε

∣∣Gv])dv be

the innovations in the gains from trade. Then,Hεi >−Hi for some positive G0-measurable

random variable Hi with E[Hi

∣∣G0]<∞ P -a.s.

Both assumptions are natural and extend standard conditions in models with completeinformation. The first condition ensures that the price of risk for the informed does not explode.When the information price is finite, the informed has an advantage, but not an arbitrage opportunity.This ensures well-posedness of the choice problem with private information. The second conditionrules out doubling strategies. In practice, brokerage houses typically impose account restrictionsof this type and issue margin calls when limits are reached.

Proof of Proposition 1. An investor with anticipative information generates wealthXa satisfyingthe forward stochastic differential equation d−Xav =Xav rvdv+Xavπvd

−Rav where d− indicatesthe forward differential operator (see Russo and Vallois 1993 and Appendix A) associated withthe forward integral. Ito’s rule for forward integrals (see Russo and Vallois 1996) gives Xaτi =

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Xaτi−1exp

(∫ τiτi−1

rvdv+∫ τiτi−1

πvd−Rmv − 1

2

∫ τiτi−1

(πvσ

mv

)2dv)

. A fund manager with logarithmic

utility, who maximizes E[

logXaτi

∣∣∣Gτi−1

], then solves

Vi−1 = logXaτi−1+E

[∫ τi

τi−1

rvdv

∣∣∣∣∣Gτi−1

]+ supπ∈G(·)

E

[∫ τi

τi−1

(πvd

−Rmv − 1

2

(πvσ

mv

)2dv

)∣∣∣∣∣Gτi−1

](A4)

where Vi−1 is the value function at time τi−1. Gains from trade with anticipative strategies are

forward integrals. Therefore, by definition,∫ τiτi−1

πvd−Rmv ≡ limε↓0

∫ τiτi−1

πvRmv,v+εε

dv for ε≥0,

where limits are in the sense of uniform convergence in probability on compact intervals (ucp)and where Rmv,v+ε≡

∫ v+εv

dRms . Using this definition, assumption (i) and the assumption that the

coefficient σm is bounded away from zero gives∫ τiτi−1

(θvσmv

)2dv<∞ where θv≡θmv +θG

v , and

∫ τi

τi−1

(πvd

−Rmv − 1

2

(πvσ

mv

)2dv

)

=− 1

2

∫ τi

τi−1

(πvσ

mv − θv

σmv

)2

dv+1

2

∫ τi

τi−1

(θv

σmv

)2

dv+∫ τi

τi−1

πvd−(Rmv −

∫ v

0θsds

)

≤ 1

2

∫ τi

τi−1

(θv

σmv

)2

dv+∫ τi

τi−1

πvd−(Rmv −

∫ v

0θsds

).

Also,∫ τiτi−1

πvd−(Rmv −∫ v

0 θsds)

=limε↓01ε

∫ τiτi−1

πv(Rmv,v+ε−E

[Rmv,v+ε

∣∣Gv])dv≡ limε↓0Hεi ,

where limits are in the ucp sense (definition of the forward integral). By the law of iteratedexpectations

E[Hεi

∣∣Gτi−1

]=E

[1

ε

∫ τi

τi−1

πv(Rmv,v+ε−E

[Rmv,v+ε

∣∣Gv])dv∣∣∣∣∣Gτi−1

]=0.

For an admissible strategy, approximate gains from tradeHεi are bounded from below. Lebesgue’s

dominated convergence theorem then ensures that expectations and limits can be exchanged, so

that E[

limε↓0Hεi

∣∣Gτi−1

]=limε↓0E

[Hεi

∣∣Gτi−1

]=0. In combination with (A4)–(A5), this gives

Vi−1 ≤ logXaτi−1+E

[∫ τi

τi−1

rvdv

∣∣∣∣∣Gτi−1

]+

1

2E

[∫ τi

τi−1

(θv

σmv

)2

dv

∣∣∣∣∣Gτi−1

]

with equality if and only if πv =θv/σmv . Thus, πv =θv/σmv and d−Rav =(θv/σ

mv

)d−Rmv with θv≡

θmv +θGv , as claimed. �

The next lemma corresponds to Theorem 1.1 in Russo and Vallois (1993). It shows that theforward integral can be interpreted as an Ito integral for certain types of anticipating integrands.

Lemma 1. Let θv (x) be an Fmv ⊗B(R)-measurable18 random field and define the random

function Kεi (x)≡∫ τi

τi−1

( 1ε

∫ vv−ε θs (x)ds

)dRmv . Assume that supε

∣∣Kεi (x)

∣∣<∞ P−a.s. Let G be

18 B(Rk)

is the Borel algebra on Rk, for k∈N.

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a random variable such that the forward integral∫ τiτi−1

θv (G)d−Rmv and the Ito integral evaluated

at the random variable G,(∫ τiτi−1

θv (x)dRmv)

|x=G, are both well defined. The equality

∫ τi

τi−1

θv (G)d−Rmv =

(∫ τi

τi−1

θv (x)dRmv

)|x=G

(A5)

then holds. That is, the forward integral of the anticipating process θv (G) is the Ito integral of theadapted process θv (x) evaluated at the random variable G.

In our context, θv (x)≡θGv (x) will represent the conditional price of risk given G=x. Thetechnical condition on Kε

i (x) ensures that there is no strategy that generates infinite wealth withpositive probability. Under this assumption, the forward integral of the informed price of riskθGv (G) corresponds to the Ito integral of the conditional price of risk θGv (x) evaluated at the signalG.

Proof of Proposition 2. DefineHεv ≡ 1

ε

∫ v+εv

((σmv

)−1dRmv −θmv dv

). Let δG (z) be the Dirac delta

function at the point G. As for arbitrary G∈Gv , F ∈Fmv and B∈B(R)

E

[∫B

1F δG (z)E[Hεv

∣∣Gv]dz]=1

εE

[∫B

1FE

[δG (z)

∫ v+ε

v

dWms

∣∣∣∣Gv]dz]

=1

εE

[∫B

1FE

[δG (z)

∫ v+ε

v

dWms

∣∣∣∣Fmv

]dz

]

=1

εE

[∫B

1FE

[δG (z)

E[δG (z)|Fm

v

] ∫ v+ε

v

dWms

∣∣∣∣∣Fmv

](A6)

× E[δG (z)|Fm

v

]dz

]

=1

εE

[∫B

1F δG (z)E

[δG (z)

E[δG (z)|Fm

v

] ∫ v+ε

v

dWms

∣∣∣∣∣Fmv

]dz

]

it follows that E[Hεv

∣∣Gv]= 1εE[

δG(z)E[ δG(z)|Fm

v ]∫ v+εv

dWms

∣∣∣Fmv

]|z=Gi

. Therefore, with pGv (z)=

Ev [δG (z)], for v∈ [τi−1,τi ) when z=Gi

θGv =lim

ε↓0E[Hεv

∣∣Gv]=limε↓0

1

εEv

[pGv+ε (z)

pGv (z)

∫ v+ε

v

dWms

]|z=Gi

=

(d[logpG (z),Wm

]v

dv

)|z=Gi

.

The first equality uses the definition of the PIPR, the second follows from (A6), and the third followsfrom the definition of the covariation. The right-hand side is equal to the Malliavin derivativeDv logpGv (z)|z=Gi . Thus, θG

v =Dv logpGv (z)|z=Gi and for any h>0

Ev

[θGv+h

]=Ev

[Ev+h

[θGv+h

]]=Ev

[∫R

Dv+hpGv+h (z)

pGv+h (z)PGv+h (dz)

]=Ev

[Dv+h

∫R

PGv+h (dz)

]

where PGv+h is the conditional cumulative distribution of the signal at v+h. The right-handside is Ev [Dv+h1]=0. For s≤v, the PIPR is a martingale with null expectation under publicinformation. �

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Proof of Proposition 3. The ex ante value functions for the skilled and unskilled managers are

Eτi−1

[logXa,sτi

]=logXa,sτi−1

+Eτi−1

[∫ τi

τi−1

rvdv

]+

1

2Eτi−1

[∫ τi

τi−1

(θmv +θG

v

)2dv

]

Eτi−1

[logXa,uτi

]=logXa,uτi−1

+Eτi−1

[∫ τi

τi−1

rvdv

]+

1

2Eτi−1

[∫ τi

τi−1

(θmv)2dv

].

Using Eτi−1

[θmv θ

Gv

]=0 for v≥τi−1, which follows from Ev

[θGv

]=0, and logXa,sτi−1

= logXu,sτi−1

gives Eτi−1

[logXa,sτi

]=Eτi−1

[logXa,uτi

]+(1/2)

∫ τiτi−1

Eτi−1

[(θGv

)2]dv and the ex ante value of

information IGi−1 =exp

(12

∫ τiτi−1

Eτi−1

[(θGv

)2]dv)

.

Using pGτi(z)=pGτi−1

(z)+∫ τiτi−1

Ev

[Dvp

Gτi

(z)]dWm

v =pGτi−1(z)+

∫ τiτi−1

DvEv

[pGτi

(z)]dWm

v =

pGτi−1(z)+

∫ τiτi−1

DvpGv (z)dWm

v =pGτi−1+∫ τiτi−1

pGv (z)θGv (z)dWmv , gives

Eτi−1

[log

pGτi(z)

pGτi−1(z)

∣∣∣∣∣Gi =z

]=Eτi−1

[∫ τi

τi−1

θGv (z)(dWm

v −θGv (z)dv)+

1

2

∫ τi

τi−1

θGv (z)2dz

∣∣∣∣∣Gi =z

]

=1

2Eτi−1

[∫ τi

τi−1

θGv (z)2dz

∣∣∣∣∣Gi =z

]where the second equality uses the Brownian motion property of Wm

v −∫ v0 θ

Gs (z)ds under the

Wiener measure conditional on Gi =z with density dPGτi /dPGτi−1

(PGτi−1,PGτi

are the conditionalcumulative distributions of the signal). Bayes’ law gives

dPτi−1 (ω|Gi =z)=dPGτi−1

(Gi =z|Fm

τi

)dPGτi−1

(Gi =z)dPτi−1 (ω), P−a.s. (A7)

As dPGτi−1

(Gi =z|Fm

τi

)=dPGτi (Gi =z), it follows that:

1

2Eτi−1

[∫ τi

τi−1

(θGv

)2dv

∣∣∣∣∣Gi =z

]=Eτi−1

[log

pGτi(z)

pGτi−1(z)

∣∣∣∣∣Gi =z

]

=Eτi−1

[dPGτi

dPGτi−1

(z)logpGτi

(z)

pGτi−1(z)

]

=Eτi−1

[dPGτi

dPGτi−1

(z)logpGτi (z)

]−logpGτi−1

(z)

and, because Eτi−1

[∫ τiτi−1

(θGv

)2dv]

=∫∞−∞Eτi−1

[ ∫ τiτi−1

(θGv

)2dv

∣∣∣Gi =z]dPGτi−1

(z),

1

2Eτi−1

[∫ τi

τi−1

(θGv

)2dv

]=∫ ∞

−∞Eτi−1

[dPGτi

(z)logpGτi (z)]−∫ ∞

−∞dPGτi−1

(z)logpGτi−1(z)

=Eτi−1

[logpGτi (Gi )

]−Eτi−1

[logpGτi−1

(Gi )]

=Eτi−1

[log

pGτi(Gi )

pGτi−1(Gi )

].

Using the definition DKL

(pGτi

(z)∣∣∣pGτi−1

(z))≡Eτi

[log

(pGτi

(Gi )/pGτi−1(Gi )

)]and the law of

iterated expectations completes the proof. �

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Proof of Proposition 4. The PIPR is θGv =θGv (Gi ) with Gi =g

(Smτi−1,τi

,ζi

). By Lemma 1

logSaτi−1,τi=− 1

2

∫ τi

τi−1

(θmv +θG

v

)2dv+

∫ τi

τi−1

θmv

σmvdRmv +

(∫ τi

τi−1

θGv (x)

σmvdRmv

)|x=Gi

.

For v∈ [τi−1,τi ), define θGv =θGv (Gi ) and let γG

v ≡θGv /σ

mv and ηG

v =(2θmv +θG

v

)θGv . Also define

maτi− ≡E[

logSaτi−1,τi

∣∣∣Fτi−]

and write logSaτi−1,τi=maτi− +

(logSaτi−1,τi

−maτi−)

. Substituting

the previous expression gives, after simplifications,

maτi− =− 1

2

∫ τi

τi−1

(θmv)2dv+

∫ τi

τi−1

θmv

σmvdRmv − 1

2

∫ τi

τi−1

E[ηGv

∣∣∣Fτi−]dv

+∫ τi

τi−1

E[γGv

∣∣∣Fτi−]d−Rmv

logSaτi−1,τi−maτi− =− 1

2

∫ τi

τi−1

(ηGv −E

[ηGv

∣∣∣Fτi−])dv+

∫ τi

τi−1

(γGv −E

[γGv

∣∣∣Fτi−])d−Rmv .

Defining δv,βv,ψi ,fmi ,ε

mi ,ε

ai as indicated in the proposition leads to the decomposition

stated. �

Proof of Corollary 1. Use the density of ζi to rewrite conditional expectations as integrals. �

Proof of Proposition 5. The state price densities are

ξGτi−1,τi

≡exp

(−∫ τi

τi−1

(rs +

1

2

(θmv +θG

v

)2)dv−

∫ τi

τi−1

(θv +θG

v

)dWG

v

)(A8)

ξmτi−1,τi≡exp

(−∫ τi

τi−1

(rs +

1

2

(θms)2)ds−

∫ τi

τi−1

θms dWms

). (A9)

Optimal policies of a manager with logarithmic utility areXa,sτi−1,τi=(ξGτi−1,τi

/x)−1

andXa,uτi−1,τi=(

ξmτi−1,τi/x)−1

. It follows that Xa,sτi−1,τi=Xa,uτi−1,τi

(pGτi

(Gi )/pGτi−1(Gi )

). Taking the log on both

sides gives Lsi =Lui +μi−1(Lui

)+εi with

μi−1(Lui)≡Eτi−1

[log

pGτi(Gi )

pGτi−1(Gi )

∣∣∣∣∣Lui], εi ≡ log

pGτi(Gi )

pGτi−1(Gi )

−Eτi−1

[log

pGτi(Gi )

pGτi−1(Gi )

∣∣∣∣∣Lui].

By Jensen’s inequality and the tower property of conditional expectations,

μi−1(Lui)

=−Eτi−1

[Eτi−1

[log

pGτi−1(Gi )

pGτi(Gi )

∣∣∣∣∣Fmτi

]∣∣∣∣∣Lui]

>−Eτi−1

[log

(Eτi−1

[pGτi−1

(Gi )

pGτi(Gi )


])∣∣∣∣∣Lui]

=−Eτi−1

[log

(∫ ∞

−∞

dPGτi−1(z)

dPGτi(z)

dPGτi(z)

)∣∣∣∣∣Lui]

=−Eτi−1

[log

(∫ ∞

−∞dPGτi−1

(z)

)∣∣∣∣Lui ]=−Eτi−1

[log(1)|Lui

]=0.

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Thus, μi−1(Lui

)>0. Moreover, the optimality of Xa,sτi−1,τi

for the informed log

manager gives Eτi−1

[Lsi

]>Eτi−1

[Lui

]. Integration by parts then yields Ti−1 (∞)=

−(Eτi−1

[Lsi

]−Eτi−1

[Lui

])=−logIG

i−1<0. The last equality holds because the log-certainty

equivalent of the information value is the relative entropy. �

Proof of Proposition 6. Let Fξ,si−1 (respectively Fξ,ui−1) be the cumulative distribution function

for the informed (respectively uninformed) state price density. Define �ξi−1 (y)≡Fξ,si−1 (y)−

Fξ,ui−1 (y) and T

ξi−1 (y)≡∫ y

−∞(Fξ,si−1 (z)−Fξ,ui−1 (z)

)dz, and let �vi−1 ≡Eτi−1

[J(ξGτi−1,τi

/x)]

−Eτi−1

[J(ξmτi−1,τi

/x)]

=∫∞

0 J (y/x)d�ξi−1 (y). Integration by parts gives

�vi−1 =J (y/x)�ξi−1 (y)|∞0 −(1/x)∫ ∞

0J ′ (y/x)�ξi−1 (y)dy =−(1/x)

∫ ∞

0J ′ (y/x)�i−1 (y)dy

=−(1/x)J ′ (y/x)T ξi−1 (y)|∞0 +(1/x)2∫ ∞

0J ′′ (y/x)T ξi−1 (y)dy

=−(1/x)J ′ (∞)T ξi−1 (∞)+(1/x)2∫ ∞

0J ′′ (y/x)T ξi−1 (y)dy.

Next, as Eτi−1

[pGτi−1 (Gi )pGτi

(Gi )


]=∫

R

dPGτi−1(z)

dPGτi(z)

dPGτi(z)=

∫RdPGτi−1

(z)=1

Eτi−1

[pGτi−1

(Gi )

pGτi(Gi )

∣∣∣∣∣ξmτi−1,τi

]=Eτi−1

[Eτi−1

[pGτi−1

(Gi )

pGτi(Gi )


]∣∣∣∣∣ξmτi−1,τi

]=1.

As ξGτi−1,τi

=ξmτi−1,τipGτi−1

(Gi )/pGτi (Gi ), Eτi−1

[ξGτi−1,τi

∣∣∣ξmτi−1,τi

]=ξmτi−1,τi

and

Eτi−1

[ξGτi−1,τi

]=Eτi−1

[ξmτi−1,τi

]. Integration by parts gives T

ξi−1 (∞)=0. Let ε

ξi ≡

ξmτi−1,τi

(pGτi−1

(Gi )/pGτi (Gi )−1)

and note that ξGτi−1,τi

=ξmτi−1,τi+εξi withEτi−1

[εξi

∣∣∣ξmτi−1,τi

]=0.

It follows that ξmτi−1,τiSSDξG

τi−1,τiin the mean-preserving spread sense (Rothschild and Stiglitz

1970). Equivalently, T ξi−1 (y)≥0 for all y∈R+. Together with Tξi−1 (∞)=0, this implies

�vi−1 = (1/x)2∫∞0 J ′′ (y/x)T ξi−1 (y)dy<0 (respectively >0) for all y∈R+ if J (·) is concave

(respectively convex). But J ′ (z)=u′ (Ia (z))I ′a (z) and J ′′ (z)=u′′ (Ia (z))I ′

a (z)2 +u′ (Ia (z))I ′′a (z),

which equals

J ′′ (z)=u′ (Ia (z))I ′a (z)2

Ia (z)

[u′′ (Ia (z))Ia (z)

u′ (Ia (z))+I ′′a (z)

I ′a (z)2

Ia (z)

].

Using I ′a (z)=1/u′′

a (Ia (z)) and I ′′a (z)=−(u′′′

a (Ia (z))/u′′a (Ia (z))

)I ′a (z)2 gives J ′′ (z)=

u′ (Ia (z))(I ′a (z)2/Ia (z)

)[Pa (Ia (z))−R(Ia (z))] where Pa (x)≡−u′′′

a (x)x/u′′a (x)

(R(x)≡−u′′ (x)x/u′ (x)) is the relative prudence (risk aversion) of the manager (investor).Condition (24) follows because Pa =2 for log utility and I ′

a (z)2/Ia (z)=1/z3. It implies that aninvestor with R(·)>2 will always prefer the unskilled return. �

Proof of Corollary 2. For v∈ [τi−1,τi ), the conditional density of the signal pGv (x)≡∂xPv (Gi ≤x)=∂xPv

(log

(Smτi−1,τi

)+σyW

ζ1 ≤ log(x)+ 1

2

(σ 2y

))is

pGv (x)=∂xPv

⎛⎝ log(Smv,τi

)−Ev

[log

(Smv,τi

)]+σyW

ζ1√

�v,τi

≤d (x,v)

⎞⎠=∂x�(d (x,v))=φ (d (x,v))

x√�v,τi

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with VARv [logGi ]≡�v,τi and

d (x,v)=logx−Ev [logGi ]√

�v,τi

=logx+ 1

2 (σy )2 −logSmτi−1,v−Ev

[logSmv,τi

]√�v,τi

.

As θGv (x)=d[logpG(x),Wm

]v/dv

θGv (x)=d[pG(x),Wm

]v

pGv (x)dv=

−φ′ (d (x,v))d[logSmτi−1,· ,W

m·]v

φ (d (x,v))√�v,τi dv

=σm(

logx−Ev [logGi ]

�v,τi

)

(because φ′(x)=−xφ (x) and d[logSmτi−1,·,W

m·]v

=σmdv) and θGv =θGv (Gi ) for v∈ [τi−1,τi ).

The optimal portfolio of the privately informed manager is πv =(θm+θG

v

)/σm =β+θG

v /σm =

β+(logGi−Ev [logGi ])/�v,τi . The random field θGv (x)/σm =(logx−Ev [logGi ])/�v,τi satisfiesthe conditions of Lemma 1. An application of Proposition 1 gives the gains from trade

d−Rav =πvd−Rmv =βdRmv +

logGi−Ev [logGi ]

�v,τid−Rmv .

Let νmiv≡ logSmτi−1,τi−Ev

[logSmτi−1,τi

]and note that E

[(logGi−Ev [logGi ])

2∣∣∣Fτi−

]=(

νmiv

)2+VAR[logζi ] where VAR[logζi ]= (σy )2. Simple calculations then show that

E[θGv

∣∣∣Fτi−]

=σmE[

logGi−Ev [logGi ]|Fτi−]

�v,τi=σm

νmiv

�v,τi

E

[(θGv

)2∣∣∣∣Fτi−

]=(σm)2

E[

(logGi−Ev [logGi ])2∣∣∣Fτi−

](�v,τi

)2= (σm)2

(νmiv

)2+(σy )2(

�v,τi

)2

so that

E[ηGv

∣∣∣Fτi−]

=2θmσm(νmiv

�v,τi

)+(σm)2

((νmiv

)2+(σy )2(

�v,τi

)2

), E

[γGv

∣∣∣Fτi−]

=νmiv

�v,τi

γGv −E

[γGv

∣∣∣Fτi−]

=logGi−Ev [logGi ]

�v,τi− νmiv

�v,τi=νmiv +σywi�v,τi

− νmiv

�v,τi=σywi

�v,τi

ηGv −E

[ηGv

∣∣∣Fτi−]

=2θmσm(θGv

σm− νmiv

�v,τi

)+(σm)2

((θGv

σm

)2

−(νmiv

)2+(σy )2(

�v,τi

)2

)

=2θmσmσyw

�v,τi+(σm)2 (σywi )

2 +2(σywi )νmiv−(σy )2(�v,τi

)2.

where wi ≡(1/

√hi)∫ τiτi−1

dWζv . Substituting in (15)–(17) gives the formulas announced. �

Proof of Corollary 3. As in the proof of Proposition 3

1

2Eτi−1

[∫ τi

τi−1

(θGv

)2dv

]=∫ +∞

−∞Eτi−1

[pGτi

(z)logpGτi (z)]dz−

∫ +∞

−∞logpGτi−1

(z)pGτi−1(z)dz

where pGτi (z)= (1/zσy )φ(z′)

and pGτi−1(z)=

(1/z

√�i−1

)φ(z′′)

with z′ ≡(logz−μGi

)/σy , z′′ ≡(

logz−μGi−1

)/√�i−1, μGi ≡ logSmτi−1,τi

− 12 (σy )2, μGi−1 ≡Eτi−1 [logGi ] and �i−1 ≡�τi−1,τi .

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Thus

Eτi−1

[∫ +∞

−∞logpGτi (z)pGτi (z)dz

]=−Eτi−1

[∫ ∞

−∞φ(z′)(

log(√

2πσy)

+σyz′ + 1

2

(z′)2 +μGi

)dz′

]

=−log(√

2πσy)− 1

2−Eτi−1

[μGi

]=−log

(√2πσy

)− 1

2−μGi−1∫ +∞

−∞logpGτi−1

(z)pGτi−1(z)dz=−

∫ ∞

−∞φ(z′′)(

log√

2π�i−1 +√�i−1z

′′ + 1

2

(z′′)2 +μGi−1

)dz′′

=−log√

2π�i−1 − 1

2−μGi−1

so that logIGτi−1

= 12Eτi−1

[∫ τiτi−1

(θGv

)2dv]

=log(√�i−1/σ

y)

=log√

1+(σm/σy )2hi . �

Proof of Corollary 4. Define the events E+ ≡{

logSmτi−1,τi>0

}and E− ≡

{logSmτi−1,τi

≤0}

. For

v∈ [τi−1,τi ), the conditional distribution of the signal Gi is pGv (y)≡Pv (Gi =y) where

pGv (y)=(Pv(E+)pi +Pv (E−)(1−pi )

)1y=1 +

(Pv(E+)(1−pi )+Pv

(E−)pi)1y=−1

=(pi +(1−2pi )Pv

(E−))1y=1 +(1−pi +(2pi−1)Pv

(E−))1y=−1

=(pi1y=1 +(1−pi )1y=−1

)+(1−2pi )

(1y=1 −1y=−1

)�(di

(Smτi−1,v

,v))

=(pi1y=1 +(1−pi )1y=−1

)+(1−2pi )y�

(di

(Smτi−1,v

,v))

≡yHv (y)−1 for y∈{−1,1}

anddi(Smτi−1,v

,v)≡−Ev

[logSmτi−1,τi

]/σm

√τi−v=−

(logSmτi−1,v

+Ev[logSmv,τi

])/σm

√τi−v.

Using d[�(di

(Smτi−1,·,v

)),Wm·

]v

=−φ(di

(Smτi−1,v

,v))dv/

√τi−v≡−Kivdv gives

θGv (y)=d[logpG (x),Wm

]v/dv=d

[pG (x),Wm

]v/(pGv (x)dv

)=2siKivHv (y) where

si ≡pi−1/2. Thus θGv =θGv (Gi )=2siKi (v)Hv (Gi ) and

πv =θm

σm+

2siσm

KivHv (Gi ), d−Rav =

(θm

σm+

2siσm

KivHv (Gi )

)d−Rmv .

Moreover, with the notation H (k)v ≡E

[Hv (Gi )

k∣∣∣Fτi−

],k=1,2,

E[θGv

∣∣∣Fτi−]

=2siKivH(1)v , E

[(θGv

)2∣∣∣∣Fτi−

]=4s2

i K2ivH

(2)v

E[γGv

∣∣∣Fτi−]

=2siσm

KivH(1)v , E

[ηGv

∣∣∣Fτi−]

=4siKiv(θmH (1)

v +siKivH(2)v

).

It remains to calculate the conditional moments H (k)v . To do this, note that the events E+,E− are

public information at time τi . Letting Lmi ≡ logSmτi−1,τi, it follows that:

H (1)v =

{Hv (1)pi +Hv (−1)(1−pi ) if Lmi >0Hv (1)(1−pi )+Hv (−1)pi if Lmi <0

=2si

(�iv−1Li<0

)(−2�ivsi +pi )(2�ivsi +1−pi ) (A10)

H (2)v =

{Hv (1)2pi +Hv (−1)2 (1−pi ) if Lmi >0Hv (1)2 (1−pi )+Hv (−1)2pi if Lmi <0

=pi (1−pi )+4s2

i

(�2iv +(1−2�iv)1Li<0

)(−2�ivsi +pi )

2 (2�ivsi +1−pi )2

(A11)

where �iv≡�(di

(Smτi−1,v

,v))

. Substituting in the general formulas from Proposition 1 for

ψi,fi ,εi leads to the expressions stated. �

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Proof of Corollary 5. Let Pτi (Gi =z)≡pGτi (z) and ki ≡sgn(

logSmτi−1,τi

). Using (11) gives

log(IGi−1

)=

1

2Eτi−1

[∫ τi

τi−1

(θGv

)2dv

]=Eτi−1

[logpGτi (Gi )

]−Eτi−1

[logpGτi−1

(Gi )]

=Eτi−1

⎡⎣ ∑z∈{−1,1}

pGτi(z)logpGτi (z)

⎤⎦−∑

z∈{−1,1}logpGτi−1

(z)pGτi−1(z)

As pGτi (z)=(1ki=1pi +1ki=−1 (1−pi )

)1z=1 +

(1ki=−1pi +1ki=1 (1−pi )

)1z=−1, this equals

log(IGi−1

)=pi logpiPτi−1

(E+)+(1−pi )log(1−pi )Pτi−1

(E−)+(1−pi )log(1−pi )Pτi−1

(E+)+pi logpiPτi−1

(E−)−logpGτi−1(1)pGτi−1

(1)−logpGτi−1(−1)pGτi−1

(−1)

where pGτi−1(z)=2(1−�(di (1,τi−1)))si (1z=1 −1z=−1)+(1−pi )1z=1 +pi1z=−1. Simplifying

yields pi logpi +(1−pi )log(1−pi )−∑z∈{−1,1} logpGτi−1

(z)pGτi−1(z). �

Proof of Proposition 7. From (42)–(43), E[RAv,v+ε

∣∣Gv]=E[ ∫ v+ε

vβsdR

Fs +

∫ v+εv

σAs dWAs

∣∣∣Gv].

For an arbitrary process X, define Hεv ≡ 1

ε

∫ v+εv

dXs . For any G∈Gv , F ∈Fmv and B∈B(Rk), the

same operations as in (A6) give

E

[∫B

1F δG (z)E[Hεv

∣∣Gv]dz]=1

εE

[∫B

1F δG (z)E

[δG (z)

E[δG (z)|Fm

v

] ∫ v+ε

v

dXs

∣∣∣∣∣Fmv

]dz

].

Therefore, with pGv (z)≡E[ δG (z)|Fmv

]for the density of the signal given public information

E

[∫ v+ε

v

βsdRFs

∣∣∣∣Gv]=E

[δG (z)

E[δG (z)|Fm

v

] ∫ v+ε

v

βsdRFs

∣∣∣∣∣Fmv

]|z=G

=E

[pGv+ε (z)

pGv (z)

∫ v+ε

v

βsdRFs

∣∣∣∣∣Fmv

]|z=G

(A12)

E

[∫ v+ε

v

σAs dWAs

∣∣∣∣Gv]=E

[δG (z)

E[δG (z)|Fm

v

] ∫ v+ε

v

σAs dWAs

∣∣∣∣∣Fmv

]|z=G

=E

[pGv+ε (z)

pGv (z)

∫ v+ε

v

σAs dWAs

∣∣∣∣∣Fmv

]|z=G

.

(A13)

The last equalities in these relations follow from the tower property of the conditional expectation.The multiplicative Clark-Ocone formula (see Detemple and Rindisbacher 2010) gives

pGv+ε (z)

pGv (z)=E[δG (z)|Fm

v+ε

]E[δG (z)|Fm

v

] ≡exp

(∫ v+ε

v

Ds logpGs (z)dWs− 1

2

∫ v+ε

v

‖Ds logpGs (z)‖2ds

)

≡ dP zv

dPv |Fv+ε

where Dv logpGv (z)=(DF

v logpGv (z),DAv logpGv (z)

)is the vector of Malliavin derivatives of

logpGv (z) with respect to WF and WA. As in (A7), the measure dP zv (ω)≡dPv (ω|Gi =z)=

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(pGv+ε (z)/pGv (z)

)dPv (ω) is a Wiener measure conditional on {G=z}. The process dWz

v =dWv−(Dv logpGv (z))′dv is a Brownian motion under dP zv (by Girsanov’s theorem). Furthermore, for

any random variable H , E [H |Gv]=EPz [H |Fm

v

]|z=G. Therefore, with θG,Fv (z)′ ≡DF

v logpGv (z),

θG,Av (z)′ ≡DAv logpGv (z) and θG,F

v ≡θG,Fv (G), θG,Av ≡θG,Av (G), it follows from (A12)–(A13) that:

1

εE

[∫ v+ε

v

βsdRFs

∣∣∣∣Gv]=1

εE

[pGv+ε (z)

pGv (z)

∫ v+ε

v

βsσFs

(θFs +θG,Fs (z)

)ds

∣∣∣∣∣Fmv

]|z=G

→βvσFv

(θFv +θG,F

v

)1

εE

[∫ v+ε

v

σAs dWAs

∣∣∣∣Gv]=1

εE

[pGv+ε (z)

pGv (z)

∫ v+ε

v

σAs θG,As (z)ds

∣∣∣∣∣Fmv

]|z=G

→σAv θG,Av

P -a.s., as ε↓0. These expressions show that θG,Fv (G) (respectively θG,A

v (G)) is the price oftiming (respectively selection) information. Expected returns in the private information filtrationbecome E

[d−RAv

∣∣Gv]=βvσFv(θFv +θG,F

v

)+σAv θ

G,Av . They include a timing premium βvσ

Fv θ

G,Fv

and a selection premium σAv θG,Av . The quantity βvσFv (respectively σAv ) is the amount of timing

(respectively selection) risk. �

Proof of Proposition 8. The optimal portfolio π ′v≡

((πFv

)′,(πAv

)′)of the informed manager is

πv =

([σFv 0βvσ

Fv σAv

]′)−1[θFv +θG,F

v

θG,Av

]

=

⎡⎢⎣((σFv

)′)−1(θFv +θG,F

v −(βvσ

Fv

)′((σAv

)′)−1θG,Av

)((σAv

)′)−1θG,Av

⎤⎥⎦.If the market is incomplete and factor-mimicking portfolios do not exist, replace θFv with θFv +λvwhere λv is the shadow price of incompleteness in the portfolio above. Then solve for λv such that

πFv =0. The demand πAv =((σAv

)′)−1θG,Av is not affected. This completes the proof. �

Proof of Proposition 9. Under the optimal policy of the informed manager, the fund’s

excess return is d−Rav =(πFv

)′d−RFv +

(πAv

)′d−RAv =

(θFv +θG,F

v

)′(σFv

)−1d−RFv +

(θG,Av

)′d−WA

v .

Defining(βFv

)′ ≡(θFv)′(σFv

)−1and

(γG,Fv

)′ ≡(θG,Fv

)′(σFv

)−1gives d−Rav =

(βFv

)′dRFv +(

γG,Fv

)′d−RFv +

(θG,Av

)′d−WA

v . As logSaτi−1,τi=∫ τiτi−1

(dRav − 1

2d [Ra]v), it follows that:

logSaτi−1,τi=∫ τi

τi−1

(βFv

)′dRFv +

∫ τi

τi−1

(γG,Fv

)′d−RFv +

∫ τi

τi−1

(θG,Av

)′d−WA

v − 1

2

∫ τi

τi−1

(δv +ηv)dv

with δv≡∥∥θFv ∥∥2and δv +ηv≡

∥∥∥(βFv +γG,Fv

)′σFv

∥∥∥2+∥∥θG,Av

∥∥2=∥∥θFv +θG,F

v

∥∥2+∥∥θG,Av

∥∥2. Decom-

posing logSaτi−1,τiin an expectation with respect to public information and an error term leads to

the formulas stated. �

Proof of Proposition 10. Applying the arguments in the proof of Proposition 4 to the case underconsideration establishes the result. �

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Proof of Proposition 11. By assumption, Gi = li(Zτi ,ζi

)is a one-to-one mapping of Zτi .

The conditional distribution of Gi is Pv (Gi ∈ (−∞,x]|ζi )=Pv(Zτi ∈ l−1

i ((−∞,x],ζi )∣∣∣ζi)=

P(Zτi ∈ l−1

i ((−∞,x],ζi )∣∣∣Zv,ζi) where the last equality uses the Markovian structure of Z.

The independence of ζi implies pGv (x)=∫

RfZτi |Zτi

(l−1i (x,ζ )

∣∣∣Zv)∣∣∣∇x l−1i (x,ζ )

∣∣∣P ζi (dζ ). The

Malliavin derivative is

DvpGv (x)=

∫R

fZτi |Zτi(l−1i (x,ζ )

∣∣∣Zv)(Dv logfZτi |Zτi(l−1i (x,ζ )

∣∣∣Zv))∣∣∣∇x l−1i (x,ζ )

∣∣∣P ζi (dζ ).

Bayes’ law (fZτi |Zv (x|z)=fZv |Zτi (z|x)(fZτi (x)/fZv (z))) and the rules of Malliavin calcu-

lus lead to Dv logfZτi |Zτi (l−1i (x,ζ )|Zv)= (∂Zv logfZv |Zτi (Zv |l−1

i (x,ζ ))−∂Zv logfZv (Zv))DvZv .

Moreover, DvZv =B(Zv)(ρ(Zv),√

1−ρ(Zv)2). Substituting these expressions in θGv (x)=(Dv logpGv (x)right)′ =(Dvp

Gv (x)right)′/pGv (x) leads to the formula stated. �

Proof of Corollary 6. Apply Proposition 11 to Gi =gi(Smτi−1,τi

,ζi

)with Smt =V (t,Dt ). �

Proof of Proposition 12. Given (62), the result is a special case of Proposition 11. �

Proof of Proposition 13. Assume an equilibrium where dSmv /Smv =σmv

(θmv dv+dWm

v

)for some

Brownian motion process Wm adapted to the equilibrium public filtration Fm(·). The variable θm is

the equilibrium market price of risk associated with the innovations inWm and σm is the volatilityof the equilibrium return. Both are adapted to Fm

(·). The proof below identifies the endogenousrestrictions on the price components.

Let pGv(x|Fm

v

)be the conditional density of the signal received by the informed, given public

information. The arguments underlying Proposition 4 lead to the PIPR relative to the publicfiltration, θGv (x)=d

[logpG

(x|Fm

v

),Wm

]v/dv. In addition to observing dividends, uninformed

agents extract some information Z from equilibrium prices and quantities, leading to an MPR

θmv that is FD,Z(·) -progressively measurable. The information process Z (possibly a vector) is

endogenous and remains to be identified. It determines the equilibrium public filtration Fm(·) =

FD,Z(·) ≡FD

(·)∨F Z

(·) and therefore the PIPR θGv relative to the equilibrium filtration. Together with

clearing in the risky asset market, Z and θGv determine the equilibrium MPR θmv .

The agents’ demand functions are Xavπav =Xav

(θmv +θG

v

)/σmv ,X

uvπ

uv =Xuv θ

mv /σ

mv , and Xnvπ

nv =

Xnv(θmv +φv

)/σmv , where θG

v ≡θGv (G). Aggregating demands and clearing the risky asset marketproduce the endogenous volatility coefficient σm and the combined holdingsHa+n≡Xaπa +Xnπn

of the active fund and the noise trader

Smv =Xavθmv +θG

v

σmv+Xuv

θmv

σmv+Xnv

θmv +φvσmv

⇐⇒σmv =δav θGv +δnv φv +θmv ≡Zv +θmv

Ha+nv ≡Xav

θmv +θGv

σmv+Xnv

θmv +φvσmv

=Smv

((1−δuv

) θmvσmv

+Zv

σmv

)=Smv

(1+δuv

(Zv

σmv−1

))where Smv =Xav +Xuv +Xnv equals aggregate wealth,

(δav ,δ

uv ,δ

nv

)≡(Xav ,X

uv ,X

nv

)/Smv is the

equilibrium distribution of wealth, and Zv≡δav θGv +δnv φv .

Let FSm

(·) be the filtration generated by the market price Sm. The market volatility coefficient

(i.e., the quadratic variation of the market return) is adapted to FSm

(·) . Thus, (Sm,σm) is adapted

to FSm

(·) ⊆Fm(·). As the uninformed agent knows Xu, the wealth share δu≡Xu/Sm is adapted to

Fm(·). He or she also observes the combined holdings Ha+n, implying that Ha+n is adapted to Fm

(·).But

(Sm,σm,δu,Ha+n

)adapted to the public filtration Fm

(·) implies that Z is adapted to Fm(·), so

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that Fm(·) ⊇FD,Z

(·) . The process Z=δaθG +δnφ provides noisy information about θG . Moreover,θm =σm−Z is adapted to Fm

(·), therefore it is known as well.To complete the characterization, note that the equilibrium wealth triplet (Xa,Xu,Xn) satisfies

dXav =Xav(θmv +θG

v

)(θmv dv+d−Wm

v

), dXuv =Xuv θ

mv

(θmv dv+dWm

v

)dXnv =Xnv

(θmv +φv

)(θmv dv+d−Wm

v

).

Aggregate wealth X=Xa +Xu+Xn =Sm satisfies dXv =Xvσmv(θmv dv+dWm

v

)with initial condi-

tionX0 =S0. The equilibrium stock price solves the backward equationdSmv =Smv σmv

(θmv dv+dWm

v

)with SmT =DT and σmv =Zv +θmv . By Ito’s lemma, equilibrium wealth shares (δa,δn,δu) solve

dδav =δav(θGv −Zv

)(−Zvdv+d−Wmv

), dδnv =δnv (φv−Zv)

(−Zvdv+d−Wmv

)dδuv =−d (δav +δnv

)=−δuvZv

(−Zvdv+dWmv

)with

(δa0 ,δ

n0 ,δ

u0

)=(na,nn,nu). The equilibrium vector (δu,δa +δn) is Fm

(·)-adapted, as it should be.

To complete the proof, assume that Fm(·) =FD,Z

(·) and note that the equilibrium Brownian motionWm can be always be written as

dWmv ≡ρDv

(dWD

v −θD,mv dv)+ρζv

(dWζ

v −θζ,mv dv)+

√1−(

ρDv)2 −

(ρζv

)2(dWn

v −θn,mv dv)

for coefficients(ρD,ρζ

)to be determined and prices of risk E

[dWD

v

∣∣Fmv

]≡θD,mv dv,

E[dW

ζv

∣∣∣Fmv

]≡θζ,mv dv and E

[dWn

v

∣∣Fmv

]≡θn,mv dv. The remainder of the proof first shows

that(ρζ ,ρD

)=(0,1) is not inconsistent with the assumption Fm

(·) =FD,Z(·) , then characterizes an

equilibrium with Fm(·) =FD,Z

(·) and(ρζ ,ρD

)=(0,1).

Solving for δa,δn and substituting in Zv =δav θGv +δnv φv establishes

δav =naE(∫ v

0

(θGs −Zs

)(−Zsds+dWms

)), δnv =nnE

(∫ v

0(φs−Zs )

(−Zsds+dWms

))(A14)

Zv =naE(∫ v

0

(θGs −Zs

)(−Zsds+dWms

))θGv +nnE

(∫ v

0(φs−Zs )

(−Zsds+dWms

))φv

(A15)

where, for processes (A,B), E (∫ v0 AsdBs)≡exp(∫ v

0 AsdBs− 12

∫ v0 A

2s d [B]s

)is the stochastic

exponential. Recall that ζ =(

1/√h)∫ T +h

Tσζv dW

ζv and φs =k

((1/

√h)∫ T +h

Tσφs (v)dWn

v +μφs)

where μφs ,σ

φs ,σ

ζs are functions of time. For this noise structure, the Malliavin derivatives

Dζs ζ =0 and Dζ

s φs =0 for s∈ [0,T ]. Using θGs =θGs (G) with θGs (x)∈Fm

(·) =FD,Z(·) (thus θGv (x)

does not directly depend on ζ,φ) and taking the Malliavin derivatives of (A15), then shows thatDζs Zv =Dn

s Zv =0 for s∈ [0,T ] if(ρζ ,ρD

)=(0,1).

The backward equation for the price gives Smv =E[E(−∫ T

vθms dW

ms

)DT

∣∣∣FD,Zv

]≡

�(D(·),Z(·)

). Taking the Malliavin derivatives on both sides yields the volatility coefficients

Smv σmv ρ

ζv =Dζ

v�(D(·),Z(·)

), Smv σ

mv

√1−(

ρDv)2 −

(ρζv

)2=Dn

v�(D(·),Z(·)

).

Given that Dζs Dv =Dn

s Dv =0 for s∈ [0,T ] and that Dζs Zv =Dn

s Zv =0 for s∈ [0,T ] if(ρζ ,ρD

)=

(0,1), it follows that(ρζ ,ρD

)=(0,1) is compatible with the conjecture Fm

(·) =FD,Z(·) .

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If an equilibrium with Fm(·) =FD,Z

(·) and(ρζ ,ρD

)=(0,1) exists, dWm

v =dWDv −θD,mv dv.

Also, θD,mv =E[dWD

v

∣∣Fmv

]/dv=E

[dWD

v

∣∣FZv

]/dv, where the second equality holds because

FDv does not provide information about the increment dWD

v , but FZv does. Thus, θD,mv =

DDv logpZ

(z|FD

v

)|z=Z≡θZ

v , as announced. The couple (Sm,σm) then solves the FBSDE

dSmv =Smv σmv

(σmv −Zv

)dv+Smv σ

mv

(dWD

v −θZv dv

),SmT =DT whereZ satisfies (A15) with dWm

v =dWD

v −θZv dv.

This completes the characterization of the equilibrium conjectured. �

Proof of Proposition 14. Let ρ =1−1/R. The optimal gross return of the fund is

XaT /Xa0 ≡exp

(∫ T0 rvdv

)Sa0,T =

(ξGT

)−1/R/E

[(ξGT

)1−1/R∣∣∣∣G0

]. By the martingale property of the

discounted optimal gross return

XaτN

Xaτ0

≡exp

(∫ τN

τ0

rvdv

)Saτ0,τN

=(ξGτ0,τN

)−1/RFτN with FτN =

E[(ξGτN ,T

)ρ ∣∣∣GτN ]E[(ξGτ0,T

)ρ ∣∣∣Gτ0]implying the excess return

logSaτ0,τN =− 1

RlogξG

τ0,τN−∫ τN

τ0

rvdv+log(FτN

)=

1

2R

∫ τN

τ0

(θmv +θG

v

)2dv+

1

R

∫ τN

τ0

(θmv +θG

v

)dWG

v −ρ∫ τN

τ0

rvdv+log(FτN

)

=− 1

2R

∫ τN

τ0

(θmv +θG

v

)2dv+

∫ τN

τ0

(θmv +θG

v

)Rσmv

d−Rmv −ρ∫ τN

τ0

rvdv+log(FτN

).

Let πav ≡πmv +πhv where πmv is the mean-variance demand and πhv the dynamic hedgingdemand. By standard arguments, πmv ≡(

θmv +θGv

)/(Rσmv

). The hedging demand πhv can

then be inferred from the expression above and logSaτ0,τN =−(1/2)∫ τNτ0

(πmv +πhv

)2(σmv

)2dv+∫ τN

τ0

(πmv +πhv

)d−Rmv . Let us now split each demand component into public and private

information parts. Thus, πmv =πm,Fm

v +πm,Gv with πm,Fm

v =θmv /(Rσmv

)and πm,Gv =θG

v /(Rσmv

).

Likewise, πhv =πh,Fm

v +πh,Gv . Collecting the private and public information components gives theportfolio decomposition πa =πFm

v +πGv , where πFm

v ≡πm,Fm

v +πh,Fm

v and πGv ≡πm,Gv +πh,Gv .

The resulting endogenous regression has the same form as (12) with coefficients

δv =(σmv π

Fm

v

)2, βv =πFm

v , γGv =πG

v , ηGv =

(σmv

)2(

2πFm

v +πGv

)πGv . (A16)

These coefficients can further be split into mean-variance and hedging parts by using the structureof the demand components πFm

v ≡πm,Fm

v +πh,Fm

v and πGv ≡πm,Gv +πh,Gv . �

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