long/short equity hedge funds and systematic ambiguity · long/short equity hedge funds and...

Working Paper Series

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National Centre of Competence in Research Financial Valuation and Risk Management

Working Paper No. 774

Long/Short Equity Hedge Funds and Systematic Ambiguity

Rajna Gibson Nikolay Ryabkov

First version: January 2011 Current version: May 2012

This research has been carried out within the NCCR FINRISK project on

“Credit Risk and Non-Standard Sources of Risk in Finance”

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Long/Short Equity Hedge Funds and SystematicAmbiguity

Rajna Gibson∗, Nikolay Ryabkov†

First draft: January 2011Current draft: May 2012

Abstract

This study first presents an optimal hedge fund portfolio choice model for aninvestor facing ambiguity or Knightian uncertainty. In the empirical section, wemeasure ambiguity as the cross-sectional dispersion in the macroeconomic forecastsand in the stock market return forecasts from the Livingston Survey and constructthe ambiguity factors for the universe of S&P500 stocks. We estimate ambiguity be-tas for long/short equity hedge funds strategies and document significant ambiguityexposures for directional L/S hedge funds. We compare the out-of-sample perfor-mance of portfolios constructed based on the L/S hedge funds alpha rankings withand without ambiguity exposures and find that the former outperform. These re-sults are robust with respect to alternative ambiguity measures, holding periods andperformance measurement models.

JEL codes: G11.Keywords: Ambiguity, Asset Allocation, Hedge Funds, Performance Measure-

ment.

∗The author is a Professor of Finance at the Geneva Finance Institute and at the Swiss Finance Institute.Address for correspondence: Rajna Gibson Brandon, Geneva Finance Research Institute, University ofGeneve, 40 bd du Pont-d’ Arve, CH-1211, Geneve 4, Switzerland or email: [email protected]†The author is at the Swiss Finance Institute PhD program, University of Zurich. Current address

for correspondence to Nikolay Ryabkov, 350 Queens Quay West Suite 1612, M5V3A7, Toronto, Ontario,Canada or email: [email protected]. Financial support by the National Centre of Competence inResearch ”Financial Valuation and Risk Management” (NCCR FINRISK) is gratefully acknowledged.

1 INTRODUCTION 2

1 Introduction

What is ambiguity? Investors act under ambiguity when they do not know the exact prob-ability measure associated with external events that may influence a decision or a choice.Ambiguity differs from the concept of risk because it deals with probabilistic uncertaintyrather than in uncertainty with respect to the realizations of an event. Knight (1921) wasthe first one who emphasized the importance of ambiguity for economic decisions. Eco-nomic agents behave differently when they know the probability distribution of uncertainoutcomes (known unknown) rather than when they do not know it and thus act under am-biguity (unknown unknowns). This finding is supported by psychological experiments suchas the Ellsberg Paradox (1961) that illustrates preference over situations in which peopleknow probabilities of uncertain events1. Such experiments demonstrated the existence ofambiguity aversion.

The notion of ambiguity should be relevant for hedge funds investors due to varioussources of ambiguity associated with hedge fund investments. First, opaqueness and dy-namic trading by hedge funds may generate ambiguity with respect to the hedge funds’risk exposures. The inability to understand the hedge funds investment strategies and tocorrectly attribute hedge fund returns to systematic risk factors is one of the main source ofambiguity for their investors. Hence, hedge funds’ investors can easily be misguided whenit comes to identifying ”pure” hedge fund alphas. There is evidence of significant ambigu-ity affecting the dynamics of equity markets as empirically documented by Anderson et al.(2009). One may conjecture that systematic stock market ambiguity - or macroeconomicuncertainty - may also affect hedge funds’ expected returns. The latter conjecture will befurther explored in this paper.

The second source of ambiguity in hedge funds’ returns relates to managerial skills. Aninvestment in hedge funds is often considered as a pure bet on the specific skills of a hedgefund manager that are to a large extent characterized by probabilistic uncertainty.

In parallel, we observe a wide debate in the academic literature and among practitionersregarding the issue as to whether hedge funds generate positive and significant alphas. Thedisagreement is caused in part by the absence of an accepted asset pricing model and well-established systematic risk factors with respect to which performance and risks of hedgefunds can be properly measured. In this study, we consider a new systematic factor whichshould be particularly important for hedge funds risk performance measurement, namelysystematic ambiguity. We postulate that beyond the traditional risk factors, hedge fundsexpected returns embed a premium for ”systematic ambiguity” exposure that should bepriced in equilibrium. If systematic ambiguity exposure is ignored, alpha estimates maybe biased and the performance of hedge funds may be misread.

The paper consists of two parts: a theoretical part and an empirical study. In thetheoretical part, we propose an asset allocation model for an ambiguity-averse hedge fundinvestor who makes her portfolio allocation decisions without relying on a single proba-

1See the paper by Epstein and Wang (1994) for a thorough discussion of this concept and its firstapplication to asset pricing.

1 INTRODUCTION 3

bility measure but rather considering all feasible alternatives. The investor allocates herwealth between a risk-free bond, a risky stock (or broad stock market index) and a hedgefund by solving an inter-temporal portfolio choice model in continuous - time. FollowingMaenhout (2004), we explicitly incorporate ambiguity aversion into the utility function andassess the impact of ambiguity aversion on the optimal allocation solution. In the generalmodel, we assume that there exists ambiguity with respect to both hedge fund and stockmarket price dynamics. The ambiguity parameter corresponding to the stock market indexprice dynamics describes systematic ambiguity. The ambiguity parameter correspondingto the hedge fund price dynamics describes the investor’s confidence about the hedge fundmanager skills.

We solve the asset allocation problem with a Max-Min utility framework to derivethe optimal portfolio weights and consumption. We observe that in general ambiguityhas a negative impact on the investor’s allocation in risky assets. We next impose amarket clearing condition and derive a two-factor risk asset pricing model for the hedgefund: where the market risk premium is due to risk aversion and, where additionally, asystematic market ambiguity premium is due to ambiguity aversion. We call this model aCAPM model with ambiguity (ACAPM).

In the empirical part of the paper, we use the Tass hedge funds’ database and focusexclusively on equity Long/Short hedge funds since the latter only have exposure to stockmarket risk(and thus potentially to systematic ambiguity). We then focus on two questions:First, do Long/Short hedge funds exhibit systematic ambiguity exposures? Second, canthe consideration of systematic ambiguity in the portfolio construction process enhancethe out-of-sample performance of Long/Short hedge fund portfolios?

Systematic ambiguity is measured by the dispersion (cross-sectional standard deviation)of the forecasts for the stock market index S&P500 (stock market ambiguity) and for theUS Industrial Production index (macro-economic ambiguity). We rely in both cases on apanel of the survey-based forecasts from the Livingston Survey obtained from the FederalReserve Bank of Philadelphia. In order to construct both systematic ambiguity factors, weestimate the stock market and macroeconomic ambiguity sensitivities of the constituentsof the S&P500 index. The ambiguity factor is defined as the spread between out-of-samplereturns of the portfolio consisting of the top decile of stocks and the portfolio consistingof the bottom decile of stocks ranked by their stock market - or by their macroeconomic-ambiguity sensitivities

In order to answer the first question, we then compute the ambiguity betas for theLong/Short hedge funds in our sample which are estimated by adding the systematic am-biguity factor to various benchmark multi-factor models: namely, to three well knownequity-based models (the CAPM, the Fama-French, and the Carhart models) and, for arobustness check, we also use a hedge fund-specific pricing model namely, the Fung andHsieh (2004) model with trend-following factors. The analysis is conducted for individualhedge funds and for equally-weighted and value-weighted portfolios of hedge funds. Wereport significant estimates of ambiguity betas across most model specifications. As ex-pected, we observe that ambiguity betas matter especially for the Long/Short hedge fundsthat pursue directional strategies.

2 RELATED LITERATURE 4

In order to answer the second question, we analyze the out-of-sample performance ofportfolios constructed with and without systematic ambiguity. We rank hedge funds basedon their alphas (the hedge funds with the top decile alphas are included in the portfolio)from the CAPM with systematic ambiguity, and compare these portfolios out-of-sampleperformance with the one of portfolios constructed using the traditional CAPM (withoutsystematic ambiguity). In other words, we compare the performance of portfolios of ambi-guity averse agents with those of risk-averse investors who ignore ambiguity concerns. Wefind that hedge funds portfolios based on the ACAPM model (with systematic ambiguity)outperform on an out-of-sample basis. The latter finding is noticeable in particular forvalue-weighted portfolios and holds for different hedge funds performance measurementmodels and holding periods.

The remainder of the paper is organized as follows. Section 2 discusses the relatedliterature. Section 3 describes the asset allocation model for ambiguity-averse hedge fundinvestors and its resulting two factor ACAPM model with systematic ambiguity. Section4 describes the data, the empirical methodology and our results. Section 5 concludes thepaper. Mathematical derivations, tables and figures are provided in the Appendix.

2 Related Literature

Our paper aims at contributing to the literature on ambiguity-averse preferences andKnightian uncertainty with a specific focus on hedge funds investments.

The formal incorporation of ambiguity into economic modeling requires constructionof ambiguity-averse preferences. This development was pioneered by Gilboa (1987) andGilboa and Schmeidler (1989) who built axiomatic foundations of multiple prior prefer-ences. Economic agents solve a Max-Min optimization by first minimizing their utilitywith respect to probability distributions from a given convex set (where set of probabilitiesconstitutes a menu of multiple priors corresponding to heterogenous beliefs regarding thestate of economy) and then maximizing with respect to the traditional choice variables(such as their consumption and investment choices). Another representation of ambiguity- averse preferences is developed in the papers by Hansen and Sargent (2001) and Andersonet al. (2003) who describe utility optimization as a robust control problem. This problemhas also a Max-Min optimization form with a minimization over alternative probabilitymeasures but the utility function contains a penalty in terms of the entropy measure rel-ative to alternative probability laws. It has been shown2 that the robust control problemis equivalent to the multiple priors setting only in the case of constrained relative entropy.This robust control approach has very appealing and intuitive interpretation of the pe-nalization term however it may fail to satisfy some axiomatic foundations of economicpreferences for general specifications of the penalty function. The paper by Maccheroniet al. (2006) resolves this potential inconsistency by constructing an ambiguity - averseutility function of a general class that encompasses both the multiple priors and robust

2See, for example, paper by Trojani and Vanini (2004).

2 RELATED LITERATURE 5

control settings and which therefore makes the latter model consistent with axioms ofeconomic preferences.

A body of the literature has examined whether ambiguity-averse preferences are ableto explain some well known financial markets’anomalies. The paper by Dow and Wer-lang (1992) utilizes the multiple prior preferences to conclude that ambiguity aversion canexplain the limited equity market participation. A recent paper by Easley and O’Hara(2009) also demonstrates that ambiguity aversion induces non-participation in financialmarkets and suggests that regulation that may decrease perceived ambiguity, especiallyduring disruptive market events, can help to mitigate the effect of ambiguity aversion andthus resolve the non-participation puzzle. The authors use as an example the 2008 creditcrunch crisis when governments all over the world increased the sums of insured depositsand indicated their the willingness to bail out major financial corporations at the vergeof bankruptcy in order to diminish ambiguity induced lack of market participation. Thepaper by Uppal and Wang (2003) applies the robust control optimization to explain thewell -known home bias effect in asset allocation. The equity premium puzzle can alsobe meaningfully addressed by ambiguity-averse preferences since ambiguity aversion raisesthe overall risk aversion and thus raises the equity premium. This result is derived in thepaper by Maenhout (2004) who obtained a closed-form solution for the portfolio choiceproblem in continuous time for i.i.d. returns. The author proposes using a state - depen-dent weighting function in the penalty term of the robust control problem in order to solvethe optimization problem analytically and to preserve wealth independence in the opti-mal solution. Maenhout (2004) study emphasizes the decrease (up to 50% for reasonablecalibration parameters) in the demand for the risky asset by robust investors that leadsto a raise in the equity premium and to a drop in the risk-free interest rate. The equitypremium is calibrated to be at 4% to 6% when both risk aversion and ambiguity aversionare considered in the model. We will rely on a similar formulation of the portfolio choiceproblem and use a similar functional form for the penalty term in our hedge fund’s assetallocation model.

As a starting point for our empirical analysis, we take the idea developed in the the-oretical model by Kogan and Wang (2003) who propose a two-factor asset pricing modelbased on a return versus risk and ambiguity relationship. To the best of our knowledge,the only previous empirical test of an ambiguity-based asset pricing model was undertakenby Anderson et al. (2009) and it was inspired by the asset pricing model derived by Koganand Wang (2003). The authors construct a measure of macro-economic ambiguity as thedispersion in the forecasts of the nominal GDP growth as well as of the corporate profitsafter taxes. The authors then find that macroeconomic ambiguity is indeed an additionalpriced factor that has a significant impact on stocks’ expected returns.

The paper by Krahnen et al. (2012) estimates ambiguity aversion by conducting ex-periments under various settings and observing how reservation prices of individuals varywith ambiguity. The authors concluded that ambiguity aversion exists and differs acrossindividuals. They also found that the functional form of the ambiguity aversion coefficientis increasing in ambiguity. The ambiguity effect could be distinctively separated from theone of risk aversion and, moreover, as the authors demonstrated, ambiguity aversion has

3 A HEDGE FUND ASSET ALLOCATION MODEL UNDER AMBIGUITY 6

a more pronounced impact on asset prices than risk aversion. Therefore, accounting forambiguity is important in many financial applications, especially in asset pricing.

Within the hedge fund literature, the identification of risk - adjusted performance isa widely discussed but still challenging topic due to the absence of a proper hedge fundpricing model and due to the dynamic and non-tractable risk exposures of these funds. Onecan compare the ambiguity approach with Bayesian methods3 to estimate alphas as shownin the paper by Kosowski et al. (2006). They apply a non-parametric bootstrap analysis toestimate alphas of hedge funds relying on the fact that most estimates of alpha fail to fitthe normal distribution and exhibit significant negative skewness and high kurtosis. Theytake the seven-factor model by Fung and Hsieh (2004) as a benchmark risk model. Thebootstrap results indicate that OLS alphas are often overstated and do not have strongpersistence patterns. The Bayesian alpha is found to be positive and statistically significantat annual horizons and thus, according to the authors, hedge funds’ performance cannot beattributed to luck. Another interesting paper which further explores the Bayesian approachis the paper by Avramov et al. (2011) that studies the performance of hedge funds whileassuming predictability in their return generating model. The authors find that strategiesthat incorporate predictability in managerial skills outperform substantially. Hence, theauthors claim that predictability in alphas explains a large component of hedge fund returnsas well as the cross-sectional dispersion observed in their performance.

Finally, Cvitanic et al. (2003) derive a closed-form solution to the optimal hedge funds’allocation problem for investors with CRRA utility and in the presence of uncertain ab-normal returns, e.g. with Gaussian priors on the abnormal (relative to the CAPM alphas)returns of risky assets. The authors estimate uncertainty risk as the standard deviation ofalpha estimates across different asset pricing models. They find that the optimal portfolioweights allocated to hedge funds should be lower under model mis-specification than underthe standard optimal asset allocation framework.

3 A Hedge Fund Asset Allocation Model under Am-

biguity

This section describes the portfolio allocation model in hedge funds for an investor withambiguity-averse preferences. The investment opportunity set consists of three assets:a risk-free asset and two risky assets: a stock representing the market portfolio and ahedge fund that can itself take long or short positions in the stock market. The investorhas CRRA preferences and displays simultaneously ambiguity aversion. We investigatethe impact of ambiguity aversion on the optimal portfolio allocation. Furthermore, wederive the equilibrium pricing implications of the model. In particular, we will see thatonly systematic ambiguity over the market portfolio returns is priced in equilibrium. Theresulting equilibrium pricing model, the Ambiguity CAPM model (ACAPM) is a two-factor

3Note that Bayesian method implies a single prior framework while ambiguity is a multiple priorapproach.


model in which systematic risk and ambiguity are both priced.

3.1 Assets

There exist three types of assets in the economy. The first asset is a risk-free bond of priceBt which yields a instantaneous riskless rate denoted by r:

dBt = rBtdt. (1)

The second asset is a risky stock market porfolio whose instantaneous returns dMt

Mtfollow

a Geometric Brownian motion with a constant drift µM , a constant volatility σM and astandard Brownian motion ZM :

dMt

Mt

= µMdt+ σMdZMt . (2)

The third asset is a hedge fund whose instantaneous returns follow a geometric Brownianmotion with constant drift µF and constant volatility σF :

dFtFt

= µFdt+ σFdZt. (3)

The hedge fund can invest in both the risky and the risk-free assets taking either long orshort positions. The total risk of investing in the hedge funds can be broken down intothe systematic risk from the stock market and into hedge fund idiosyncratic risk. Thestandard Brownian motion Zt represents the total risk of the hedge fund and satisfies

dZt = ρdZMt +

√1− ρ2dZF

t , (4)

where ρ is the correlation coefficient between the hedge fund and the stock market returns,ZFt is a Brownian motion related to the idiosyncratic (hedge fund specific) risk, ZM

t is aBrownian motion related to systematic risk (market portfolio).

We assume that the correlation coefficient between hedge fund and the stock marketreturns is constant:

E(dZFt dZ

Mt ) = ρdt, (5)

This may be a strong assumption given the time-varying risk exposures typically takenby hedge funds. Even though stochastic correlation risk models might better describe thedynamics of the hedge fund returns under partially observable and dynamic risk exposures,we try to keep the modeling framework parsimonious in order to focus on the ambiguity-related implications.

Different hedge funds investment strategies can be distinguished from each other by val-ues of the correlation coefficient. A higher correlation is attributed to directional strategieswhile a lower value of the coefficient would characterize ”market neutral” strategies.


3.2 Model Misspecification

We assume that the model or the probability law which characterizes the stochastic dy-namics of the risky assets returns is not correctly specified. Let P be the initial probabilitymeasure under which assets’returns dynamics in the economy are specified. We refer to thestochastic equations describing the dynamics of the assets returns under this probabilitymeasure as the reference model. Denote the alternative probability measures by QH whichis parameterized by an appropriately adapted process Ht. The existence of the processHt is ensured by Girsanov’s theorem. The process Ht uniquely defines the alternativeprobability measures. Assume that QH is an absolute continuous measure with respect tothe reference probability measure P . Then, the Radon-Nykodim derivative or the densitydQHdP exists and is correctly defined. Moreover, it coincides with its conditional expectation.

Under the Novikov’s condition, the density is an exponential martingale which is equal to

dQH

dP= exp−

∫ T

0

|Ht|2

2dt−

∫ T

0

HtdZt, (6)

where Zt = (ZMt , Z

Ft ) is a vector of Brownian motions and Ht = (hMt , h

Ft ) is a vector

of the ambiguity parameters related to the corresponding sources of ambiguity. Ht is avector and thus expression |Ht|2 implies |Ht|2 = (hMt )2 + (hFt )2. Indeed, we assume thatthere is no correlation between the two sources of ambiguity hMt and hFt . This equationdetermines the parametrization of alternative probabilities. Since we have two sources ofrisk: the stock market and the hedge fund idiosyncratic risks, we also obtain two sources ofambiguity: the systematic stock market ambiguity and hedge fund idiosyncratic ambiguity.Ht determines the relationship between the Brownian motions related to the reference andthe alternative models:

ZMt = Zt +

∫ t

0

hMs ds, ZFt = Zt +

∫ t

0

hFs ds. (7)

Defining the stochastic dynamics of the stock market and of the hedge fund returnsunder the alternative probability measure, we note that the model misspecification is de-termined only by the drift change of each corresponding process. To show this, one canwrite the dynamics of the stock market Mt and of the hedge fund Ft under the alternativeprobability measure substituting the previous equation for ZH

t . The true dynamics of themarket and of the hedge fund returns thus jointly satisfy:

dMt

Mt

= (µM + σMhMt )dt+ σMdZ

Mt (8)

dFtFt

= (µF + ρσMhMt + σF

√1− ρ2hFt )dt+ σF (ρdZM

t +√

1− ρ2dZFt ) (9)

Note that given our specification of ambiguity, the distortion only affects the expectedreturns vector. Moreover, the size of the adjustment is scaled by the volatility coefficients.The more volatile the asset returns, the higher the potential impact of ambiguity on theexpected returns.


3.3 Investor Preferences and The Portfolio Optimization Prob-lem

The investor is characterized by a constant relative risk averse (CRRA) inter-temporalutility over an infinite time horizon, with a discount factor δ and a risk aversion coefficientγ. The optimization problem for such an agent is the following:

maxθM ,θF ,Ct

E∫ ∞

0

e−δtC1−γt

1− γdt, (10)

where Ct is an investor’s instantaneous consumption, θM is the fraction of his wealthinvested in the stock market, θF is the fraction of his wealth invested in the hedge fundand the residual 1− θM − θF is being allocated to the instantaneously risk-free bond.

We use robust control optimization with a relative entropy penalty term to specify theambiguity-averse preferences. The relative entropy measures the size of model misspecifi-cation which is the ”distance” between two probability laws QH and P . Entropy is definedas:

Ent(QH |P) = EH ln(QH

P) = EH(−

∫ T

0

|Ht|2

2dt−

∫ T

0

HtdZt) = EH1

2

∫ t

0

H2sds

Compounding the entropy, we obtain the value of global entropy as follows:

Entglob(QH |P) = δ

∫ ∞0

e−δtEH∫ t

0

(hMs )2 + (hFs )2

2ds (11)

This entropy in the intertemporal utility of an ambiguity-averse agent constitutes apenalty term for any deviation from the reference model. The utility optimization problemhas a Max-Min form where we minimize first with respect to parameters of the alternativeprobability laws hM and hF and then maximize utility with respect to the consumptionand the portfolio weights:

maxθM ,θ,C

minhM ,hF

EH∫ ∞

0

e−δt(C1−γt

1− γ+

(hMs )2

2ψM+

(hFs )2

2ψF

)dt (12)

The positive vector parameter ψ = (ψM , ψF ) is an agent-specific weight indicating howmuch he penalizes the alternative scenarios. This parameter represents the ambiguity-aversion coefficient. If ψ = 0, the deviation penalty is infinite and the agent chooses toremain under the reference model. If ψ = ∞, the penalty term goes to zero in the limitand the agent does not restrict herself in the choice of alternative probability measures.This is a myopic solution.

The optimization problem with penalty function is solved subject to the stochasticwealth dynamics of the investor endowed with initial wealth W0 as in the standard Merton’smodel. The wealth dynamics under the reference probability measure is the following:

dWt =((θM(µM − r) + θF (µF − r) + r)Wt − Ct

)dt+Wt(σMθM + σFρθF )dZM

t +

+WtσF√

1− ρ2θFdZFt . (13)


The stochastic process for the wealth dynamics distorted by ambiguity concerns satis-fies:

dWt =(

(θM(µM − r + σMhMt ) + θF (µF − r + ρσMh

Mt + σF

√1− ρ2hFt ) + r)Wt − Ct

)dt+

+Wt(σMθM + σFρθF )dZM

t +WtσF√

1− ρ2θFdZFt . (14)

3.4 The Optimal Portfolio Choice

The solution of the optimization problem for the ambiguity averse agent is achieved with theindirect utility function J(W, t) that should satisfy the Hamilton-Jacobi-Bellman equation:

δJ = maxCt,θF ,θM

minhMt ,h

Ft

C1−γt

1− γ+

(hMt )2

2ψM+

(hFt )2

2ψF+ Ah(J), (15)

where Ah(J) is a generator as in Merton’s intertemporal asset allocation model under thealternative probability measure:

Ah(J) = JW

((θM(µM − r + σMh

Mt ) + θF (µF − r + ρσMh

Mt +

√1− ρ2σFh

Ft ) + r)Wt − Ct

)+

1

2JWWW

2t

((θMσM + θFσFρ)2 + θ2

Fσ2F (1− ρ2)

). (16)

The specification of the ambiguity aversion coefficient proposed by Maenhout (2004)defines ψ as a function of the indirect utility in the following form:

ψM =ΩM

WJW, ψF =

ΩF

WJW, (17)

where Ω = (ΩM ,ΩF ) is a time-invariant vector proportional to ambiguity aversion vector-coefficient ψ = (ψM , ψF ) and J is the indirect utility function. This functional specificationwill allow us to obtain a closed-form solution to the portfolio choice problem. Note thatdue to the choice of the CRRA utility function, the previous expression reduces to

ψM =ΩM

J(1− γ), ψF =

ΩF

J(1− γ). (18)

The minimization problem gives a unique solution for the optimal ambiguity parametersdue to the convexity of the function with respect to hMt and hFt :

hMt = −ΩMσM(θM + θFρ), hFt = −ΩF θFσF√

1− ρ2. (19)

After solving the minimization part, we substitute the expressions for hMt and hFt intothe objective function and solve the maximization problem that determines the optimalportfolio weights θM and θF and the optimal consumption Ct.


The optimal consumption in terms of indirect utility satisfies:

C∗ = [JW ]−1γ . (20)

The implicit expressions for the optimal fraction of wealth invested in the stock marketand in the hedge fund satisfy:

θM =µM − r − θFρσM(ΩMσM + γσF )

(γ + ΩM)σ2M

. (21)

θF =µF − r − θMρσM(ΩMσM + γσF )

γσ2F + ΩMρ2σ2

M + ΩFσ2F (1− ρ2))

. (22)

Full details of the explicit solution and several special cases can be found in the Ap-pendix. Here, we illustrate the solution to the optimal asset allocation problem in simpli-fied terms. Let us consider the equations for the optimal θ∗M and θ∗F as a system of linearequations:

AθM +BθF = µM − r (23)

CθM +DθF = µF − r (24)

which is in matrix notations:Mθ = µ− r, (25)

where M =

[A BC D

], θ =

(θMθF

)and µ − r =

(µM − rµF − r

). Coefficients A,B,C, and D are

parameters dependent on risk σM and σF , ambiguity ΩM and ΩF , and on the correlationρ.

A = (γ + ΩM)σ2M (26)

B = ρσM(ΩMσM + γσF ) (27)

C = B (28)

D = γσ2F + ΩMρ

2σ2M + ΩFσ

2F (1− ρ2)). (29)

The explicit solution can be expressed in terms of the inverse matrix4 as

θ = M−1(µ− r) (30)

or equivalently

θM =(µM − r)− B

D(µF − r)

A− B2

D

(31)

θF =(µF − r)− C

A(µM − r)

D − B2

A

. (32)

4Under assumption of existence of inverse matrix, e.g. non-zero determinant AD 6= BC.


Systematic stock market ambiguity and idiosyncratic hedge fund ambiguity interactwith each other and their relevant importance depends on the correlation coefficient ρ. Onthe one hand, when ρ approaches 1, there is no impact of hedge fund ambiguity becauseΩF disappears from coefficient D. Therefore only stock market ambiguity influences theoptimal weights either through increasing overall risk aversion (effect from denominator)or through relative betas (coefficient at the excess return in the numerator). On the otherhand, when ρ approaches 0, the relative betas are zero (no correlation between hedge fundand stock market), B = C = 0 and the optimal allocation weights reduce to the followingexpressions:

θM =µM − r

(γ + ΩM)σ2M

(33)

θF =µF − r

(γ + ΩF )σ2F

, (34)

These expressions for ambiguity-averse preferences show that ambiguity aversion amplifiesthe risk aversion in the case of orthogonal assets.

3.5 Equilibrium hedge funds’ pricing model

In order to derive the equilibrium pricing relationship, we now impose the market clearingconditions on the optimal portfolio weights. In equilibrium, the representative investorholds the market portfolio. The market clearing conditions on the proportion of portfolioweights thus satisfy:

θF = 0 and θM = 1. (35)

Substitution of these values into the implicit formulas for the optimal portfolio weights(23) and (24) yields the following formulas for the equilibrium expected excess returns:

µM − r = (γ + ΩM)σ2M (36)

µF − r = ρσM(ΩMσM + γσF ) (37)

Therefore the CAPM model with ambiguity factor (ACAPM from here onwards) has thefollowing representation for the stock market expected returns:

µM = r + γσ2M + ΩMσ

2M (38)

and for the hedge fund expected returns:

µF = r + γσFρσM + ΩMρσ2M . (39)

Those formulas correspond to the two-factor relationship between equilibrium expectedreturns versus risk and ambiguity that have been first studied in the paper by Kogan andWang (2003) and empirically tested in the paper by Anderson et al. (2009) for the equity

4 EMPIRICAL ANALYSIS 13

market. These equations show that the investor is compensated for both market risk andacting under market ambiguity.

The risk and ambiguity premia for the hedge fund investor depend on the correlationcoefficient between the hedge fund and the stock market returns. Under the assumptionof zero correlation, hedge funds investors would not earn a premium neither for stockmarket risk nor for stock market ambiguity. Such a relationship is characteristic of non-directional hedge fund strategies and absolute return strategies with a zero market beta. Ifthe correlation has a non-zero value, the investor in the hedge fund also earns an ambiguitypremium reflected in the term ΩMρσ

2M as well as a stock market risk premium expressed by

the well known term γσFρσM . Note that in equilibrium, only stock market or ”systematicambiguity” is priced.

In the next section, we shall study the properties of the ACAPM model applied toLong/short (thereafter L/S) equity hedge funds returns and thereafter examine whetherthe ACAPM can be used to select superior L/S equity hedge funds’ portfolios.

4 Empirical Analysis

The objective of the empirical analysis is twofold. First, we estimate the ambiguity betasof equity L/S hedge funds with various benchmark models in order to characterize thesystematic ambiguity exposure of this specific class of hedge funds. Second, we constructlong-only portfolios of L/S hedge funds relying on the ACAPM model alphas in the port-folios’ formation and compute these portfolios’ out-of-sample returns. We then comparethe performance of these portfolios with the one generated by portfolios formed withouttaking into account stock market ambiguity.

This empirical study is, to our knowledge, the first attempt to estimate the impactof stock market ambiguity on equity L/S hedge funds’ return properties. We proposeto construct the ambiguity factor based on the cross-sectional dispersion of professionalforecasts regarding the stock market index returns and a specific macroeconomic indicator,namely, industrial production growth.

The analysis is restricted to the equity L/S hedge funds investment strategy. First, thisis the largest category of hedge funds - in terms of assets under management - within ourhedge fund sample. Second, the primary investment instruments used in this strategy areequity-linked and thus allow us to meaningfully focus on the role of stock market ambiguityon the returns of hedge funds belonging to this strategy. We do not exclude the possibilitythat ambiguity also affects other hedge fund strategies. However, the return generatingmodel would then require different non-CAPM types of benchmark pricing models andmore detailed knowledge of the specific investment instruments used by those hedge funds.This is left for future research.


4.1 Hedge Funds Data

We use the TASS Hedge Fund database equity L/S hedge funds data during the timeperiod from January 1994 through November 2007. The database contains monthly dataon the rate of returns, the assets under management, and other characteristics of live anddefunct hedge funds. A defunct hedge fund is a fund that stopped reporting to the TASSdatabase due to liquidation, merger or to any other reason. These funds are contained inthe Graveyard module which is available from January 1994 onwards5. Note, that reportingto the database is a voluntarily decision of a fund’s manager and thus we are unable toknow the exact reason of non-reporting. Nonetheless , the combination of defunct and livefunds allows us to correct for the presence of the survivorship bias.

We impose various filters to refine our data sample. First, as mentioned, we combinethe data for live and graveyard funds in order to account for the survivorship bias. Second,we exclude the first 12 months observations from the sample in order to account for theinstant history bias. Furthermore, only hedge funds with returns reported net of fees inUS dollars and at monthly tracking frequency are selected. These filters should reducethe above-mentioned biases common to hedge fund data. The final sample of equity L/Shedge funds over the period between January 1994 and November 2007 consists of 2070observations. The average life time of a L/S equity hedge fund in the sample is 60 monthswhile the median lifetime extends over 50 months.

Panel A of Table 1 presents descriptive characteristics of our sample of hedge funds.We report cross-sectional average values computed from the monthly data for the wholesample and for each year6. The number of L/S equity hedge funds was growing from 206in 1994 to 986 in 2007 with a peak of 1168 hedge funds in 2005. The positive trend inthe sample size reflects rapid development of this segment of the hedge funds industryover the past 13 years. The drop in the sample size during the last year may indicatethe realization of capacity constraint in the L/S equity hedge funds strategy. Moreover,many quantitative equity funds experienced large losses during August 2007. The averageassets under management (AUM thereafter) of hedge funds were growing over time from60 mln.USD to 152 mln.USD until the end of 2007 with troughs in 1995 and 1999.

The fee structure consists of a fixed management fee, an incentive fee and of a high watermark provision. The management fee is on average 1.20 percent, the average incentive feereaches18 percent during the sample period. In parallel, the lock up period increasedfrom 2.8 to 5.5 months during the sample period. Growing demand for L/S hedge fundsproducts may explain their ability to charge higher fees and impose stricter liquidity rules.The leverage ratio reached its peak of 140% in 1999 and 2000 after the financial crisis of1998 and then started decreasing to a 130 % range. However the number of hedge fundsthat report non-zero value of leverage considerably increased since the beginning of oursample period. Roughly, one third of L/S equity hedge funds reports non-zero leverage.

Panel B of Table 1 illustrates descriptive statistics of L/S equity hedge funds returns.L/S hedge funds display large variability in their monthly returns: the annualized return

5This is the reason why most research using this database starts in January 1994.6Please note that we only cover the first eleven months in 2007.


of an average hedge fund is 12.9 percent with an annualized standard deviation of 15.92percent. The annualized Sharpe ratio of an average hedge fund (where the risk free rate isequal to the return of the 1-month Treasury bill) is 0.70. The annual return distributionis positively skewed (skewness coefficient is 0.76) however the annual kurtosis coefficient isclose to the value 3 for the normal distribution: 4.25 for an average hedge fund and 3.56 for amedian hedge fund with a standard deviation of 1.58. In addition, we performed a Jarque-Bera goodness-of-fit test of the normality of hedge funds returns. The null hypothesisindicates that the returns are normally distributed against the alternative hypothesis thatthe returns are not normally distributed (in the Pearson family of distributions). We runthe test for each hedge fund and assign value JBi = 0 if the null hypothesis cannot berejected and JBi = 1 otherwise. We report cross-sectional average of JB value as wellits median and standard deviation. JB = 0.66 means that we reject the null hypothesisof a normal distribution of returns for 66% of the L/S equity hedge funds in the sample.Moreover, we report the cross-sectional average, median and standard deviation for thep-values and actual values of the Jarque-Bera statistic.

4.2 Construction of the stock market ambiguity factor

We construct two systematic ambiguity factors using the dispersion (standard deviation)in the cross-section of the survey-based forecasts for the S&P500 stock market index andfor the growth in the Industrial Production index. The data source for the forecasts isthe Livingston Survey which started in June 1946 by Joseph Livingston and was takenover in 1990 by the Federal Reserve Bank of Philadelphia. The survey is conducted twicea year each June and December and asks participants to provide forecasts for the keyeconomic variables for the end of current month, six months, and 12 months ahead. Wewill consider only the mid-term 6-months-ahead forecasts for the analysis. On one hand,the forecasts for the end of current month might be inappropriate as some informationabout the actual values of the variables in the current month may be learned during themonth by a forecaster. On the other hand, the cross-section of the longer-term forecastssuch as 12-months ahead forecasts might have higher dispersion not because of ambigu-ity regarding the state of economy but rather due to a greater inability to forecast longterm trends. The individual forecasters have all different affiliations including nonfinancialbusiness (30%), investment banking (29%), commercial banking (20%), academic institu-tions (13%), government, insurance companies and labor organization (the remaining total8%)7. According to Croushore (1997), the number of participants in the mailing list forthe survey is about 90 and about 60% of them return the survey each time.

We will, for robustness purposes, rely on two different ambiguity measures: macroeco-nomic ambiguity and stock market ambiguity. We use these two measures for two reasons.On the one hand, using two different variables would contribute to the robustness of theresults and mitigate potential data problems with survey-based forecasts. On the otherhand, we believe that both macroeconomic and stock market ambiguity may significantly

7Please see article by Croushore (1997) for the reference to those estimates


affect the returns of L/S equity hedge funds and we are interested in the comparativeanalysis of those two ambiguity measures.

As far as the macro-economic ambiguity measure is concerned, the Livingston Surveyprovides values for the 6-months ahead forecast for the Industrial Production Index withseasonal adjustments. The data are taken from December 1989. Actual values of theIndustrial Production index are available from the Federal Reserve Board statistical releaseG17 series. On average, the panel of respondents consists of 37 forecasters with a standarddeviation of 12. We compute the expected semi-annual index return as the percentagechange between a forecast and a base period index level. The macroeconomic ambiguitymeasure is defined as the cross-sectional standard deviation of the forecasted returns.

The stock market ambiguity measure is defined as the standard deviation in the fore-casts of the S&P500 stock market index returns. The stock market forecast is a 6-monthsahead forecast for the S&P500 index level on the last trading day of June or December.The underlying data source is the New York Times and the data is available from Decem-ber 1990. On average, the panel of respondents consists of 25 forecasters with a standarddeviation of 6. We compute a semi-annual expected return for each forecaster as the pricereturn in percent between a forecasted index value and a base period index level. Theambiguity measure is defined as the cross-sectional standard deviation of the forecastedreturns.

In order to construct each ambiguity factor, we proceed as follows: we rank all stocks,constituents of the S&P500 index8 according to their sensitivity to the specific ambiguitymeasure (S&P500 or industrial production) by running the following OLS rolling regressionat semiannual frequency and fixed-size estimation window:

StockReturni,t −RiskFreet = α + βmkt(SP500t −RiskFreet) + βamb ∗ Ambt + εt (40)

The ambiguity factor is then computed as the out-of-sample return of a long/shortequally-weighted portfolio of stocks where long positions are taken in the top decile ofstocks and short positions are taken in the bottom decile of stocks ranked by their ambi-guity sensitivities βamb. The size of estimation window is 4 years. The initial estimationwindow consists of all data prior to January 1994:from December 1990 for the stock marketambiguity and from December 1989 for the macroeconomic ambiguity. In both cases, thefirst observation for the ambiguity factor refers to January 1994. We generate monthly am-biguity factor returns while we re-estimate the regression for sensitivities every 6 months.

The time series for the stock market ambiguity and for the macroeconomic ambiguityfactors are displayed in Figures 1 of the Appendix. In both cases, the ambiguity factorexperiences the highest volatility over the period between 2000 and 2003. The maximumabsolute values of the ambiguity factors are reached in 2000 during the spikes in uncertaintyjust before the dot-com bubble burst. The lowest values of the ambiguity factors are usuallyobserved during periods of recovery.

8Data for stock returns of the S&P500 constituents stocks is downloaded from the CRSP database


4.3 L/S Hedge Funds’ Ambiguity Betas

In this section, we estimate the ambiguity exposures of L/S equity hedge funds adding amacroeconomic ambiguity factor or a stock market ambiguity factor to traditional linearmulti-factor hedge funds return generating models. The regressions are estimated forportfolios of equally-weighted and value-weighted hedge funds as well as for individualhedge funds. All the coefficients are estimated with standard OLS regressions.

We will rely on three benchmark risk factor models: the CAPM, the Fama-French modelaugmented with the HML and SMB factors, and the Carhart model with an additionalmomentum factor. The data for the equity factors are taken from the K.French datalibrary9. The market factor is the return on the market (which is the value-weightedreturn on all NYSE, AMEX, and NASDAQ stocks) in excess of the risk - free rate. Therisk free rate is the 1-month T-bill rate taken from Ibbotson Associates.

Before proceeding with the estimation, we investigate potential multi-collinearity be-tween the two ambiguity factors and the other risk factors. We calculate the pairwisecorrelation coefficients such as the Pearson correlation, the Kendall correlation and theSpearman correlation for both macroeconomic ambiguity ambIP and the stock marketambiguity ambSP factors and the other risk factors. The coefficients are displayed inTable 2. We observe that both ambiguity factors only display models correlation levelswith the market factor. It is worth noting however that we found different signs for thecorrelations with the market factor for the stock market ambiguity factor (positive) andmacroeconomic ambiguity factor (negative) respectively. Higher correlations are observedwith the Fama-French factors HML and SMB. SMB and HML have relatively highPearson correlations with the stock market ambiguity factor 0.37 and -0.37 correspond-ingly however, in the case of the Kendall or Spearman coefficients, the value drops to anegligible level. Hence it is likely that the high value of the Pearson correlation is dueto some outliers and does not present a matter of concern. The momentum factor hasthe highest correlation coefficient with macroeconomic ambiguity: 0.65 for the Pearsoncoefficient ( the Spearman coefficient is still high at 0.43). Hence, the momentum factorexhibits a significant correlation with the macroeconomic ambiguity factor. In order toaddress the potential multi-collinearity issue between the momentum and the ambiguityfactors, we will ”orthogonalize” the momentum factor replacing it by its orthogonal coun-terpart, namely the OLS residual from a univariate regression of the momentum factor oneach of the ambiguity factors. We will report both non-adjusted and adjusted results inseparate tables.

Let us start by examining the estimated ambiguity betas for the portfolios of equally-weighted (EW ) and value-weighted (VW ) long/short equity hedge funds portfolios. Theestimates are presented in Table 3 for the CAPM, the Fama-French and the Carhartmodels augmented by the ambiguity factor. Under the augmented CAPM model, we findsignificant estimated ambiguity betas in the range of 0.07 to 0.15 depending on the portfolioformation and type of ambiguity factor chosen. Value-weighted portfolios tend to have

9The K. French data library is available under the following web page: http ://mba.tuck.dartmouth.edu/pages/faculty/ken.french/datalibrary.html.


higher ambiguity betas suggesting that larger hedge funds display higher stock marketor macro economic ambiguity exposures. Macroeconomic ambiguity is characterized byslightly higher values of ambiguity betas in comparison with the stock market ambiguity.The market betas for the portfolios of hedge funds are fairly stable with values around0.50.

The Fama-French and the Carhart models both have higher explanatory power forlong/short equity hedge funds portfolios returns than the one-factor CAPM: 0.85 versus0.68 for the equally-weighted portfolios and 0.77 versus 0.50 for the value-weighted portfo-lios. The ambiguity betas in the Fama-French specifications are lower than in the CAPMand are significant only for the macroeconomic ambiguity factor. The likely cause of theinsignificance is the high correlation of the stock market ambiguity factor with the SMBand HML risk factors. However, due to the high correlation with the momentum riskfactor, the significance of the macroeconomic ambiguity disappears in the Carhart model.We adjust for the multi-collinearity in the case of the momentum factor substituting theoriginal MOM factor in Carhart model with OLS-residuals from the regression of the mo-mentum factor on the ambiguity factor and report these results in Table 4. We obtainestimates of ambiguity betas at 0.02 and 0.04 values for the equally-weighted portfolio andthe value-weighted portfolio respectively, the coefficient being significant only in the caseof the latter portfolio. Finally, we conduct the same procedure for the macroeconomicambiguity AmbIP which give us significant ambiguity beta estimates of 0.05 and 0.10 forthe equally-weighted portfolio and value-weighted portfolio respectively.

From this analysis of the ambiguity betas of the long/short equity hedge funds portfo-lios, we can conclude that the ambiguity exposure is a statistically significant risk exposurefor these portfolios independently of the pricing model considered. Value-weighted port-folios have higher exposure to ambiguity and hence hedge funds with higher assets undermanagement are more exposed to stock market or macroeconomic ambiguity. During mar-ket disruptions when uncertainty is rising, larger hedge funds seem to be more sensible tosystematic ambiguity which we see reflected in their higher realized ambiguity exposures.

We next extend the measurement of ambiguity exposures to individual L/S hedge fundsrunning each of the three multi-factor risk models with both ambiguity factors for eachhedge fund in the sample. Table 5 reports the mean and median estimates of alphas andambiguity betas that are significant at the 10 percent level as well as the adjusted R-squared coefficients for these regressions in the case of both macroeconomic ambiguity andstock market ambiguity factors augmented asset pricing models. We observe that about27 percent of the L/S hedge funds have significant stock market ambiguity exposures and32 percent have significant macroeconomic ambiguity betas under the augmented CAPMregression. The median estimate of the ambiguity beta equals 0.13 for the stock marketambiguity and 0.15 for the macroeconomic ambiguity factors. The median explanatorypower of the models with ambiguity for individual hedge funds reach approximately 0.30in the case of the augmented CAPM. Ambiguity betas and the number of hedge funds withsignificant ambiguity betas relative to the augmented Fama-French and Carhart modelsare lower. However, in these cases, we did not adjust the results for the multi-collinearitybetween the ambiguity factors and the momentum factor.


Finally, we differentiate hedge funds by their market betas and examine separately theambiguity betas for high stock market beta and low stock market market betas hedgefunds. Those results are reported in the bottom two panels of Table 5. The threshold forthe market betas is set at 0.1, e.g. high market betas hedge funds are those with statisticallysignificant at 10% level market betas higher than 0.1 and low market beta hedge fundscontain the rest of the sample: with lower than 0.1 market betas or insignificant at 10% levelmarket betas. There are 838 hedge funds in the high market beta category and 1232 in thelow market beta category. About 50-56 percent of the high market beta hedge funds (465hedge funds for the ACAPM model) have significant ambiguity exposures with estimatedambiguity betas coefficients similar to those observed in the overall sample: 0.13 for thestock market ambiguity and 0.12 for the macroeconomic ambiguity. Only 12-15 percents ofthe low market beta hedge funds exhibit significance of the ambiguity factor with medianestimates of the ambiguity beta of 0.04 for the macroeconomic ambiguity factor. Theseresults confirm the hypothesis stated in the theoretical section and according to whichsystematic ambiguity matters the most for L/S equity hedge funds that pursue directionalstrategies and thus have higher stock market betas.

4.4 Asset Allocation with Ambiguity

This section examines whether it is important to account for systematic ambiguity in theL/S hedge funds’ portfolio construction process. We estimate alphas for individual hedgefunds in our sample under both the ACAPM and the CAPM models. These estimatedalphas are then used to select hedge funds and construct long-only portfolios whose out-of-sample returns performance is examined. Portfolios consist of the top 10 hedge fundsranked by their estimated alphas under both the ACAPM and CAPM models. We as-sess the risk-adjusted performance of those hedge funds portfolio during the whole sampleperiod (except of the observations from the first estimation window). The portfolio con-struction methods was motivated by Avramov and Wermers (2006) study that assessedthe ex-post out-of-sample performance of various portfolio strategies with monthly rebal-ancing. The initial estimation window size is 60 months in that paper and an additionalmonth is added at each realigning point. In our study, we conduct the rolling estimationwith a fixed-size window of 60 months and rebalance the portfolio each 6 months. Thesemi-annual rebalancing approach corresponds to an average lockup period. For the ro-bustness check, we also provide results with the rebalancing frequency of 1 month and 12months. The portfolios are constructed on both an equally-weighted and a value-weightedbasis. We collect the out-of-sample returns of these portfolios (based on the ambiguity andmarket models) from January 2000 (61st month) to November 2007 (end of the sample pe-riod) and compute their realized alphas relative to three benchmark multi-factors models:the CAPM, the Fama-French three-factor model, and the Carhart four-factor model.

Table 6 reports the risk-adjusted performance or monthly alphas in percentage pointsof the portfolios of L/S hedge funds. Comparison across performance measurement is or-ganized in columns. EW or VW stands for whether the portfolio is constructed on anequally-weighted or a value-weighted basis. If the portfolio construction involves estimat-


ing alphas utilizing models with systematic ambiguity, we add the term AmbSP for thestock market ambiguity and AmbIP for the macroeconomic ambiguity factor respectively.We find that the alphas of the portfolios formed with the ambiguity factors are positive,statistically significant and are higher than the alphas estimated for the portfolios formedwithout the ambiguity factors. For example,with the CAPM performance measurementmodel over a 12 months holding period, the estimated alpha equals 1.01 for the equally-weighted portfolio formed without the ambiguity factor versus 1.23 (1.29) for the modelswith the stock market ambiguity factor (macroeconomic ambiguity factor). Value-weightedportfolios have higher alphas (except over the one month holding period) but similar rank-ings: for instance under the CAPM benchmark performance model and over the one yearholding period, alpha equals 1.09 (with a t-statistic of 1.6) for the no-ambiguity model andincreases to 1.54 (with a t-statistic of 2.86) for the AmbSP and to 1.92 (with a t-statistic of3.86) for the AmbIP models. The abnormal performance of all hedge funds portfolios de-creases but remains significant over longer and thus more realistic (given hedge funds’lockup periods) holding periods.

Thus, we can conclude from these out-of-sample performance tests, that selecting longonly L/S hedge funds portfolios based on hedge funds’ ambiguity alphas can generate su-perior performance that remains statistically and economically significant across all pricingmodels even when yearly are considered.

4.5 Robustness Checks

In this section, we perform various tests in order to examine the robustness of the empiricalresults presented so far.

First, in the entire analysis, we rely on two ambiguity factors: the first factor whichassesses the ambiguity surrounding the forecasts of the S&P500 rate of return and thesecond one which assesses the ambiguity surrounding the forecasts of the Industrial Pro-duction index growth. The results are to a large extent consistent across those two factors.We measured ambiguity in both cases as the forecasts’s dispersion using the cross-sectionalstandard deviation. We also studied alternative measures of dispersion such as the range( i.e the difference between the maximum and minimum forecasted values) and the meanabsolute deviation (MAD) computed as the average of the absolute values of deviations ofthe individual forecasts from the arithmetic mean. We found consistent results and no sig-nificant difference when using those measures as opposed to the standard deviation for theestimation of ambiguity betas and for testing the portfolios’ out-of-sample performance.

Second, we always consider equally-weighted portfolios as well as value-weighted port-folios to illustrate the impact of large funds. We find that generally the value-weightedportfolios generate superior abnormal performance. Moreover, value-weighted portfoliostend to have higher estimated ambiguity betas regardless of the benchmark risk models.In the portfolio allocation section, our results clearly show that accounting for ambiguityexposure is important to filter out less performing large hedge funds.


Third, we vary the estimation window for the portfolio construction (36 months10versus60 months and the latter is reported). We find no significant impact of the rebalancingwindow on the results.

Fourth, we enlarge the scope of performance models used in order to also accountfor the seven risk factors of Fung and Hsieh (2004)model that encompasses three trendfollowing factors. The factors for the Fung and Hsieh (2004) model are available from theD. Hsieh’s data library11. The first three factors are trend-following risk factors : firstthe PTFSBD - a Bond Trend-Following Factor, constructed as the return of a PTFSBond lookback straddle; second, the PTFSFX - a Currency Trend-Following Factor,which is constructed as the return of a PTFS Currency Lookback straddle; and finallyPTFSCOM - a Commodity Trend-Following Factor, which is constructed as the returnof a PTFS Commodity Lookback Straddle. The next two factors are equity-oriented riskfactors: the equity market factors are the the S&P500 monthly total return index; andthe size spread factor Size defined as the Russell 2000 index monthly total return minusS&P500 monthly total return12. The last two factors are bond-oriented risk factors: thebond market factor Bond defined as the monthly change in the 10-year treasury constantmaturity yield (month end-to-month end), available from the St.Louis FED Economic Datadatabase (FRED); and the credit spread factor Credit defined as the monthly change inthe Moody’s Baa yield less 10-year treasury constant maturity yield (month end-to-monthend), also available at the St.Louis FED Economic Data database (FRED).

The bottom panel of Table 2 reports the correlation coefficients of the stock marketambiguity and the macroeconomic ambiguity factors with the Fung and Hsieh (2004) fac-tors. We found no significant correlation between both ambiguity factors and the Fungand Hsieh (2004) factors that would raise any multi-collinearity issues in the subsequentregression analysis. Table 7 reports the factor loadings when using the Fung and Hsieh(2004) model with ambiguity factors in the case of equally-weighted and value-weightedportfolios of L/S equity hedge funds. The total return of the S&P500 index and the sizespread are the only significant factors. However, the ambiguity factor is always significantwith OLS estimates in the range of 0.11 to 0.17. The explanatory power of the model Fungand Hsieh (2004) measured by adjusted R-squared coefficient increases noticeably afteradding either of the ambiguity factors. Table 8 reports ambiguity betas and hedge fundsalphas for the individual hedge funds and for hedge funds sorted by their market betas.High market beta hedge funds once again tend to have higher ambiguity exposure esti-mates and about 50 percent of high market beta hedge funds have statistically significantambiguity exposures versus only about 10 percent for the low market beta hedge funds.

Finally, we analyze the time series properties of ambiguity betas conducting a rollingregression estimation. Hedge funds risk and ambiguity exposures are very likely to be time-varying due to the use of dynamic trading strategies. In order to assess the time-variationin the ambiguity beta estimates, the rolling regressions with 36-months fixed-size sample

10These last results are not reported for parsimony but are available upon request.11Please refer to the following website http : //faculty.fuqua.duke.edu/ dah7/HFRFData.htm.12W. Fung updates on his web page this factor. In the original paper they used Wilshire Small Cap

1750 minus Wilshire Large Cap 750 monthly return

5 CONCLUSION 22

window were estimated for the CAPM and the Fama-French models and in the case ofvalue-weighted portfolios13. At the end of Appendix, Figures 2A and 2B illustrate rollingstock market ambiguity betas while Figure 3A and 3B illustrate rolling macroeconomicambiguity betas. In all figures, the stock market ambiguity exposures are growing duringthe post-LTCM time period with a peak around 2000 during the dot-com bubble burst.Afterwards, we experienced a strong bull market, and the estimated stock market ambiguityexposures decreased. When we look at the macroeconomic ambiguity exposures, we observethat these betas reached a peak around the turn of the millennium and then increasedgradually since 2003 and more rapidly so since 2005 which is in contrast with stock marketambiguity exposures that are characterized by a negative trend towards the end of theperiod. Thus, L/S equity hedge funds’ sensitivity to fundamental uncertainty increased atthe same time as their sensitivity to stock market uncertainty decreased.

5 Conclusion

Following the paper by Maenhout (2004), this study derives the optimal portfolio choiceof a hedge fund investor who is sensitive to ambiguity. Ambiguity arises when economicagents do not know the exact probability laws governing the state processes. The solutionof the asset allocation model reveals the impact of stock market ambiguity on the optimalportfolio weights in that the investor tends to reduce his or her allocation to risky assets.Furthermore, imposing market clearing conditions, we obtain an equilibrium asset pric-ing model with stock market ambiguity (ACAPM). In equilibrium, only systematic stockmarket ambiguity exposure is priced.

In the empirical section, we focus on L/S equity hedge funds and start by estimatingtheir exposures to stock market and macro-economic ambiguity factors. For that purpose,we measure macroeconomic (and stock market) ambiguity as the cross-sectional dispersionin the forecasts for the Industrial Production index growth (and for the S&P500 indexreturn) from the Livingston Survey and construct the ambiguity factors for the universe ofS&P500 stocks. We estimate ambiguity betas for the long/short equity hedge funds strat-egy and document significant ambiguity exposures irrespective of the asset pricing modelused, specially for those L/S hedge funds that follow more directional -higher stock marketbeta - strategies. We then compare the out-of-sample performance of hedge funds portfo-lios constructed following their alpha rankings obtained from two pricing models with andwithout a systematic ambiguity factor (the ACAPM and the CAPM respectively). Theout-of sample performance analysis of the portfolios constructed based on their ACAPMalphas displays superior abnormal risk-adjusted returns especially for the value-weightedportfolios and this performance is robust to alternative rebalancing horizons and to alter-native performance measurement models.

This study offers an insight into a so far neglected dimension of L/S equity hedge fundrisk profiles, namely their stock market (or macroeconomic) ambiguity exposures. Ourempirical results suggest that for this large category of hedge funds, systematic ambiguity

13Rolling ambiguity betas for alternative model specifications are available upon request.

5 CONCLUSION 23

is economically and statistically significant and could be meaningfully exploited by hedgefund investors and fund-of-hedge fund managers. Interesting extensions of this study wouldinvolve examining on one hand if systematic ambiguity matters for understanding othertypes of hedge funds’ strategies and on the other to examine if ambiguity regarding theskills of individual hedge funds managers should also be taken into account during theasset allocation process. These issues are left for further research.

REFERENCES 24

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Kosowski, R., Naik, N., and Teo, M. (2006). Do Hedge Funds Deliver Alpha? A Bayesianand Bootstrap Analysis. Working Paper. Forthcoming in Journal of Financial Eco-nomics.

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6 APPENDIX: OPTIMAL ASSET ALLOCATION UNDER AMBIGUITY 26

6 Appendix: optimal asset allocation under ambigu-

ity

The asset allocation problem for the ambiguity-averse investor has a form of Max-Minoptimization:

maxθM ,θ,Ct

minHt

EH∫ ∞

0

e−δt(C1−γt

1− γ

)dt+

Entglob(QH |P)

ψδ=

= maxθM ,θ,Ct

minHt

EH∫ ∞

0

e−δt(C1−γt

1− γ+

(hMt )2

2ψM+

(hFt )2

2ψF

)dt (41)

The sources of ambiguity hMt and hFt are uncorrelated with each other.The optimization problem with penalty function is solved subject to the stochastic

wealth dynamics of investor endowed with initial wealth W0 as in the standard Merton’smodel. The wealth dynamics under the reference probability measure is the following:

dWt =((θM(µM − r) + θF (µF − r) + r)Wt − Ct

)dt+

Wt(σMθM + σFρθF )dZMt +WtσF

√1− ρ2θFdZ

Ft . (42)

The indirect utility function J(W, t) satisfies the Hamilton-Jacobi-Bellman equation:

δJ = maxCt,θF ,θM

minhMt ,h

Ft

C1−γt

1− γ+

[hMt ]2

2ψM+

[hFt ]2

2ψF+ Ah(J), (43)

Applying first order condition to HJB equation, we find the optimal consumption interms of indirect utility as follows

C∗ = [JW ]−1γ , (44)

and the implicit expressions for the optimal fraction of wealth invested in the stock marketand in the hedge fund in terms of expected returns on securities as follows:

θM =µM − r − θFρσM(ΩMσM + γσF )

(γ + ΩM)σ2M

. (45)

θF =µF − r − θMρσM(ΩMσM + γσF )

γσ2F + ΩMρ2σ2

M + ΩFσ2F (1− ρ2))

, (46)

The explicit formulas are more cumbersome and therefore we present them in severalsteps. First let us consider the equations for θ∗M and θ∗F as system of linear equations:

AθM +BθF = µM − r (47)

CθM +DθF = µF − r (48)

6 APPENDIX: OPTIMAL ASSET ALLOCATION UNDER AMBIGUITY 27

which is in matrix notations:Mθ = µ− r, (49)

where M =

[A BC D

], θ =

(θMθF

)and µ − r =

(µM − rµF − r

). Coefficients A,B,C,D are

parameters dependent on risk σM and σF , ambiguity ΩM and ΩF , and correlation ρ.

A = (γ + ΩM)σ2M (50)

B = ρσM(ΩMσM + γσF ) (51)

C = B (52)

D = γσ2F + ΩMρ

2σ2M + ΩFσ

2F (1− ρ2)). (53)

The explicit solution can be expressed in terms of inverse matrix14 as

θ = M−1(µ− r) (54)

or equivalently

θM =D(µM − r)−B(µF − r)

AD −BC(55)

θF =A(µF − r)− C(µM − r)

AD −BC. (56)

This is equivalent to

θM =(µM − r)− B

D(µF − r)

A− B2

D

(57)

θF =(µF − r)− C

A(µM − r)

D − B2

A

. (58)

The portfolio weights for both securities are proportional to expected excess returns andinversely proportional to the risk and ambiguity associated with those securities expressedby combination of A − B2

Dand D − B2

A. Moreover portfolio weights for each asset are

adjusted by the expected excess return of other asset with adjustment coefficient BD

and CA

.Those coefficient could denote relative betas of securities with respect to another securityβM,F and βF,M correspondingly. The higher expected excess return of another asset, thelower allocation to the security.

In order to understand the impact of ambiguity on the optimal asset allocation, weconsider several special cases of above-mentioned formula which will illustrate the compar-ative static analysis. First it is easy to see that the portfolio weights under no ambiguitycase, e.g. when ΩM = 0 and ΩF = 0, are equivalent to the standard Merton’s portfolio

14Under assumption of existence of inverse matrix, e.g. non-zero determinant AD 6= BC.

7 TABLES AND FIGURES 28

weights for risky securities. In this case A = γσ2M , B = C = ργσMσF and D = γσ2

F thatleads to the following solution:

θM =µM − r − βM,F (µF − r)

γσ2M(1− ρ2)

(59)

θF =µF − r − βF,M(µM − r)

γσ2F (1− ρ2)

, (60)

where

βM,F =ρσMσF

(61)

βF,M =ρσFσM

. (62)

In general case, if we substitute values for A,B,C,D the optimal weights are as follows:

θM =µM − r − ρσM (ΩMσM+γσF )

γσ2F+ΩMρ2σ

2M+ΩF σ

2F (1−ρ2))

(µF − r)

(γ + ΩM)σ2M −

ρ2σ2M (ΩMσM+γσF )2

γσ2F+ΩMρ2σ

2M+ΩF σ

2F (1−ρ2)

(63)

θF =µF − r − ρσM (ΩMσM+γσF )

(γ+ΩM )σ2M

(µM − r)

γσ2F + ΩMρ2σ2

M + ΩFσ2F (1− ρ2)− ρ2σ2

M (ΩMσM+γσF )2

(γ+ΩM )σ2M

. (64)

The systematic market ambiguity and idiosyncratic hedge fund ambiguity interact witheach other and its relevant importance depends on the correlation coefficient ρ. On theone hand, when ρ goes to 1 there is no impact of hedge fund ambiguity as ΩF disappearsfrom coefficient D. Thus only stock market ambiguity influences the optimal weights eitherthrough increasing overall risk aversion (effect from denominator) or through relative betas(coefficient at the excess return in the numerator). On the other hand, when ρ goes to 0relative betas are zero (no correlation between hedge fund and stock market), B = C = 0and optimal allocation are reduced to the following formulas:

θM =µM − r

(γ + ΩM)σ2M

(65)

θF =µF − r

(γ + ΩF )σ2F

, (66)

This is exactly formulas for Merton’s weights for ambiguity-averse preferences and showsthat in the case of orthogonal assets ambiguity adds to risk aversion coefficient.

7 Tables and Figures


Table 1: This table reports descriptive summary of both live and defunct long/short equityhedge funds over the period between January 1994 and November 2007. Panel A reportshedge funds characteristics. Annual average values (Total) as well as averages over thewhole sample period (computed from the monthly data) are reported. Sample Size is av-erage number of live hedge funds. Fees describes the fee structure in percentage pointswhich consists of management fee and incentive fee correspondingly. Lock up is averagelock up period reported in months. Leverage is reported in percentage of capital (AUM).NumbLev is average number of hedge funds that report non-zero leverage. AUM is assetsunder management of a hedge funds in a current year reported in USD millions. Averagelife of hedge funds is 60 months, the median value is 50 months. Total number of everexisted hedge funds over the period is 2070. Panel B reports hedge funds’ return statis-tics: cross-sectional mean, median and standard deviation for the annualized hedge fundsreturns, annualized standard-deviation, annualized Sharpe ratio, annual skewness and an-nual kurtosis coefficients (kurtosis of normal distribution is 3), and Jacques-Bera test withthe Null hypothesis that distribution is normal (JB = 0 if the Null cannot be rejected vs.JB = 1 if the Null is rejected at 5 % significance level), p-value and value of JB statistic

Panel A

Year Sample Size Fees Lock Up Leverage NumbLev AUM

1994 206 1.09/ 17.00 2.84 127 54 581995 265 1.11/ 17.58 2.92 126 70 491996 358 1.11/ 18.04 3.17 127 95 531997 462 1.11/ 18.44 3.40 131 137 571998 555 1.12/ 18.64 3.55 138 178 581999 653 1.13/ 18.76 4.05 140 229 562000 778 1.15/ 18.97 4.60 141 287 772001 886 1.18/ 19.10 5.12 138 306 782002 959 1.21/ 19.08 5.27 137 312 772003 1014 1.24/ 19.00 5.34 135 305 742004 1101 1.28/ 19.08 5.41 135 310 922005 1168 1.33/ 19.12 5.46 133 310 1072006 1143 1.36/ 19.00 5.54 133 296 1302007 986 1.38/ 18.91 5.41 133 247 152Total 752 1.20/18.60 4.43 134 224 80

Panel B

Mean Median Std Dev Max MinRet 12.90 11.74 13.19 136.29 -54.46Std Dev 15.92 12.76 11.77 112.84 0.08Sharpe 0.70 0.68 0.84 5.89 -5.46Skewness 0.76 0.72 0.61 3.21 -1.40Kurtosis 4.25 3.56 1.58 15.70 2.10JB 0.66 1.00 0.47 1.00 0.00JB pvalue 0.07 0.01 0.13 0.50 0.00JB stat 49.13 14.83 68.12 666.83 0.06


Table 2: This table reports pairwise Pearson, Kendall and Spearman correlation coeffi-cients between ambiguity factors and excess market return (emkt), Fama-French factors(SMB and HML), Momentum (Mom) and Fung and Hsieh hedge funds factors. ambIPis macroeconomic ambiguity factor based on the Industrial Production Index growth fore-casts. ambSP is the stock market ambiguity factor based on the S&P500 index returnforecasts.

ambSP ambIP

FF Pearson Kendall Spearman Pearson Kendall Spearman

emkt 0.07 0.03 0.05 -0.11 -0.09 -0.13SMB 0.37 0.08 0.13 0.26 0.07 0.10HML -0.37 -0.12 -0.17 -0.16 0.02 0.02Mom 0.17 0.05 0.07 0.65 0.31 0.43

Fung Hsieh

ptfsbd 0.14 0.12 0.17 -0.02 -0.01 -0.02ptfsfx -0.12 -0.15 -0.21 0.00 -0.01 -0.02ptfscom 0.07 0.02 0.04 0.19 0.11 0.16sp500tr -0.01 0.01 0.02 -0.19 -0.12 -0.17size spread -0.01 0.02 0.02 0.02 -0.04 -0.04gs10 -0.01 0.02 0.03 0.04 0.04 0.06credit spread -0.11 -0.07 -0.10 -0.11 0.01 0.01


Table 3: This table reports multi-factor risk model for the hedge funds portfolios. EWdenotes equally-weighted portfolio of hedge funds and VW denotes value-weighted portfolioof hedge funds. OLS coefficients for the CAPM, Fama-French and Carhart models arereported. AmbSP is ambiguity factor based on the S&P500 forecasts. AmbIP is themacroeconomic ambiguity factor based on the Industrial Production forecasts. 1% (5%)statistical significance is marked by an two (one) asterisks. The value of t-statistic is inbrackets below estimated coefficients.

CAPM EW VW EW,AmbSP VW,AmbSP EW,AmbIP VW,AmbIP

Alpha 0.62** 0.60** 0.67** 0.67** 0.62** 0.59**(5.36) (3.77) (6.00) (4.43) (5.76) (4.20)

emkt 0.51** 0.49** 0.51** 0.48** 0.53** 0.52**(18.73) (13.05) (19.19) (13.38) (20.64) (15.29)

Amb 0.07** 0.11** 0.09** 0.15**(3.91) (4.34) (5.24) (6.63)

AdjR2 0.68 0.50 0.70 0.55 0.72 0.61

FF EW VW EW,AmbSP VW,AmbSP EW,AmbIP VW,AmbIP

Alpha 0.62** 0.62** 0.63** 0.63** 0.61** 0.60**(7.18) (4.90) (7.21) (5.00) (7.30) (5.08)

emkt 0.46** 0.41** 0.46** 0.42** 0.48** 0.45**(19.61) (11.89) (19.50) (12.00) (20.50) (13.53)

SMB 0.26** 0.31** 0.26** 0.30** 0.25** 0.28**(10.64) (8.65) (10.17) (8.12) (10.06) (8.05)

HML -0.01 -0.05 0.00 -0.03 0.01 -0.02(-0.23) (-1.05) (-0.02) (-0.62) (0.24) (-0.41)

Amb 0.01 0.03 0.05** 0.10**(0.76) (1.42) (3.42) (5.02)

AdjR2 0.83 0.70 0.83 0.70 0.84 0.74

Carhart EW VW EW,AmbSP VW,AmbSP EW,AmbIP VW,AmbIP

Alpha 0.53** 0.45** 0.54** 0.45** 0.54** 0.46**(6.48) (3.98) (6.47) (4.04) (6.48) (4.07)

emkt 0.49** 0.47** 0.49** 0.47** 0.49** 0.47**(21.67) (15.29) (21.51) (15.29) (21.61) (15.32)

SMB 0.25** 0.28** 0.24** 0.27** 0.24** 0.27**(10.52) (8.76) (10.18) (8.35) (10.32) (8.53)

HML 0.01 -0.01 0.01 0.00 0.01 -0.01(0.34) (-0.34) (0.41) (-0.08) (0.38) (-0.25)

MOM 0.08** 0.17** 0.08** 0.17** 0.08** 0.15**(5.11) (7.55) (5.04) (7.43) (3.70) (5.34)

Amb 0.01 0.02 0.01 0.02(0.33) (0.93) (0.53) (1.00)

AdjR2 0.85 0.77 0.85 0.77 0.85 0.77


Table 4: This table reports multi-factor risk model for the hedge funds portfolios: ambi-guity is adjusted for the collinearity with momentum factor (original Mom is substitutedby OLS-residuals from the univariate regression of Mom on AmbSP and AmbIP ). Allother parameters are identical to the previous table. 1% (5%) statistical significance ismarked by an two (one) asterisks. The value of t-statistic is in brackets below estimatedcoefficients.

Carhart EW VW EW VW

Alpha 0.60** 0.59** 0.61** 0.60**(7.45) (5.28) (7.51) (5.45)

emkt 0.49** 0.46** 0.49** 0.47**(21.57) (14.98) (21.51) (15.29)

SMB 0.25** 0.29** 0.24** 0.27**(10.74) (9.02) (10.18) (8.35)

HML 0.00 -0.03 0.01 0.00(0.10) (-0.70) (0.41) (-0.08)

MOM 0.08** 0.16** 0.08** 0.17**(4.97) (7.22) (5.04) (7.43)

AmbSP 0.02 0.04*(1.15) (2.15)

AdjR2 0.85 0.77 0.85 0.77

Alpha 0.61** 0.61** 0.60** 0.58**(7.31) (5.06) (7.45) (5.32)

emkt 0.47** 0.43** 0.49** 0.47**(20.54) (13.24) (21.61) (15.32)

SMB 0.26** 0.31** 0.24** 0.27**(10.90) (9.08) (10.32) (8.53)

HML 0.00 -0.04 0.01 -0.01(-0.13) (-0.96) (0.38) (-0.25)

MOM 0.07** 0.15** 0.08** 0.15**(3.43) (4.71) (3.70) (5.34)

AmbIP 0.05** 0.10**(3.69) (5.63)

AdjR2 0.84 0.73 0.85 0.77


Table 5: This table reports ambiguity betas and alphas relative to CAPM, Fama-French,and Carhart models for individual long/short equity hedge funds. AmbSp is the stockmarket ambiguity. AmbIp is the macroeconomic ambiguity. Results are reported for thewhole sample as well as for the high market betas subsample of hedge funds( market betais statistical significant at 10 % level and absolute value is higher than 0.1) and low marketbeta hedge funds (the rest of the sample). Number of hedge funds with high market betais 838 out of total 2070 observations.

All hedge funds AmbSp AmbIpCAPM FF Carhart CAPM FF Carhart

Mean Alpha 0.99 0.87 0.75 1.09 1.00 1.11Med Alpha 0.87 0.81 0.75 0.90 0.90 0.95PrctofSgn Alpha 47.58 44.09 41.52 47.26 45.94 40.80Mean Beta 0.11 -0.02 -0.01 0.13 0.08 0.08Med Beta 0.13 -0.08 -0.07 0.15 0.12 0.10PrctofSgn Beta 26.91 15.12 13.96 31.79 24.40 14.44Mean adjR2 0.35 0.39 0.44 0.33 0.41 0.39Median adjR2 0.33 0.39 0.46 0.32 0.42 0.38

Low Mkt Beta AmbSp AmbIpCAPM FF Carhart CAPM FF Carhart

Mean Alpha 0.03 -0.16 -0.23 0.16 0.03 -0.14Med Alpha -0.23 -0.30 -0.30 -0.20 -0.30 -0.31PrctofSgn Alpha 14.36 14.45 16.62 11.98 13.04 10.58Mean Beta 0.00 -0.01 -0.02 0.03 0.02 0.00Med Beta -0.03 -0.03 -0.03 0.04 0.03 0.03PrctofSgn Beta 14.36 14.45 16.62 11.98 13.04 10.58NumSgn Beta 175 174 198 146 157 126Mean adjR2 0.08 0.11 0.13 0.09 0.15 0.12Median adjR2 0.06 0.09 0.10 0.07 0.13 0.08

High Mkt Beta AmbSp AmbIpCAPM FF Carhart CAPM FF Carhart

Mean Alpha 0.27 0.27 0.09 0.22 0.33 0.25Med Alpha 0.46 0.40 -0.28 0.43 0.46 0.34PrctofSgn Alpha 56.16 37.94 36.53 51.81 47.76 25.85Mean Beta 0.13 0.06 0.06 0.11 0.05 0.04Med Beta 0.13 0.08 0.08 0.12 0.08 0.07PrctofSgn Beta 56.16 37.94 36.53 51.81 47.76 25.85NumSgn Beta 465 313 301 429 394 213Mean adjR2 0.25 0.32 0.35 0.26 0.34 0.34Median adjR2 0.21 0.29 0.32 0.23 0.32 0.33


Table 6: This table illustrates comparative performance analysis of the portfolios formedwith and without ambiguity factor. The portfolios consist of 10 highest ranking equally-weighted or value-weighted hedge funds. The ranking is based on the historical alphasestimated by CAPM one-factor model versus two-factors model with the ambiguity factor:either the stock market ambiguity AmbSp or macroeconomic ambiguity AmbIp. Theestimation window of regressions is 60 months. The out-of-sample portfolio formationperiod is 1 month, 6 months or 12 months such that we rebalance the portfolio everymonth, every 6th month or every 12th month. The table reports risk-adjusted portfoliosperformance. e.g. estimated alphas across different models: CAPM, Fama-French (FF),and Carhart model. 1% (5%) statistical significance is marked by an two (one) asterisks.The value of t-statistic is in brackets below estimated coefficients.

1month 6monthsPF CAPM FF Carhart CAPM FF Carhart

EW alpha 2.21** 1.98** 1.93** 1.44** 1.22** 1.16**(4.6) (4.72) (4.56) (3.4) (3.57) (3.38)

EW.AmbSP alpha 2.43** 2.25** 2.23** 1.56** 1.3** 1.29**(5.72) (5.5) (5.39) (3.92) (3.73) (3.66)

EW.AmbIP alpha 1.92** 1.76** 1.77** 1.69** 1.46** 1.47**(4.89) (4.41) (4.38) (4.86) (4.32) (4.29)

VW alpha 2.09** 1.71** 1.67** 1.47* 1.14* 1.00(3.15) (2.8) (2.7) (2.15) (2.14) (1.88)

VW.AmbSP alpha 2.26** 2.04** 1.97** 1.69** 1.42** 1.39**(3.28) (3.07) (2.93) (3.1) (2.75) (2.67)

VW.AmbIP alpha 1.83** 1.58** 1.5** 1.94** 1.67** 1.66**(3.32) (2.97) (2.81) (3.92) (3.5) (3.45)

12monthsPF CAPM FF Carhart

EW alpha 1.01* 0.85** 0.78*(2.52) (2.7) (2.47)

EW.AmbSP alpha 1.23** 1.04** 1.04**(3.3) (3.23) (3.18)

EW.AmbIP alpha 1.29** 1.05** 1.08**(3.99) (3.3) (3.35)

VW alpha 1.09 1.07* 0.85(1.6) (2.06) (1.69)

VW.AmbSP alpha 1.54** 1.52** 1.45**(2.86) (2.91) (2.76)

VW.AmbIP alpha 1.92** 1.8** 1.77**(3.86) (3.64) (3.54)


Table 7: Robustness check: this table reports OLS coefficients of the Fung-Hsieh multi-factor risk model for the hedge funds portfolios. EW denotes equally-weighted portfolio ofhedge funds. VW denotes value-weighted portfolio of hedge funds. AmbSP is ambiguityfactor based on the S&P500 index return forecasts. AmbIP is the macroeconomic ambigu-ity factor based on the Industrial Production index growth forecasts. 1% (5%) statisticalsignificance is marked by an two (one) asterisks. The value of t-statistic is in bracketsbelow estimated coefficients.

EW VW EW,AmbSP VW,AmbSP EW,AmbIP VW,AmbIPAlpha 0.29 0.39 0.01 0.03 -0.03 -0.09

(0.53) (0.56) (0.02) (0.04) (-0.06) (-0.14)ptfsbd -1.19 -2.03 -1.96* -3.03* -0.90 -1.60

(-1.20) (-1.61) (-2.09) (-2.57) (-0.98) (-1.41)ptfsfx 1.27 1.15 1.93* 2.00* 1.49* 1.47

(1.59) (1.13) (2.55) (2.11) (2.00) (1.61)ptfscom 0.86 1.87 0.44 1.32 -0.14 0.39t-stat (0.77) (1.32) (0.43) (1.01) (-0.13) (0.30)sp500tr 0.46** 0.42** 0.47** 0.42** 0.50** 0.47**t-stat (12.88) (9.19) (13.96) (10.05) (14.67) (11.32)size spread -0.01 -0.01* 0.00 -0.01 -0.01 -0.01t-stat (-1.56) (-2.10) (-1.37) (-1.94) (-1.47) (-2.07)gs10 0.29 -0.19 0.34 -0.13 0.15 -0.40t-stat (0.45) (-0.24) (0.57) (-0.18) (0.25) (-0.56)credit spread 0.02 -0.14 0.22 0.11 0.19 0.11t-stat (0.08) (-0.38) (0.81) (0.34) (0.72) (0.34)Amb 0.12** 0.16** 0.11** 0.17**t-stat (5.15) (5.33) (5.21) (6.29)AdjR2 0.50 0.35 0.57 0.45 0.57 0.48


Table 8: Robustness check: this table reports ambiguity betas and alphas for the Fung-Hsieh multi-factor risk model for individual long/short equity hedge funds. AmbSp isthe stock market ambiguity. AmbIp is the macroeconomic ambiguity. Mean, medianand percentage of significant values for the OLS estimates of alpha, beta and adjustedR-squared coefficient from regression with significant ambiguity betas at 10% significancelevel are reported. The results are reported for the whole sample as well as for the highmarket betas subsample of hedge funds( market beta is higher than 0.1) and low marketbeta hedge funds. Number of hedge funds with high market beta is 838 out of total 2070observations

AmbSP All hedge funds Low Mkt Beta High Mkt Beta

Mean Alpha -1.63 -0.52 -2.36Med Alpha -2.36 -0.96 -2.14PrctofSgn Alpha 58.25 11.29 56.57Mean Beta 2.48 0.01 0.20Med Beta 1.70 0.03 0.17PrctofSgn Beta 19.90 11.29 56.57Mean adjR2 0.32 0.12 0.26Median adjR2 0.29 0.10 0.24

AmbIp All hedge funds Low Mkt Beta High Mkt Beta

Mean Alpha -3.61 -0.90 -2.17Med Alpha -3.23 -1.06 -2.13PrctofSgn Alpha 52.72 13.38 51.22Mean Beta 2.99 0.03 0.16Med Beta 1.83 0.03 0.14PrctofSgn Beta 19.52 13.38 51.22Mean adjR2 0.30 0.14 0.25Median adjR2 0.29 0.11 0.24

1994 1996 1998 2000 2002 2004 2006 2008−0.4

−0.2

0

0.2

0.4Figure 1A. Stock Market Ambiguity Factor

year

Am

bigu

ity F

acto

r, m

onth

ly %

1994 1996 1998 2000 2002 2004 2006 2008−0.4

−0.2

0

0.2

0.4Figure 1B. Macroeconomic Ambiguity Factor

year

Am

bigu

ity F

acto

r, m

onth

ly %

Jan 97 Jan 99 Jan 01 Jan 03 Jan 05 Jan 07−0.2

−0.1

0

0.1

0.2

0.3

Figure 2A. 36−m rolling stock market ambiguity beta

year

CA

PM

, VW

pf o

f HF

s


−0.1

0

0.1

0.2

0.3

Figure 2B. 36−m rolling stock market ambiguity beta

year

Fam

a−F

renc

h, V

W p

f of H

Fs


−0.1

0

0.1

0.2

0.3

Figure 3A. 36−m rolling macroeconomic ambiguity beta

year

CA

PM

, VW

pf o

f HF

s


−0.1

0

0.1

0.2

0.3

Figure 3B. 36−m rolling macroeconomic ambiguity beta

year

Fam

a−F

renc

h, V

W p

f of H

Fs

long/short equity hedge funds and systematic ambiguity · long/short equity hedge funds and...

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