Electronic copy available at: 1785995 Incentives and Endogenous Risk Taking: A Structural View on Hedge …

Electronic copy available at: http://ssrn.com/abstract=1785995

Incentives and Endogenous Risk Taking:

A Structural View on Hedge Fund Alphas

ANDREA BURASCHI, ROBERT KOSOWSKI and WORRAWAT SRITRAKUL�

ABSTRACT

Hedge fund managers are subject to several non-linear incentives: (a) performance fee options

(call); (b) equity investor�s redemption options (put); (c) prime broker contracts allowing for forced

deleverage (put). The interaction of these option-like incentives a¤ects optimal leverage ex-ante, de-

pending on the distance of fund�s value to high-water mark. We study how these endogenous e¤ects

in�uence traditional measures of risk-adjusted performance. We show that structural measure that

account for these features are less subject to false discovery biases. The result is stronger for low

quality funds and is economically important for the way we assess the performance of hedge funds.

Finally, we show that out-of-sample portfolios based on structural dynamic measures dominate

portfolios based on traditional reduced-form rules.

JEL classi�cation: D9, E4, G11, G14, G23

Keywords: Optimal portfolio choice, Euler equation, hedge fund performance

First version: November 10th, 2010, This version: May 25th; 2013

�Andrea Buraschi is at The University of Chicago Booth School and Imperial College Business School; Robert Kosowski

is at Imperial College Business School and the Oxford-Man Institute of Quantitative Finance; Worrawat Sritrakul is

at Imperial College Business School. We would like to thank participants at the American Finance Association 2012

meeting, in particular, Ralph Koijen, the discussant; participants at the December 2011 Imperial College London hedge

fund conference; the Gerzeensee Asset Pricing Symposium; Rene Garcia, Lubos Pastor and Jens Jackwerth for their

constructive comments. Robert Kosowski acknowledges the British Academy�s support via the Mid-Career Fellowship

scheme. The usual disclaimer applies.


This paper studies the implications of non-linear managerial incentives and funding con-

tracts for measures of risk-adjusted performance of hedge funds. We consider a structural model of

a hedge fund manager who is subject to several of the contractual features that usually affect his

payoff. These include: (i) performance fee-based incentives, (ii) funding options by the prime broker,

and (iii) equity investor’s redemption options. Together, these features create a non-linear payoff

structure that affects his leverage decision. The resulting optimal portfolio choice is state-dependent

due to the time-varying endogenous incentives perceived by the manager and it varies depending on

the distance of the assets under management from the high-water mark. This has important implica-

tions for the correct way to compute risk-adjusted measures of performance. As optimal leverage is

endogenous and state-dependent, traditional performance regressions with constant coefficients are

potentially misspecified. The call option-like performance fee incentive motivates the manager to

use more leverage, while put option-like features (together with the concern about the future value

of the incentive options) induce the manager to reduce leverage. The relative importance of these

two effects depends on the moneyness of these options, thus the distance of net asset value (NAV)

to high-water mark. We use the solution of the structural model to document, using a large panel

dataset of hedge fund returns, the difference of estimates of skill implied by traditional reduced-form

alpha and those implied by a structural estimation, which corrects for endogenous risk taking. We

find that the difference is economically significant and often changes the inference about good ver-

sus bad performing funds on a risk-adjusted basis. Moreover, our empirical method allows us to

use previously unexploited information about second moments that improves the efficiency of the

estimators, leading to improved out-of-sample performance.

Although we focus on hedge funds, our results have a broader economic motivation and impli-

cations. Separating the effect of risk taking and skill, based on observed investment performance,

is a fundamental problem that not only affects investors in alternative investment funds, but also

investors in (and regulators of) levered financial institutions such as investment banks which employ

incentive contracts. A Financial Times article in 2009 quotes a Bank of England official as saying

that “The superior performance of the financial services sector in the years leading up to the credit

crisis was almost entirely due to luck rather than skill – and banks increasingly gambled on luck

in an effort to keep up with their peers. [...]Good luck and good management need to be better

distinguished”.1 This quote illustrates not only the risk management challenges that policy makers

face but also the performance attribution problem that investors face. Using traditional tools it is

1‘Bank profits were due to ’luck, not skill’, By Norma Cohen (July 1, 2009)

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difficult to distinguish performance that is due to true investment skill from that attributable to

excessive risk taking.

An extensive literature has investigated the return characteristics of hedge funds. Fung and Hsieh

(1997 and 2004) and Agarwal and Naik (2004) highlight the importance of taking into account the

ability of hedge funds to invest in derivatives and suggest empirical methods to correct performance

attribution measures by using dynamic option strategies. These extensions are economically and

statistically important. A recurrent result, however, is that even after these corrections, reduced-

form (after-fees) alphas in hedge funds are large and positive (Fung and Hsieh, for instance, report a

monthly alpha of 0.78% for the TASS dataset). This evidence contrasts with the negative (after-fees)

reduced-form alphas obtained for other delegated managers such as mutual funds. The result could

be due to a mispecification of the time-varying investment opportunity set. Patton and Ramadorai

(2011) allow for dynamic risk exposure using threshold conditioning information based on market-

wide economic conditions. They find that their dynamic approach succeeds in reproducing realized

hedge fund returns with substantial increased accuracy. However, they find that even after controlling

for time-varying macroeconomic beta the reduced-form alphas are similar in size to Fung and Hsieh

(2004).

In this paper, we argue that reduced-form alphas can be affected by an endogeneity bias: fund

performance is the outcome of a portfolio management decision in which the incentive contracts

of the manager makes optimal leverage endogenous. Since hedge funds have substantial discretion

in selecting their leverage, the practical effect is potentially economically significant. In a simple

Merton-type economy with CRRA preferences the optimal portfolio choice is time-invariant. In this

context traditional reduced-form alphas are both correctly specified. When the hedge fund includes

non-linear incentives (both with the prime-broker and the investor), the manager is endogenously

induced to dynamically adjust the fund’s leverage depending on the distance of the asset values from

the fund’s high-water mark. Traditional reduced-form estimates of alpha make it difficult, therefore,

to disentangle whether a manager’s performance stems from his genuine skill or if it has just been

the result of the interaction of luck and excessive endogenous risk taking.

The first advantage of our structural approach is to explicitly take into account this endogeneity.

The second advantage is in terms of the efficiency of the estimator for skill measure. Our structural

approach allows us to use information in time-varying second moments, thus improving the statistical

efficiency of the estimator of genuine skill. This is potentially important since reduced-form alphas

2

are based on relatively short samples of data and thus likely to be imprecisely estimated.

We draw from the work of Koijen (2010), who shows how it is possible to use a structural approach

to estimate skill and risk preferences of mutual funds. Koijen (2010)’s novel approach allows to recover

cross-sectional information of mutual fund manager’s managerial ability and risk preferences by

directly using the Euler conditions. He shows that the restrictions derived from structural portfolio

management models can be used to recover both attributes from the data. However, extending

Koijen’s approach to hedge funds is not straightforward since incentive contracts that govern hedge

funds are significantly more complex than those of mutual funds and the endogenous optimal portfolio

choice becomes potentially highly nonlinear

While both mutual funds and hedge funds are delegated managers, they differ in terms of most

other institutional dimensions: legal framework, capital structure, investment mandate and business

model. First, hedge funds’ limited partnership structure allows for both asset segregation and liquid-

ity lock-ups imposed on investors, which facilitate prime brokerage contracts.2 The private nature of

their legal structure grants them the contractual flexibility in setting long-term lock-ups to investors,

whose legal rights are those of a (limited) partner, as opposed to a client. Second, investment advi-

sors that manage hedge funds typically receive an asset management fee and a performance fee. The

performance fee is typically subject to a high-water mark, which means that the manager receives

a compensation only on increases in the NAV of the fund in excess of the highest net asset value it

has previously achieved. The high-water mark provisions can be viewed as a call option issued by

the investors to the fund manager (Goetzmann, Ingersoll, and Ross (2003)). However, the funding

role played by the prime-broker also makes the capital structure of hedge funds potentially fragile:

As the 2007-2008 experience shows, when counterparty risk becomes acute during systemic events,

prime brokers tend to increase hedge funds’ collateral requirements and mandate haircuts in response

to higher perceived counterparty risk, thus inducing forced deleveraging of risky positions. Dai and

Sundaresan (2010) note that a hedge fund’s contractual relationships with its equity investors and

prime broker can be considered as short option positions. Therefore, these two option-like features

affect the hedge fund’s balance sheet, both from the perspective of its assets and its liabilities. Any

structural model of hedge fund behavior must account for these features as they affect the objective

function of the fund manager.

2The prime broker plays an essential role in the capital structure of a hedge fund. By contrast, most mutual funds,as Almazan et al. (2004) document, are restricted (by government regulations or investor contracts) with respectto using leverage, holding private assets, trading OTC contracts or derivatives, and short-selling; see also Koski andPontiff (1999), Deli and Varma (2002) and Agarwal, Boyson, and Naik (2009).

3

The first contribution of this paper is to account for contractual features such as (i) incentive op-

tions, (ii) equity investor’s redemption options and (iii) prime broker contracts, which jointly creates

option-like payoffs and to show how this affect endogenous risk taking and generates a misspecifi-

cation in the reduced-form alpha. The solution to the optimal investment problem is a dynamic

investment strategy in which optimal leverage is endogenous and depends on the distance of the

NAV from the high-water mark.

The second contribution is to study the implications of the model for a large sample of hedge

funds and investigate the difference between structural and reduced-form estimates of alpha. We

impose the structural restrictions implied by economic theory to estimate the deep parameters of the

model. We document the differences between reduced-form regression estimates of performance and

the estimates based on the structural true skill and investigate the features that contribute to the

misspecification. We find strong direct evidence that leverage is endogenous and state dependent.

The functional form is not monotone in performance, but it is bell-shaped and centered just below

the high-water-mark. This is consistent with the model and with the fact that hedge funds are both

long a call option and short a put option. We find that failure to account for these features leads to

a substantial amount of false discovery: attributing a positive alpha to funds that instead have no

superior ability to generate excess returns.

Finally, we conduct a simple out-of-sample test and compare the performance of portfolios of

hedge funds constructed by ranking funds on the basis of different ex-ante performance measures.

We find that structural estimates lead to superior out-of-sample performance and thus dominate

reduced-form ones. This is interesting since it shows the practical importance of controlling for

managerial incentives and endogenous risk taking and provides a useful methodology that can be

used in a variety of contexts including, but not limited to, funds of hedge funds. Moreover, it can

assist regulators in accounting for the effects of agency contracts in an attempt to limit a manager’s

excessive risk taking behavior while providing them the flexibility to deploy their managerial abilities.

1 Related Literature

Our work is related to three streams of the literature. First, we build on the work of Carpenter

(2000) who examines the dynamic investment problem of a risk averse manager compensated with

a call option on the assets he controls. She shows that under the manager’s optimal policy, as the

asset value goes to zero, volatility goes to infinity. While the importance of convex incentives has

4

been long noted, the unbounded leverage result (as the asset value converges to zero) does not hold

in the case of performance fee incentives with high-water mark features. Panageas and Westerfield

(2009) account for high-water mark restrictions and show that while risk-taking may be increasing

in the convexity of incentives in the context of a static two period model, in an intertemporal model

this may no longer be the case. Since any loss of fund assets reduces the value of the anticipated

sequence of future call options, higher risk taking today increases the risk of losing future options.

This is due to the fact that the current high-water mark remains as the effective strike price: all later

call options must have strike prices higher than the current high-water mark. The authors show that

even risk-neutral managers do not take unbounded risk, despite convex payoffs, when the contract

horizon is in(de)finite, thus reversing some of the results obtained in static two period models. Other

important contributions include Hodder and Jackwerth (2007), Guasoni and Obloj (2011), and Basak,

Pavlova, and Shapiro (2007). The last study the implications of a non-linear relationship between

mutual fund flows and performance on the manager’s risk taking behavior. Even though the mutual

fund fee structure is linear in assets under management, the incentive to attract more funds creates

convexities in the manager’s objective function which leads to risk-shifting. Cuoco and Kaniel (2011)

investigate the asset price implications in equilibrium when portfolio management contracts link the

compensation of fund managers to the excess return of the managed portfolio over a benchmark

portfolio. They show that performance fees can distort the allocation of managed portfolios in a way

that induces significant effects on prices and Sharpe ratios of the assets included in the benchmark.

In this paper, we build on these earlier contributions and further develop the analysis of optimal risk

taking by considering not only the payoff convexity related to the performance fee structure, but

also the capital structure fragility induced by the possibility of investors withdrawing capital and

prime-brokers forcing deleveraging. In general, this additional ingredient does not allow for closed-

form solutions. However, by restricting the incentive contract to a two-period problem we can obtain

an expression which is in closed-form up to the solution of a differential equation. This allows us

to implement a maximum likelihood estimation of the model, which we use to study empirically the

difference between reduced-form and structural alphas. An important difference with the previous

literature is that the optimal portfolio choice is not constant, but it is state dependent and function

of the distance to high-water mark. The interaction of capital structure fragility and high-water

mark restrictions gives rise to important non-linearities that affect ex-ante measures of risk-adjusted

performance.

A second stream of the literature, studies the role of incentive arrangement and discretion in

5

hedge fund performance. Agarwal, Daniel, and Naik (2009) conclude that the level of managerial

incentives affects hedge fund returns. In particular, funds with better managerial incentives (such as

higher option deltas, greater managerial ownership and the presence of high-water mark provision)

delivers better performance. Aragon and Nanda (2012) empirically analyze risk shifting by poorly

performing hedge funds and test predictions on the extent to which risk choices are related to

the fund’s incentive contract, risk of fund closure and dissemination of performance information.

Both papers compute measures of risk-adjusted performance using reduced-form regressions. In our

paper, we document the empirical effect of the misspecification generated by agency effects using a

structural approach. Then, we document the value of explicitly accounting for these agency effects

by studying the ex-post performance of funds selected based on their high reduced-form alpha versus

structural alpha. We find that selecting funds based on structural measures of skill leads to superior

out-of-sample performance.

Our paper is also related to several studies that shed light on the fragility of the capital structure

of levered financial institutions, such as hedge funds (Dai and Sundaresan (2010), Liu and Mello

(2011) and Brunnermeier and Pedersen (2009)). We build on their insights and incorporate some of

these features into our structural model.

The paper is structured as follows. In Section 2 we develop a structural model of the hedge fund

manager’s objective function. Section 3 presents the estimation methodology. Section 4 applies the

estimation methodology to a simulated economy. Section 5 describes the data that we use in this

study. Section 6 and 6.4 presents empirical results for the application of the structural model to

hedge fund returns. Section 7 concludes.

2 Structural Model and Manager’s Investment Problem

In this section we present our structural model by first defining the investment opportunity set of

a hedge fund manager. We then state and solve the manager’s portfolio problem. That solution

allows us to derive restrictions on the traditional αOLS βOLS, and σε,OLS that are estimated from

reduced-form ordinary least-squares (OLS) regressions. Finally, we study the model implications for

reduced-form performance and compare them with structural estimates of “true” managerial ability.

6

2.1 Trading Technology

We assume that the hedge fund manager can trade in a money market account with a constant

interest rate r:

dS0t = S0

t r dt. (1)

He can also invest in a strategy-specific benchmark asset that follows

dSBt = SB

t (r + σBλB) dt+ SBt σB dZB

t . (2)

The first two assets are commonly available to all managers of a given investment style. However,

individual fund managers differ in terms of their pure alpha-generating asset, which is assumed to

be fund specific.3 We interpret the pure alpha investment technology as an unleveraged return-

generating process SAt that is orthogonal to SB

t :

dSAt = SA

t (r + α∗) dt+ SAt σA dZA

t with E(dZAt dZB

t ) = 0. (3)

The larger is α∗, the better is the true alpha-generating skill. Let λA be the Sharpe ratio of the

alpha-generating asset, so that λA ≡ α∗/σA.

2.2 Manager’s Portfolio Problem

The hedge fund manager determines what proportion θt of his fund’s wealth will be devoted to the

investment opportunity set. Thus, the fund asset value evolves as

dXt = Xt(r + θ′tµ) dt+Xtθ′tΣ dZt, (4)

where µ ≡ (σAλA, σBλB)′, σA ≡ (σA, 0)

′, σB ≡ (0, σB)′, Σ ≡ (σA, σB), and Zt = (ZA

t , ZBt )′.

The manager’s portfolio investment problem is to identify the optimal trading strategy—in other

words, the θt that maximizes the objective function

max(θs)s∈[0,T ]

E0[U(p(XT −HWM)+ +mXT − c(K −XT )+)] (5)

3In order to minimize notation, in this section we avoid specifying (a) the return processes as style-j dependent(where dSj,B

t = Sj,Bt (r + σj

BλjB) dt + Sj,B

t σjBdZ

j,Bt for j = 1, . . . , J) and (b) the alpha technology as fund-i specific

within style j (where dSij,At = Sij,A

t (r + αijB) dt + Sij,A

t σijBdZij,A

t for i = 1, . . . , I). However, these considerations willbecome relevant in the empirical section.

7

subject to the budget constraint (4) and the restrictions that 0 ≤ Xt for all t ∈ [0, T ] and K ≤ HWM.

The utility function is assumed to be of the CRRA form, with U(W ) = W 1−γ/(1−γ). The parameter

p is the performance fee, m is the management fee, and c is the level of concern about the short put

option positions (the funding and redemption options).

The high-water mark level is defined as HWMt = maxs≤tXs. This formulation implies that

future high-water mark values are affected by managerial decisions, so that the manager faces a

sequence of call options that depend on past actions. Examples of studies that explicitly account

for this effect include Panageas and Westerfield (2009) and Guasoni and Obloj (2011). In different

settings, they discuss how the risk-seeking incentives of option-type compensation contracts affect

an infinite-horizon investor. In their contexts, the optimal portfolio solution can be derived in closed

form because the optimal portfolio weights are constant and independent of the distance between

fund value Xt and the HWM.

Our interest here is in studying the implications of the finite-horizon problem, since in that case

the risk-shifting behavior can become state dependent. On the one hand, for tractability we assume

that the high-water mark is constant and prespecified at the beginning of a contract’s evaluation

period. On the other hand, we allow the hedge fund manager to derive her utility from three

components. The first two of these (the management fee and the performance fee) are common in

most hedge fund compensation contracts. The management fee, which constitutes a fraction m of the

fund’s value, is a claim to a fraction p of a call option–like payoff with the fund’s high-water mark as

its strike price. The performance fee is a linear claim on the fund’s value at the end of the evaluation

period. The third of the objective function aims to capture the nature of the relationship between

the hedge fund and its prime broker. Typically, a hedge fund’s contract with its prime broker gives

the latter the right to reduce funding supply or increase margin calls in adverse economic states or

when the hedge fund’s counterparty risk is considered to be dangerously high. According to Dai and

Sundaresan (2010), the relationship between a hedge fund and its prime broker can be viewed as

a short put option position. Thus the parameter c captures the deadweight costs induced by any

de-leveraging required by the prime broker when the fund value falls below K.

The parameter p may also capture broader nonlinear effects that transcend the fees generated

by current assets. Indeed, a vast empirical literature has offered compelling evidence of nonlinearity

in fund flows. Top-performing funds receive substantial net cash inflows at the expense of bottom-

performing ones. Hence the call option can be interpreted as the fund manager’s upside of receiving

8

high flows due to superior performance. Similarly, the put option can be interpreted also as the

additional effect of the downside of net cash outflows due to inferior performance. Poor performance

has negative career implications for managers whose funds are liquidated. If a fund is unwound, its

manager may incur a deadweight loss owing to the difficulty of starting a new fund or finding another

job. The parameter c may be able to capture this additional effect.4

2.3 Model Solution to the Problem

If markets are complete, then one can use the martingale approach developed in Cox and Huang

(1989) to solve for the optimal investment problem. In this case we can solve an easier static problem

rather than the complex dynamic investment problem. Let φt be the state price process, which follows

dφt

φt= −r dt− λ′ dZt; (6)

here λ ≡ (λA, λB)′. When markets are complete, there exists a unique nonnegative state price process

φt such that the solution to the optimal investment problem is

maxXT

E0[U(p(XT −HWM)+ +mXT − c(K −XT )+)] (7)

subject to

E0[φTXT ] = X0 (8)

and 0 ≤ XT . The utility function is defined as

u(XT ,HWM,K, p,m, c) =

U(p(XT −HWM) +mXT ) for XT > HWM,

U(mXT ) for K < XT ≤ HWM,

U(mXT − c(K −XT )) for b < XT ≤ K.

(9)

We make the technical assumption that U(XT ) = −∞ for XT ≤ b, where b is the fund value at

which the hedge fund manager would otherwise begin to receive negative utility. See Figure 1.

[[ INSERT Figure 1 about Here ]]

4See Hodder and Jackwerth (2007) for the implications of the manager’s endogenous decision to close a hedge fund.In contrast, we consider the implication of options whose exercising decision is outside the manager’s control. Jurekand Stafford (2011) find supporting evidence that the risk profile of hedge funds resembles that of a short index putoption strategy during negative systemic shocks.

9

Observe that, because managerial incentives here are nonlinear, the objective function u(XT ,

HWM, K, p, m, c) is not globally concave in XT . A randomized strategy will therefore yield higher

expected returns. To solve the manager’s portfolio investment problem, we use the “concavification”

techniques of Carpenter (2000) and Basak, Pavlova, and Shapiro (2007).

Condition 1 XHWM and XK exist and satisfy the following conditions:

(a) XHWM > HWM and XK ≤ K;

(b) U ′(p(XHWM −HWM) +mXHWM) = U ′(mXK − c(K − XK));

(c) U(p(XHWM−HWM)+mXHWM) = U(mXK −c(K−XK))+U ′(mXK −c(K−XK))(XHWM −

XK);

(d) U(p(XT −HWM)+mXT ) ≤ U(mXK − c(K − XK)) +U ′(mXK − c(K − XK))(XT − XK) for

all XT ∈ [HWM, XHWM].

Condition 1 provides the conditions that identify the concavification points, XHWM and XK . Figure 2

displays the concavified utility function. The initial utility function is dominated ex ante by the

convex combination marked by the dotted line, which can be interpreted as a randomization of

the initial investment strategy between XHWM and XK . Additional details and discussion of the

concavification (and of Condition 1) are given in Appendix B. It can be verified that Condition 1

is satisfied for the following (empirically plausible) parameter values: m = 0.02, p = 0.2, c = 0.02,

K = 0.6(HWM), and γ = 1.5.5


Therefore, the concavified objective function reads

u(XT ,HWM,K, α,m, c) =

U(p(XT −HWM) +mXT ) for XT > XHWM,

U(mXK − c(K − XK)) + ν(XT − XK) for XK < XT ≤ XHWM,

U(mXT − c(K −XT )) for b < XT ≤ XK ;

(10)

5In some of the calibrations that follow, we assume rates of 2% and 20% for (respectively) the management fee mand the performance fee p. The deadweight cost (c) and strike price (K) of the put option are unobservable, but weassume that the former is equal to one year’s management fee (i.e., 2%). We follow Basak, Pavlova, and Shapiro (2007)in assuming that γ = 1.5.

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here ν = U(p(XHWM−HWM)+mXHWM)−U(mXK−c(K−XK))

XHWM−XK.

We then solve the hedge fund manager’s problem described in function (7) with the concavified

objective function (10). In terms of the new objective function, the manager solves maxXTE0[u(XT ,

HWM, K, α, m, c)] subject to E0[φTXT ] = X0. The first-order conditions of the optimization

problem require (i) that the manager’s marginal utility be proportional to the level of the state price

at the terminal time and (ii) that the budget constraint be satisfied. These conditions uniquely

determine the optimal terminal fund value X∗T ≡ I(ζφT ), where ζ is the Lagrange multiplier corre-

sponding to the static constraint I ≡ (u′)−1. Then the stochastic process of the optimal fund value

can be determined by X∗t = Et

(φTφt

X∗T

). Using Ito’s lemma, we can derive the dynamics of dX∗

t and

compare the coefficients of its diffusion term to equation (4); thus we obtain the optimal allocation

θ∗t . The result is summarized in Proposition 1. We simplify the derivation of Et

(φTφt

X∗T

)by mapping

optimal terminal fund values to terminal state prices as in Carpenter (2000). Appendix A gives the

detailed proof.

Proposition 1 Under Condition 1, the optimal allocation of a hedge fund manager with nonlinear

incentives is given by

θ∗t = −∂X∗(t, φt)

∂φt

φt

X∗t

(ΣΣ′)−1Σλ, (11)

where

∂X∗(t, φt)

∂φ= − ζ−1/γGt

φ(1+γ)/γt (p+m)1−1/γ

(N(dzHWM

2,t )

γ+

N ′(dzHWM2,t )

∥λ∥√T − t

)− αHWM

φt(α+m)

HtN′(dzHWM

1,t )

∥λ∥√T − t

− XHWMHt

φt ∥λ∥√T − t

[N ′(dzK1,t )−N ′(dzHWM1,t )]

− ζ−1/γGt

φ(1+γ)/γt (m+ c)1−1/γ

[N ′(dzb2,t)−N ′(dzK2,t )

∥λ∥√T − t

+N(dzb2,t)−N(dzK2,t )

γ

]− cKHt

φt(m+ c) ∥λ∥√T − t

(N ′(dzb1,t)−N ′(dzK1,t )) + bHtN(−dzb1,t)


. (12)

Here dzi1,t ≡ ln(zi/φt)+(r−0.5λ′λ)(T−t)

∥λ∥√T−t

and dzi2,t ≡ dzi1,t +∥λ∥

√T−t

γ ; Ct ≡ exp(ln(φt) − r(T − t)), Dt ≡

exp(1−γ

γ ln(φt)− 1−γγ (r+0.5λ′λ)(T − t)+0.5

(1−γγ

)2 ∥λ∥2 (T − t)), Gt ≡ exp

(− 1−γ

γ (r+0.5λ′λ)(T −

t) + 0.5(1−γ

γ

)2 ∥λ∥2 (T − t)), and Ht ≡ exp(−r(T − t)); and N(x) denotes the cumulative Normal

distribution function. Appendix A gives the explicit characterization of X∗t .

11

Although the optimal allocation described in equation (11) might look complex, it can be rewritten

as

θ∗t =(ΣΣ′)−1Σλ

γt(13)

where γt can be interpreted as the effective level of risk aversion. The key result of Proposition 1 is

that γt is endogenous and state dependent:

γt(γ, λ,Xt,HWM,K, p,m, c) ≡ −∂X∗(t, φt)

∂φt

φt

X∗t

.

Equation (13) reveals that, even if the investment opportunity set were constant (as in Merton),

agency contracts make the optimal allocation state dependent. The optimal allocation is driven not

only by the characteristics of the investment opportunity set and the risk aversion of the hedge fund

manager but also by such contractual parameters as fee rates (p, m, and c), the distance between

the fund’s current value and its high-water mark, and the existence (and perceived strike value K)

of short put option–like positions.


Figure 3 displays the results implied by (13). When fund value exceeds the high-water mark, the

manager allocates a lower amount of his funds to the risky asset than would be implied by the Merton

(1969) benchmark solution.6 When his call option starts being in the money, the manager has an

incentive to reduce leverage (which reduces his fund volatility) in order to lock in his performance

fee. However, the results suggest that a manager begins to increase his leverage with respect to

Merton’s allocation when his fund value is considerably higher than the HWM. Our findings in this

region are similar to the analytical result of the problem studied by Carpenter (2000) and also to

the numerical solution to the problem in Hodder and Jackwerth (2007). However, our result implies

different dynamics when the fund value is less than the high-water mark. Unlike the manager who is

maximizing only performance fees (dashed line in Figure 4), the fund manager in our setup doesn’t

monotonically increase his allocation to the risky asset as fund value decreases; instead, he reduces

his leverage as fund value decreases toward the put option’s strike price . The presence of put options

induces the manager to reduce leverage quite aggressively. The optimal leverage can fall below what

is implied by the Merton benchmark solution (solid line in Figure 4), and the manager continues

6The Merton (1969) constant allocation is µ/γσ2.

12

to reduce leverage until just before receiving negative utility. At that point, the manager begins to

increase leverage.7


A simple calibration of the model establishes the considerable economic magnitude of the differ-

ence in endogenous leverage induced by realistic parameters for the agency contracts. The empirical

implications of these effects and their role in performance attribution measures are the questions

that we study next.

2.4 Structural and Reduced-Form Alphas Compared

The closed-form solution given by equation (11) is reminiscent of the classical Merton (1969) optimal

portfolio choice solution. However, it depends on the fund-specific structural parameters that describe

the agency contracts and the realization of the state variable (i.e., the distance between the NAV

and the HWM). We use this solution to derive explicit restrictions and then compare the parameters

obtained from a standard reduced-form regression and the structural parameters (Koijen (2010)).

First consider the reduced-form regression with constant coefficients, where fund performance is

regressed on the benchmark excess return. If we substitute the benchmark return process, then the

standard reduced-form regression becomes:

dXt

Xt= (r + αOLS + βOLSσBλB) dt+ βOLSσB dZB

t + σε,OLS dZAt , (14)

where αOLS denotes reduced-form alpha, βOLS denotes reduced-form beta, and σε,OLS is the standard

deviation. According to the fund value process (4) and its optimal investment (13), the optimal fund

value evolves according to

dX∗t

X∗t

=

(r +

α∗2

γtσ2A

+λ2B

γt

)dt+

λB

γtdZB

t +α∗

γtσAdZA

t . (15)

Let θB,t and θA,t be the portfolio allocations to the investment opportunities described, respectively,

in (2) and (3). If we match the drift and diffusion terms, then the cross restrictions implied by the

7See also the numerical results in Hodder and Jackwerth (2007).

13

structural model are given by

αt,OLS = θ∗Atα∗ =

α∗2

γtσ2A

=λ2A

γt, (16)

βt,OLS = θ∗Bt =λB

γtσB, (17)

σt,OLS,ε = θ∗AtσA =α∗

γtσA=

λA

γt. (18)

Restriction (16) provides a link between traditional reduced-form alpha (αOLS) and true alpha

(α∗). The link reveals that the former is typically proportional to the latter but in a nonlinear way.

This restriction naturally raises the question of how well a reduced-form alpha measures the true

managerial skill of a hedge fund manager. An answer to this question depends on the determinants

of the optimal allocation θ∗At made by that manager. If the optimal allocation is constant and

determined exclusively by the risk and return characteristics of the investment opportunity set (as

in a traditional Merton model without agency distortions), then reduced-form alpha is an unbiased

estimate of managerial skill. However, if the optimal allocation is influenced by nonlinear agency

contracts, then reduced-form alpha is a misspecified estimate of true skill. For instance, a high

reduced-form alpha could be the fortunate result of too much leverage as managers aim to maximize

their incentive options. Of course, high leverage increases not only the manager’s expected return

(because of the call option) but also the likelihood of large negative returns.

We document some preliminary evidence of this effect in Figure 5. The graph plots empirical

estimated values of alpha (using traditional OLS methodology) for different values of the distance

from high-water mark (we provide detailed information on the data set in subsequent sections). The

cross-sectional dispersion reaches its maximum value when the NAV is equal to the HWM; in that

case, OLS alphas range from −15% to +20%. The cross-sectional dispersion of OLS alphas are much

smaller (ranging between −0.5% and +0.5%) for fund values that are well below the high-water

mark. This evidence is consistent with the main result of the model, which indicates that in this case

it is optimal for the manager to decrease leverage, thereby reducing the portfolio’s volatility and so

discouraging investors and prime brokers from exercising their redemption and funding options. This

structural effect is not captured by an unconditional measure of ex ante risk-adjusted performance,

such as the traditional OLS alpha.


14

The restriction (16) also reveals that reduced-form alpha varies over time, since effective (en-

dogenous) risk aversion γt is state dependent. This variance implies that reduced-form specifications

with constant coefficients may be misspecified, and it motivates our use of a structural approach

that employs fund-specific information to help identify γt. To clarify the state-dependent nature of

reduced-form alpha—that is, the link between a fund’s value and its high-water mark—we calibrate

the model and plot both the reduced-form alpha implied by the model (αOLS) and true managerial

skill (α∗) against different fund values.


Figure 6 shows that standard reduced-form alpha, αOLS, significantly overestimates true active

managerial skill, α∗, when the fund value is about 20% below the high-water mark. In contrast, αOLS

underestimates α∗ in the region where the fund value is about 50% below the put option’s strike

price (K) and about 18% above the HWM. Thus the standard reduced-form alpha overestimates

true alpha when the short option positions are deep out-of-the-money. In the region where fund

value is far above the high-water mark, the call option is deep in-the-money and so the effect of

misspecification is reduced. In general, the effect’s impact is significant exactly when the NAV is

close to the HWM, which is usually the region where hedge funds spend most of their time. This

finding demonstrates the importance of adopting a methodology that explicitly accounts for these

endogenous effects.

3 Estimation Methodology

When estimating a manager’s true alpha per unit of his investment technology’s volatility, it becomes

evident that fund performance (15) is driven by α∗/σA and not by each of the two parameters

separately; for this reason, we proceed by estimating that ratio, denoted λA, and using it as our

primitive measure of true skill (i.e., the Sharpe ratio of the manager’s alpha technology, which is

insensitive to the endogenous risk taking induced by agency contracts). We then estimate true

managerial skill using a structural approach and compare the values so obtained to the reduced-

form values. After investigating the results in the context of a simulated economy, we apply the

methodology to a comprehensive panel data set of hedge funds.

15

3.1 Structural Estimation

The structural estimation procedure is articulated in two steps. First we estimate the set of bench-

mark asset parameters ΘB ≡ {σB, λB}. Second, because the fund’s conditional return-generating

process satisfiesdX∗

t

X∗t

=

(r +

α∗2

γtσ2A

+1

γtλ2B

)dt+

1

γtλB dZB

t +α∗

γtσAdZA

t , (19)

we can estimate the Sharpe ratio of the manager’s investment technology λA ≡ α∗/σA and use this

as our proxy for the manager’s true managerial skill. Assuming that asset prices are log-normal, we

estimate the structural parameters by maximizing log-likelihood: argmax∑T/h

t=h ℓ(rXt | rBt ;λA, ΘB).

Under the assumption of log-normality, the joint dynamics of benchmark and asset returns can be

written in discrete time as (respectively)

rBt =

(r + σBλB − 1

2σ2B

)h+ σB∆ZB

t and (20)

rXt =

(r +

λ2B

γt+

λ2A

γt− 1

2

λ2B

γ2t− 1

2

λ2A

γ2t

)h+

λB

γtdZB

t +λA

γt∆ZA

t , (21)

where(∆ZB

t ∆ZAt

)∼ N(0, hI). Thus, the distribution of rXt satisfies

rXt | rBt ∼ N(µt, σ2t ), (22)

where

µt ≡(r +

λ2B

γt+

λ2A

γt− 1

2

λ2B

γ2t− 1

2

λ2A

γ2t

)h+

λB

γtσB

(rBt −

(r + σBλB − 1

2σ2B

)h

), (23)

σ2t =

λ2A

γ2th. (24)

By Proposition 1, the solution of γt(γ, λ,Xt,HWM,K, p,m, c) is available in closed form. There-

fore, when drawing inferences about true managerial skill, we use not only fund returns generally

but also fund-specific information about current high-water marks and contractual arrangements.

Applying the estimation procedure just described to a large hedge fund data set is computationally

intensive owing to the complex functional form of γt. The reason is that we must calculate the

Lagrange multiplier ζ∗, which is not available in closed form. Moreover, ζ∗ is also a function of a key

parameter that we aim to estimate, λA. Hence the econometrician faces numerical iterations for solv-

ing ζ∗ within the usual maximum log-likelihood iterations, which significantly slows the estimation

16

process.

We address this numerical issue by using a two-step procedure. First, we solve for the exact

functional form of γt in terms of the state variables and parameters (Θf ≡ {Xt/HWMt;λA,K}); here

Xt/HWMt can be observed directly whereas the other two parameters, λA and K, are estimated.

We allow the put option’s strike price to take a finite set of discrete values: 50%, 60%, 70%, and 80%

of the fund’s high-water mark. Then we use semi-parametric methods to estimate the parameters

of the exponential polynomial function that minimizes the distance from this HWM to the exact

functional form 1/γt:

1

γt=

a

γ+

b ∗ exp(−(c ∗ Xt

HWMt− (d+ e(f ∗K + g))

)2/(h− iK)2

)Xt

HWMt∗ λ ∗ γ

+exp

(−(j XtHWMt

)2)Xt

HWMt∗ λ ∗ l

. (25)

Second, given an estimate for the approximated function 1/γt, we can estimate the structural pa-

rameters λA,K; we initialize the parameters for management and performance fees to 2% and 20%,

respectively, while fixing γ = 1.5. We proceed by iterating between step 1 and step 2 until conver-

gence.

Figure 7 and Figure 8 plot the exact functional form 1/γt against, respectively, structural skill

λA and put option strike price K. Remarkably, we find that 1/γt is not constant—which highlights

the importance of considering explicitly the effects of any contract incentives. Yet at the same time

we find that the functional form is relatively simple and almost Gaussian in the region between the

put and call strike prices. Panel A of Figure 7 and Figure 8 compare the exact functional form with

its functiona approximation (Panel B) for different levels of the parameters λA and K. We find

that a second-order exponential polynomial is accurate to capture the dependence 1/γt on these the

distance to high-water mark for different levels of these two parameters.

[[ INSERT Figure 7 about Here ]] [[ INSERT Figure 8 about Here ]]

4 Learning from a Simulated Economy

4.1 Reduced-Form and Structural Estimation

We aim to investigate not only the precision of our estimation method but also the differences

between a reduced-form alpha estimated from an OLS regression with constant coefficients (αOLS)

and one that is implied by the theoretical (structural) restriction (αt,OLS = λ2A/γt), which is state

17

dependent. Toward that end, we proceed by simulating the economy and computing the model-

implied estimates. First, the innovation terms ∆ZAt and ∆ZB

t are independently simulated 2,500

times. Each simulation is then used to calculate 36 monthly returns. The simulated value of ∆ZBt

is used to calculate rBt according to equation (20). We calibrate the parameters to the following

values: r = 0.05, σB = 0.05, λB = 0.2, and γ = 1.5. Similarly, rXt is calculated according to

equation (21), where the true skill per unit of volatility (α∗/σA) is assumed to be based on values

λA ∈ {0.5, 1, 1.5} and σA = 0.05. We next estimate the variable λA via maximum likelihood and

compare the estimates to the true values. Finally, we compute γt by using our estimate of true skill

(λA), together with information about the fund’s current value (Xt) and high-water mark (HWMt).

The current high-water mark is defined as HWMt = maxs≤tXs and is updated on an annual basis.

We assume an initial investment of $1.

The distribution of the estimated parameters as implied by the structural model is expressed

in (22). Table 1 reports the estimation results. Panel A of the table gives the means and (in

parentheses) the standard deviations of our 2,500 reduced-formed alpha estimates αOLS computed

from OLS regressions, βOLS and information ratios. Panel B shows the means and standard deviations

of the skill and alpha estimates, respectively λA and αt,OLS, implied by the structural restriction (18)

as well as βt,OLS.

[[ INSERT Table 1 about Here ]]

As revealed by Panel B, structural estimation accurately recovers the true value of λA: the

means of the estimates are close to the corresponding true values and their standard deviations are

low. These results demonstrate that the estimation methodology performs well even when discretely

sampled (monthly) data are used. The structural estimation of alpha in the last column of Panel B

also confirms that alpha is time varying even for a constant investment opportunity set, λA. This

variance is due to the effective risk aversion (γt), which is state dependent even when the risk aversion

parameter γ is constant.

The second result concerns the efficiency of these estimators. In finite samples, the structural

estimator of alpha (αt,OLS) has a lower standard deviation than the reduced-form estimator of alpha

(αOLS). This finding is robust to different levels of true skill, as indicated by the standard deviations

of cross-sectional estimates. When λA = 1.5, for instance, the standard deviation of αOLS is 18.20

whereas that of αt,OLS is only 9.28. The reason for this improved efficiency is that the structural

18

estimator of alpha uses additional information about skill which is based on the second moments of

fund returns; see (18).

4.2 Simulated Out-of-Sample Tests

An important use of performance attribution measures is for selecting the components of a hedge

fund portfolio. To investigate the different implications of structural versus reduced-form estimates

of alpha, we run some out-of-sample experiments based on simulated economies. Suppose 90% of

the funds have no access to alpha-generating technologies and that all funds are subject to the same

incentive contracts. How much false discovery (i.e., attributing a positive alpha to what is actually

a zero-alpha fund) could then result from reduced-form techniques? What is the difference in the

out-of-sample performance as assessed while accounting (or not) for endogenous risk shifting?

To answer these questions, we start by simulating returns for 10,000 hedge funds. We assume

that only 10% of the funds have access to an alpha-generating technology (with a Sharpe ratio of

λA = 0.6) and that 90% of the funds cannot generate alpha. We assume that all other parameters

remain as in Section (4.1), and the simulations are based on equation (15). Then, each year we

form equal-weighted portfolios of the top-decile hedge funds as ranked either by the structural skill

estimates or the OLS alpha estimated using returns over the preceding three years. Each portfolio

is held for one year before it is rebalanced.

Two sets of results emerge. First, we find that the average probability that using reduced-form

OLS techniques will falsely identify unskilled funds as skilled funds is 59.53% (see Table 2, Panel B).

Thus, well over half of the funds ranked in the top decile by OLS alpha actually have zero alpha.

In contrast, the structural approach leads to much less false discovery. This finding is important;

it shows that reduced-form regressions are associated with a significant risk of false detection when

incentive contracts affect the behavior of portfolio managers.


Second, the portfolio that is rebalanced on the basis of structural estimates of skills outperforms

the one that is rebalanced on the basis of OLS estimates of alpha (Table 2, Panel A). This finding

is attributable to the high frequency of false discovery induced by the reduced-form regressions,

which impairs average portfolio performance. In simulated samples, this out-of-sample difference in

performance is large and statistically significant.

19

4.3 Risk Shifting in a Simulated Economy

A third set of questions concerns how the extent of risk shifting varies as a function of fund quality.

To address this question, we simulate an economy with 10,000 funds. One half of the funds are

assumed to have low skill (λA = 0.01) and the other half to have high skill (λA = 0.6). Then we

investigate the risk-shifting incentives as a function of the distance between the net asset value and

the high-water mark. We consider two measures of risk shifting (Ri,t), one based on standardized

βOLS and one based on standardized σOLS,ε. We standardize both measures to account for the

possibility that certain funds have betas and residual volatilities that are unconditionally different

from those of other funds. Both measures are computed from the simulated economy using the

model-implied cross restrictions 17 and 18. Moreover, Dist2HWMt ≡ (Xt −HWMt)/HWMt.

Ri,t = u+ c1 × 1{Dist2HWMi,t>0} ×Dist2HWMi,t

+ c2 × 1{Dist2HWMi,t<0} ×Dist2HWMi,t.

The existence of risk shifting implies the null hypothesis H0 : c1 < 0, c2 > 0. Table 3 reports the

results.


We test the null hypothesis and find evidence of risk shifting when either measure of risk shifting

is used. As predicted by the model, the magnitude of risk shifting depends on a fund’s quality. Indeed,

high-skill funds are less prone to risk shifting, as shown by both the c1 and the c2 coefficient being

smaller for both measures of risk shifting. The intuition is simple: it is more costly for high-quality

funds to gamble away their alpha-generating technology.

In the next section we compare these results with those obtained using “live” data.

5 Data and Benchmark Assets

We analyze the performance of hedge funds using monthly returns (net of fees) of live and dead

hedge funds as reported in the BarclayHedge database from January 1994 through December 2010.

There are several reasons for basing our analysis on this particular database. A recent compre-

hensive study of the main commercial hedge fund databases by Joenvaara, Kosowski, and Tolonen

(2012) (abbreviated JKT) compares the BarclayHedge, TASS, HFR, Eurekahedge, and Morningstar

20

databases and finds that BarclayHedge includes the largest number of funds (9,719; as compared

with 8,220 funds in the TASS database). Moreover, BarclayHedge has the highest percentage (65%)

of dead or defunct funds, which makes it the least likely to suffer from survivorship bias. The Bar-

clayHedge database constitutes the largest contribution to the aggregate database created by JKT.

Those authors remark that BarclayHedge is superior also in terms of assets under management

(AuM) coverage; it has the longest AuM time-series (57% of return observations contain a corre-

sponding AuM observation), which suggests different behavior when aggregate returns are calculated

on a value-weighted basis. The amount of missing AuM observations varies significantly across data

vendors—from 11% for BarclayHedge and 19% for HFR to 32% for Morningstar, 34% for TASS, and

37% for Eurekahedge. However, JKT do find that economic inferences based on the BarclayHedge

and TASS databases are similar in a number of respects. For instance, both BarclayHedge and TASS

(as well as HFR) reveal economically significant performance persistence for the equal-weighted port-

folios at semi-annual horizons; JKT find only limited evidence of performance persistence when using

the Eurekahedge and Morningstar databases.

A key distinguishing feature of the BarclayHedge database is its detailed cross-sectional infor-

mation on hedge fund characteristics. It is critical for our application that the database also includes

monthly AuM and information about the fund’s HWM provisions, both of which are key determi-

nants of a hedge fund manager’s incentives, performance, and risk. Our initial fund universe contains

more than 16,000 live and dead funds. To ensure that we have enough observations to estimate our

model precisely, we exclude hedge funds with fewer than 36 monthly return observations.8 These

restrictions lead to a sample of 4,828 hedge funds.

We group funds into eleven categories according to each one’s primary investment objective:

“commodity trading advisor” (CTA), convertible arbitrage, emerging markets, equity long/short,

equity market neutral, equity short bias, event driven, fixed-income arbitrage, global macro, multiple

strategies, or “other”. Table 4 displays summary statistics for the sample of funds, including medians

as well as the first and third quartiles of the first four moments of each fund type’s excess returns. The

average excess return across all funds is 6.38% per year. Funds that focus on emerging markets and

those that focus on equity short bias exhibit the highest and the lowest average return (respectively)

as well as the most volatility.

8In unreported results we show that our results remain qualitatively unaffected when we instead exclude funds withfewer than 24 monthly observations.

21


We also collect hedge fund investment style benchmark return indices compiled by Dow Jones

Credit Suisse and use them to benchmark hedge fund returns in our estimation. According to our

assumption about the investment opportunity set, all hedge fund managers (of a given investment

style) invest in the same strategy-specific asset (SB) even though they differ in terms of their alpha-

generating asset (SA).

Since the strategy-specific benchmark assets are unobservable, we proxy SBt by using the style

index returns, adjusting for any potential alpha with respect to seven Fung and Hsieh factors. The

strategy-specific proxy can be seen as a portfolio of basis assets in the investment opportunity set

of hedge funds in the same investment style. Thus, first we regress each style index’ returns on the

seven Fung and Hsieh factors (including an intercept). We then use only statistically significant

betas from the regression to compute our proxy for the benchmark asset SBt .

Table 5 reports summary statistics for hedge fund investment style index returns and their

maximum drawdowns. The “equity short bias” index has the largest drawdown of all the benchmarks;

the “event driven” index has the smallest drawdown. The “equity market neutral” index exhibits

extreme values for higher-order moments owing to a large loss in November 2008, when this strategy

lost nearly 40% in a single month.


Table 6 reports the results of regressing style index returns on the seven Fung–Hsieh factors.

In this table, one can see the large positive reduced-form alphas discussed in the Introduction:

4.33% (global macro), 4.34% (equity long/short), and 4.11% (event driven). Equity short bias

exhibits the lowest Fung–Hsieh alpha. Table 7 reports summary statistics of the proxies we use for

the benchmark asset returns for each investment objective.

[[ INSERT Table 6 about Here ]] [[ INSERT Table 7 about Here ]]

6 Empirical Analysis

The empirical analysis is articulated in three parts. First, we investigate the evidence for risk shifting

by using a model-independent methodology. Next, we estimate the structural model and discuss the

22

properties of alternative measures of risk-adjusted performance. Finally, we conduct an out-of-sample

analysis.

6.1 Structural versus Reduced-Form Models

Implementing the structural estimation methodology described in Section 3 requires, as one of the

inputs, the fund-specific time-varying moneyness Xt/HWMt of the agency contracts. This quantity is

not straightforward to compute because a hedge fund’s high-water mark is unobservable; it depends

on the specific timing of fund flows at the individual investor level. We carefully build an empirical

proxy for each fund’s current high-water mark by using fund-specific fund flow information.9 At

the start of each month, when the fund receives net positive inflows, we assume that a new share

class is created that has a high-water mark equal to the fund’s current value. The total HWM of

a fund manager is then the value-weighted average of HWMs of all existing and new share classes.

When the fund flow is negative, we assume that the outflows occur as a result of investors holding

the average vintage share class.10 On the one hand, it is intuitive that hedge funds within the same

investment category may have more similar capital structures in terms of credit lines provided by

prime brokers, redemption and lockup periods. On the other hand, these features may well vary

across different strategies. The estimated value of K maximizes the in-sample likelihood. These

results are summarized in Table 8. We find that, for most investment objectives, the put option

strike price is about 80% of the high-water mark. This result implies that hedge funds have on

average 20% drawdown tolerance before being negatively affected by large outflows or forced de-

leveraging. However, we sometimes find that certain strategies have larger drawdown tolerance (as

much as 40%). This may also be affected by the existence of exogenous restrictions concerning lockup

and redemption periods—as for example, with funds that focus on convertible arbitrage and fixed

income arbitrage.


We next estimate the reduced-form OLS alpha (αOLS) and the structural OLS alpha that (16)

implies (αt,OLS). Table 9 summarizes the main results by investment objectives; the results across

all investment objectives are shown in the last row. Each column reports the cross-sectional median

9See Agarwal, Daniel, and Naik (2009) for the fund flows calculation.10To improve this last approximation, we optimize further by calibrating the position of the peak of the function (25)

through the parameter d; this has the effect of maximizing the average in-sample likelihood by investment objectives.

23

of the skill measures as well as their first and third quartiles (in parentheses). Because the structural

alpha estimate (αt,OLS) varies over time as a result of time-varying effective risk aversion (γt), we

report the mean and standard deviation in the last two columns. We observe that OLS alpha (αOLS),

which by construction is constant in reduced-form regressions, differs from the median value of the

time-varying structural alpha (αt,OLS). In our sample period, the overall median of the former is

greater than that of the latter. The same relation holds with respect to investment objectives with

only two exceptions: the investment types of equity short bias and multiple strategies.


The difference in ex ante measures of risk-adjusted performance is such that the ranking of

hedge funds by investment objectives is substantially affected. The OLS (resp., mean structural)

alpha measure implies that funds focusing on emerging markets (resp., equity short bias) have the

highest skill. As for the least skillful funds, these are the ones that focus on fixed income arbitrage

(by the OLS measure) or those that focus on convertible arbitrage (by the structural measure);

compare the second and third data columns of Table 9. The difference is due to the endogenous

effect that time-varying risk aversion (γt) has on risk taking. That effect can cause OLS alpha to

be misleadingly high in some periods. For instance, a low-skill hedge fund manager who observes

his fund value to be below the high-water mark could temporarily increase leverage; doing so would

increase his fund’s volatility in the short term—and thereby increase the probability of that fund’s

value surpassing the HWM and thus generating performance fees. If this gamble is a successful one

then the result could be an artificially high alpha value attributable to pure luck and not to genuine

skill.

There are two simple ways for the fund manager to increase risk and deploy leverage: by increasing

beta and/or by increasing residual risk. We next explore whether the data indicate that fund

managers prefer one of these two channels as a means of dynamically shifting risk.

6.2 Empirical Evidence of Risk Shifting

How much risk shifting by hedge funds is evidenced by the data? The model predicts a bell-shaped

link between (endogenous) leverage and current distance from the NAV to the HWM (Figure 3).

Moreover, in the context of the structural model, restrictions (17) and (18) show that both beta and

residual risk are state dependent as well as endogenous. Since each is driven by the same factor (γt),

24

they are also perfectly correlated. More importantly, however, the model predicts that the effect of

risk shifting is stronger for lower-quality funds. To study this prediction empirically and to compare

the role of these two channels (i.e., risk aversion and quality), we first dynamically estimate betas

using 24-month rolling windows. For each rolling window we regress each fund’s excess returns on

its style’s hedge fund benchmark excess returns (recall that constructing the style benchmark was

discussed in Section 5);11 thus,

dXt

Xt− r dt = αt dt+ βt

(dSB

t

SBt

− r dt

)+ εt. (26)

The regression also provides a time-varying estimate of the residual risk εt. We then investigate the

risk-shifting incentives as a function of distance to high-water mark, Dist2HWMi,t.

As mentioned in Section 4.3, we consider two measures of risk shifting Ri,t, one each based on

standardized βOLS and σOLS,ε to account for the possibility of some funds having unconditionally

different betas and residual volatilities than others.

Ri,t = u+ c1 × 1{Dist2HWMi,t>0} ×Dist2HWMi,t

+ c2 × 1{Dist2HWMi,t<0} ×Dist2HWMi,t + εi,t

We sort hedge funds according to their structural skill measure λA. We define high-skill (resp. low-

skill) funds as those in the top (resp. bottom) quartile of the λA distribution. Then we compare the

parameters c1 and c2 for these two sets of funds.

Two results emerge (see Table 10). First, when we test the null hypothesis that H0 : c1 < 0, c2 >

0, we indeed find stronger evidence of risk shifting for low-quality funds. For this subgroup, we find

evidence of risk shifting both when using changes in beta and when using changes in residual risk.

The absolute values of the respective coefficients c1 and c2 are larger for low-quality funds and are

always statistically significant at the 1% level. This finding supports the null hypothesis of the model

that low-skill managers have more incentive to engage in risky behavior when the fund value is closer

to its high-water mark. Second, for high-skill managers we find mild evidence of risk shifting in terms

of changes in beta; however, we find no evidence of risk shifting by way of idiosyncratic risk.


11We also estimate beta more traditionally as exposure to returns on the S&P500. The results are not substantiallychanged by this variation.

25

Another way to investigate the implications of these results is by conducting an out-of-sample

analysis of the different empirical skill measures. That will be the focus of the next section.

6.2.1 Alternative Risk-Shifting

We investigate the robustness of the previous result by using an alternative specifications of the

relationship between incentives and leverage. In this section, instead of sorting with respect to

estimated parameters, such as λA, we sort by investment style. Moreover, we study the effect on

the overall volatility of fund returns, given that a manager’s compensation is commonly linked to

total portfolio value (for another study that uses overall volatility of fund returns as a proxy for risk

shifting, see Aragon and Nanda (2012)). To test the hypothesis of a bell-shaped link, we run a panel

regression similar to the one used in the previous section, while controlling for firm fixed effects.12

We define risk shifting as the change in volatility between period [t, t+ T ] and period [t− T, t]:

σi,[t,t+T ] − σi,[t−T,t] = ui + β1 × 1{Dist2HWMi,t>0} ×Dist2HWMi,t

+ β2 × 1{Dist2HWMi,t<0} ×Dist2HWMi,t

+ β3 × (AVGVIX[t,t+T ] −AVGVIX[t−T,t]) + εi,t. (27)

Here Dist2HWMt ≡ (Xt−HWMt)/HWMt and 1{X} is an indicator variable set equal to 1 if X = true

(and to 0 otherwise); AVGVIX denotes the average VIX (volatility index). The model predicts that

maximum volatility should be observed when Dist2HWMt = 0. Using regression (27), we investigate

the null hypothesis by testing H0 : β1 < 0, β2 > 0. This scenario should be compared to the

implication of a traditional Merton economy, for which neither β1 nor β2 is significantly different

from zero (i.e., H0 : β1 = 0, β2 = 0). That implication holds in the economies discussed in Panageas

and Westerfield (2009) and Guasoni and Obloj (2011). However, the manager studied in Carpenter

(2000) would exhibit H1 : β2 < 0, β1 < 0 because optimal leverage is unbounded when fund value

approaches zero.

It could be that overall volatility is affected by marketwide shocks, so we control for change in

the VIX between [t, t+ T ] and [t− T, t].13 Panel A of Table 11 summarizes the regression results for

12If funds are heterogeneous in terms of their level of volatility, controlling for firm-fixed is redundant in a regressionrun in changes. The inclusion of a control in a regression in differences is done to control for potential unconditionalheterogeneity in the changes in volatility during the sample.

13It is also possible that some hedge funds might, on average, have relatively high constant realized volatility and/orthat their fund asset values happen to be concentrated near their high-water marks. These funds provide limitedinformation about the functional form of risk shifting in terms of distance to the HWM. Therefore, to address the

26

T = 12 months; Panel B reports similar results for T = 6 months. The findings are robust across

the two panels. In both cases, on average the hedge funds exhibit risk shifting near the high-water

mark (as predicted by our theory). In the comparison of regression results across different hedge

fund investment categories, the values in Panel A are strong evidence of risk shifting at a 12-month

frequency for all investment categories (excepting only equity market neutral). Even though the

results are marginally weaker when we use 6-month windows (Panel B), strong evidence in risk

shifting is still exhibited by the fund investment types of convertible arbitrage, equity long/short,

equity short bias, event driven, fixed income arbitrage, multiple strategies, and “other”.


6.3 Cross-sectional Variation in Alphas

According to equation (16), hedge fund αt,OLS can be decomposed in two components, λA and γt.

To study the extent to which incentives help to explain the cross-sectional variation in empirical

αt,OLS, we run the following regression:

αOLS,i,t = const + b1 × λi,t + b2 × γi,t + εi,t. (28)

and in logs, with log(1+αOLS,i,t) =const + b1×log(λi,t)+b2×log(γi,t)+εi,t. Table 12 summarizes the

results. In both specifications, we find that αOLS,i,t is positively related to skill and negatively related

to effective risk aversion. The slope coefficient are statistically significant at the usual confidence

levels. We also compute the extent to which the two components help to explain cross-sectional

difference in alphas. We find that a 1 standard deviation increase in γi,t implies 190 basis point

decrease in αOLS,i,t (based on the coefficients of specification (2)). At the same time, a 1 standard

deviation increase in λi,t implies a 356 basis point increase in αOLS,i,t. The results suggest that while

both components are important, endogenous incentives are economically very significant. The results

are also robust to controlling for fund size, AUMi,t. In unreport results, we find similar findings across

investment styles. Investment styles which are known to be more discretionary are found to be those

in which the effect due to γi,t is stronger. This confirms the importance of accounting explicitly for

the role played by endogeneous incentive on risk taking.

possible resulting bias, we condition on those funds that have experienced large negative or positive distance to HWMat some stage in the overall sample. The deviation threshold for inclusion is set at 15% below and above the fund’shigh-water mark.

27


Another way to investigate the implications of these results is by conducting an out-of-sample

analysis of the different empirical skill measures. That will be the focus of the next section.

6.4 Out-of-Sample Analysis

The implication of Section 6.1 is that structural estimates of skill, λA, have better in-sample prop-

erties than do reduced-form estimates for two reasons: reduced misspecification bias and improved

efficiency. This superiority stems from the structural estimation’s controlling for endogenous leverage

and using information from the second moments of fund returns when estimating skill. In this sec-

tion, we test the hypothesis (that structural estimates are superior) by comparing the out-of-sample

performance of a portfolio of hedge funds formed on the basis of these two measures.

For each year we sort all hedge funds into deciles according to (a) their structural true skill

measure λiA and (b) their reduced-form alpha αi

OLS. We then form decile portfolios. These portfolios

are rebalanced every January (2001–2010) based on the skill measures estimated using hedge fund

monthly returns (net of fees) and high-water marks during the 36-month window that precedes the

portfolio rebalancing month. Portfolios are equally weighted on a monthly basis, so that weights are

re-adjusted whenever a fund disappears.14 Finally, we compute the out-of-sample performance of

these portfolios.

We use two tables to summarize the out-of-sample results conditional on different skill measures.

Table 13 reports the ranking results based on reduced-form alpha, and Table 14 reports results based

on a structural estimate of skill. Both tables include several other portfolio performance measures,

including alphas based on the Fung–Hsieh seven-factor model.


Table 13 shows that OLS alpha, αiOLS, does not distinguish well between good and bad out-of-

sample funds. The out-of-sample alpha performance is 6.28− 4.43 = 1.85% higher for the top-decile

portfolio (6.28) than for the bottom-decile one (4.43%); this is seen in column [2] of the table. We

test the significance of the difference in alphas by computing the Fung–Hsieh seven-factor alpha of

the spread returns between top- and bottom-decile portfolios. The alpha performance of the spread

14We follow the same portfolio construction procedure as described in Carhart (1997) and Kosowski, Naik, and Teo(2007).

28

portfolio is −0.46 and insignificantly different from zero (with a t-statistic of −0.14); this is shown in

the table’s last row. By some performance measures—such as portfolio average return (Mean Ret)

and growth of a one-dollar investment ($1 growth)—the top-decile portfolio slightly outperforms the

bottom-decile portfolio. In contrast, the information ratio (IR) measure suggests that the bottom

decile outperforms the top decile. The two portfolios perform similarly in terms of Sharpe ratio (SR).

These results also show that the OLS alpha’s out-of-sample performance is not robust, especially

when agency contracts are nonlinear.


Table 14 displays the out-of-sample performance of a portfolio constructed on the basis of struc-

tural estimates of skill, λiA. These results demonstrate that structural skill measures perform well

in distinguishing good from bad hedge funds, since the top-decile portfolio outperforms the bottom-

decile portfolio by 11.44 − 0.43 = 11.01% in terms of the Fung–Hsieh seven-factor alpha. The

difference is statistically significant: the alpha of the spread portfolio (reported in the table’s last

row) is 8.70% and significantly different from zero at the 5% confidence level. Other performance

measures also suggest that the top-decile portfolio significantly outperforms the bottom-decile one.

For instance, a hedge fund investor who in January 2001 invested $1 in the top-decile portfolio

ranked by structural skill, λiA, would receive $4.32 at the end of the sample period; this compares

most favorably with the $1.46 received after investing in the bottom-decile portfolio. The Sharpe

ratio of the top-decile portfolio over the ten-year period is 1.05 − 0.54 = 0.51 higher than that of

the bottom-decile portfolio. According to this evidence, the structural skill measure performs well

in distinguishing good from bad hedge funds.


Table 15 reports the Fung–Hsieh seven-factor alphas of the decile portfolio returns conditional

on hedge fund investment objectives and skill measures. Similarly to the results reported in Table 13

and Table 14, these results suggest that the structural skill measure is better than OLS alpha at

distinguishing good from bad hedge fund managers for eight out of eleven investment objectives

(Fung–Hsieh seven-factor alphas). For instance, the difference between the alphas of decile-1 and

decile-10 portfolios formed in terms of structural skill measures is 10.93%, compared with only 0.89%

for the counterpart portfolios formed by OLS alpha (see the table’s last column). The difference is

29

even more significant for fixed income arbitrage and global macro funds: the bottom-decile portfolios

of these two strategies outperform the top-decile portfolios by 6.98% (fixed income arbitrage) and

5.37% (global macro). In contrast, for portfolios of fixed income arbitrage funds and global macro

funds sorted by the structural skill measure, the top decile outperforms the bottom by (respectively)

8.74% and 17.59%.

One other result should be highlighted. The top-decile portfolio across all hedge funds formed

on the basis of structural skill delivers a Fung–Hsieh seven-factor alpha of 11.44%; this compares

well with the top-decile portfolio formed on the basis of OLS alpha, which delivers a (risk-adjusted)

measure of only 6.28% (see the last row of Table 15). This relation holds also within strategies in eight

of the eleven strategies. For example, the top-decile portfolio of fixed income arbitrage funds formed

by the structural skill measure is 12.87 − 1.54 = 11.33% higher than the corresponding portfolio

formed based on OLS alpha. The large difference is likely related to this investment strategy’s

tendency to use leverage dynamically. If risk aversion is low and leverage is high, then reduced-form

performance can be driven by low risk aversion and not by true skill, which may explain why ranking

these funds in terms of the structural skill estimate leads to superior out-of-sample performance.

Another investment objective for which we find similar results is global macro, a strategy whose

funds are often viewed as a directional bet on macroeconomic themes. Hence global macro funds

have more potential for discretionary leverage variations than do more systematic investment funds—

such as CTA, whose leverage is (supposedly) determined independently of the difference between the

NAV and the HWM.

Figures 9-11 plot the growth of $1 invested in the top-decile portfolio (OLS alpha versus structural

skill measure) for all hedge funds and each investment objective. The first figure reveals that investing

in the top decile of hedge funds selected by structural skill would have generated more final-period

wealth than would investing in the OLS alpha counterpart portfolio (from January 2001 through

December 2010); this is shown in the “ALL” subfigure. The same observation can be made for six

of the eleven investment objectives when the analysis is conditioned by hedge fund strategies.




It is noteworthy also that, despite the similarity in performance of the two skill measures for

30

convertible arbitrage funds prior to the second half of 2008, the structural skill measure significantly

outperforms OLS alpha after 2008. The probable cause for this difference is that, unlike the struc-

tural skill measure, OLS alpha is sensitive to the dynamics of leverage before, during, and after

the 2008 financial crisis. An example is offered by convertible funds. Managers in this strategy

experienced severe volatility in their NAV as a result of the prime broker crisis and the consequent

forced de-leveraging, and that volatility made them extremely sensitive to accounting for endogeneity

issues. Conditioning on their structural skills, these hedge fund managers perform better (see the

“Convertible Arbitrage” subfigure). We observe a similar phenomenon for the categories of emerging

markets, long/short equity, fixed income arbitrage, global macro, and “ALL” funds.

7 Discussion and Conclusions

In this paper we develop a comprehensive structural approach to measure and predict performance of

leveraged financial institutions that employ complex incentive contracts. The empirical application

of this structural model allows us to use previously unexploited “second moments” information in a

novel way to draw inferences about the risk-adjusted performance of investment funds with option-

like contractual features.

Intuitively, we are able to better distinguish the effects of risk taking versus skill on past fund

performance. In the structural model, we carefully motivate our assumptions about the manager’s

objective and trading technology before deriving both the optimal investment strategy and the

implied dynamics of assets under management. We build on and extend the work of Koijen (2010)

on mutual funds by explicitly modeling such hedge fund–specific contractual features as high-water

marks, equity investors’ redemption options, and prime broker contracts, which combine to create

option-like payoffs that affect the hedge fund manager’s risk taking.

Our second contribution is in applying the model to a large sample of hedge funds. We impose

the structural restrictions implied by economic theory when estimating the model’s deep parameters.

These restrictions allow us to distinguish true skill from endogenous appetite for risk. We document

differences between in-sample reduced-form regression estimates of performance and estimates based

on the structural model.

Our third contribution is to present empirical evidence of the hedge fund risk shifting predicted

by our theoretical model. The findings reject the hypothesis of constant risk taking, a notion that

is implied by models with an infinite horizon or with monotonically decreasing optimal leverage

31

in distance between NAV and HWM. Thus the results reported here underscore the importance of

accounting for short option positions when leverage is reduced because fund value is below the high-

water mark. Such options serve to discipline the manager, who might otherwise increase leverage

unboundedly with decreasing fund value.

Finally, we document the economic impact of using structural versus reduced-form estimates. In-

sample estimates of managerial ability are shown to be more accurate when based on our structural

model than when based on reduced-form models. Moreover, out-of-sample portfolios of hedge funds

formed using our structural measure of skill outperform portfolios based on reduced-form alphas.

Although we use hedge funds in our empirical application, these results have broader economic

implications. Separating the effects of risk aversion and skill on investment performance is a crucial

task faced by many parties. These parties include investors in alternative investment funds as well

as the investors in (and regulators of) leveraged financial institutions, such as banks, that employ

incentive contracts.

The findings reported in this paper suggest a number of possible extensions. First, our study

assumes a frictionless economy. The existence of both market liquidity risk and lockup periods im-

posed on the investor can interact with incentive contracts and affect optimal leverage in a nontrivial

way. Second, we assume a homoskedastic (log-normally distributed) benchmark. A vast literature

has provided empirical evidence that volatility is not constant, which means that the investment op-

portunity set is time varying. Both features would affect the value of options studied in this analysis,

amplifying or attenuating the endogeneity effect in a state-dependent manner. Finally, Chapman,

Evans, and Xu (2010) discuss the optimal portfolio choices of young and old fund managers in a dy-

namic life-cyle model. In their context, the optimal policies of the manager depend on age, the ratio

of wealth to labor income, the value of the manager’s private information, and the portfolio’s recent

past performance relative to the benchmark. A promising direction for future research would be to

account more directly for some of these features in structural estimations of performance attribution

metrics.

32

A Appendix A: Proof of Proposition 1

Under Condition 1, a hedge fund manager facing non-linear incentives maximizes the following

concavified objective function:

maxXT

E0[U(XT , HWM,K, p,m, c)], (29)

subject to

E0[φTXT ] = X0, (30)

where U(XT ,HWM,K, p,m, c) is defined as:

U(XT ,HWM,K,α,m, c) =

U(p(XT −HWM) +mXT )

U(mXK − c(K − XK)) + ν(XT − XK)

U(mXT − c(K −XT ))

for XT > XHWM

for XK < XT ≤ XHWM

for b < XT ≤ XK

,

(31)

We then proceed to define the Lagrangian problem, which can be written in terms of the optimal

control XT applied to the concavified utility function (31). Let ζ be the Lagrange multiplier applied

to the constraint (30), the Lagrangian reads:

maxXT

U(XT ,HWM,K, p,m, c)− ζφTXT . (32)

From the first order conditions, it is possible to show that optimal terminal fund value is given by:

X∗T =

(ζφTp+m

)−1/γ+ pHWM

p+m1{XT>XHWM} + XHWM1{XK<XT≤XHWM} (33)

+

(ζφTm+c

)−1/γ+ cK

m+ c1{b<XT<XK} + b1{XT≤b},

where ζ solves E[φTX∗T ] = X0 and 1z = 1 when z occurs and 1z = 0 otherwise.

We can simplify the derivation of the expectation of φTX∗T by mapping the set of terminal fund

value to the state price. This is similar to the solution method used in Carpenter (2000). Therefore

equation (33) is equivalent to :

33

X∗T =

(ζφTp+m

)−1/γ+ pHWM

p+m1{φT<zHWM} + XHWM1{zK>φT>zHWM} (34)

+

(ζφTm+c

)−1/γ+ cK

m+ c1{zb>φT>zK} + b1{φT>zb},

where state price, zi, solves U′(Xi,HWM,K, p,m, c) = ζzi . This is the first order condition of the

Lagrangian. As a results, critical value for the state prices are :

zHWM = (p(XHWM −HWM) +mXHWM )−γ(p+m)/ζ (35)

zK = (mXK − c(K − XK))−γ(m+ c)/ζ (36)

zb = (mb− c(K − b))−γ(m+ c)/ζ (37)

Notice that there is an inverse relation between the state price and fund value. This is intuitive

since the state price is closely related to marginal utility which is a decreasing function of the level

of consumption. According to the martingale property, the optimal fund value is the process:

X∗t = Et(

φT

φtX∗

T ) (38)

This can be analytically derived: the optimal fund value is given by:

X∗t =

ζ−1/γ

φt(p+m)1−1/γI1,t +

pHWM

φt(p+m)I2,t +

XHWM

φtI5,t (39)

+ζ−1/γ

φt(m+ c)1−1/γI3,t +

cK

φt(m+ c)I4,t +

b

φtI6,t

where: I1,t ≡ Et(φ1−1/γT 1{φT<zHWM}) =DtN(dzHWM

2,t ), I2,t ≡ Et(φT 1{φT<zHWM}) =CtN(dzHWM1,t ),

I3,t ≡ Et(φ1−1/γT 1{zK≤φT<zb}) = Dt(N(dzb2,t) − N(dzK2,t )), I4,t ≡ Et(φT 1{zK≤φT<zb}) = Ct(N(dzb1,t) −

N(dzK1,t )), I5,t ≡ Et(φT 1{zHWM≤φT<zK}) = Ct(N(dzK1,t ) − N(dzHWM1,t )), I6,t ≡ Et(φT 1{zb≤φT }) =

CtN(−dzb1,t). Where N(x) denotes the cdf function, with dzi1,t ≡ ln(zi/φt)+(r−0.5λ′λ)(T−t)

∥λ∥√T−t

, dzi2,t ≡

dzi1,t +∥λ∥

√T−t

γ , and Ct ≡ exp(ln(φt) − r(T − t)), Dt ≡ exp(1−γγ ln(φt) − 1−γ

γ (r + 0.5λ′λ)(T − t) +

0.5(1−γγ

)2∥λ∥2 (T − t)).

According to the optimal fund value in equation (39), we can use Ito’s lemma to derive the

34

dynamic of dX∗(t, φt) and then compare a coefficient of its diffusion term to equation (4) to obtain

the following optimal allocation:

θ∗t = −∂X∗(t, φt)

∂φt

φt

X∗t

(ΣΣ′)−1Σλ (40)

where

∂X∗(t, φt)

∂φ= − ζ−1/γGt

φ1+γγ

t (p+m)1−1/γ

(N(dzHWM

2,t )

γ+

N ′(dzHWM2,t )

∥λ∥√T − t

)− αHWM

φt(α+m)

HtN′(dzHWM

1,t )

∥λ∥√T − t

(41)

− XHWMHt


[N ′(dzK1,t )−N ′(dzHWM

1,t )]

− ζ−1/γGt

φ1+γγ

t (m+ c)1−1/γ

[N ′(dzb2,t)−N ′(dzK2,t )

∥λ∥√T − t

+N(dzb2,t)−N(dzK2,t )

γ

]

− cKHt

φt(m+ c) ∥λ∥√T − t

(N ′(dzb1,t)−N ′(dzK1,t )) + bHtN(−dzb1,t)


(42)

with Gt ≡ exp(−1−γγ (r + 0.5λ′λ)(T − t) + 0.5

(1−γγ

)2∥λ∥2 (T − t)) and Ht ≡ exp(−r(T − t)).

B Appendix B: Concavification

Since the objective function we study is not globally concave, we solve the optimization problem

by convexification. This requires solving for the two contact points XK and XHWM such that a

randomization between U(XK) and U(XHWM ) dominates the convex region. Condition 1 guarantees

the existence of the concavification points and the tangent line that dominate the region. In general,

the tangent point XK may occur inside the range (K, HWM) hence the chord between XK and

XHWM does not dominate the entire region between K and HWM . This happens when Condition

1 is violated. Figure 12 illustrates the concavified utility function U , with the tangent dotted line

linking U(XK) and U(XHWM ). Panel (a) depicts the transformed utility function when Condition 1

is satisfied. While panel (b) illustrate the transformed utility function when Condition 1 is violated.

In this case, XK is inside (K,HWM), so that the tangent line doesn’t dominate the convex region

of the utility function. In general, this happens when the risk aversion parameter γ is high.

[Insert Figure 12 here ]

35

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38

.

Tab

le1:

Reduced-Form

andStructuralSkillEstim

ates

inSim

ulatedEconomies.

Thistable

displaysmeansan

dstan

dard

deviation

s(in

parentheses)of

skillestimates

from

2,500simulation

sof

36monthly

returns.

Panel

Ashow

stheestimate

ofreduced-form

alpha,αOLS,

reduced-form

beta,

βOLSan

dinform

ationratio,IR,whilePanel

Bshow

stheestimatesoftrueskillmeasure,λA,meanan

dstandard

deviation

oftimevaryingreduced-form

alpha,

αt,OLS,whichis

implied

from

thetheoreticalrestriction:αt,OLS=

λ2 A/γtandmeanan

dstan

darddeviation

oftimevaryingreduced-form

beta,

βt,OLS,whichis

implied

from

thetheoreticalrestriction:βt,OLS=

λB/(γtσ

B).

The36-m

onth

returnsaresimulatedwiththefollow

ingparameters;λA∈{0

.5,1,1.5},γ=

5,r

=0.05,λB=

0.2σB=

0.05an

dσA=

0.05,

whereλAdenotes

trueskilll,γdenotes

constan

tcoeffi

cientofrelativerisk

aversion,rden

otes

risk

free

rate,whileλBan

dσBden

otethe

SharpeRatio

andvolatilityof

thebenchmarkassetB.

Trueparam

eter

Pan

elA:Reduced-form

Estim

ation

Pan

elB:StructuralEstim

ation

λA

αOLS

βOLS

IR

λA

Mean(αt,OLS)

Std

(αt,OLS)

Mean(βt,OLS)

Std

(βt,OLS)

0.5

5.93

1.82

0.29

0.49

5.39

3.12

1.76

1.26

(10

.15)

(0.74)

(065

)(0.09

)(2.44

)(1.49)

(0.57)

(0.50)

119.67

2.04

0.85

0.99

19.12

8.05

2.03

1.60

(14

.99)

(0.92)

(0.65)

(0.15

)(5.95

)(2.60)

(0.58)

(0.41)

1.5

39.82

2.14

1.37

1.48

38.80

13.68

2.12

1.83

(18

.20)

(1.11)

(0.64)

(0.19

)(9.28

)(3.13)

(0.52)

(0.31)

39

Table 2: Out-of-sample analysis - Simulated Economy. Panel A displays the out-of-sample perfor-mance of the top decile portfolio ranked by true skill measure, λA, and OLS alphas in a simulatedeconomy. The out-of-sample performance metrics are annualised alpha relative to simulated bench-mark (Alpha), t-statistic of the alpha (t-stat), mean annualised returns (Mean Ret), growth of 1dollar investment ($1 growth), annualised information ratio (IR) and Sharpe Ratio (SR). The simu-lated hedge funds are sorted at the beginning of every year (10 years in total) into deciles based on thetrue skill measure, λA, and OLS alphas estimated using 36-month rolling window preceding portfoliorebalancing time. The portfolios are equally weighted on a monthly basis, with the weights beingre-adjusted whenever a fund disappears. Panel B document the percentage of unskilled managersbeing identified as skilled managers (i.e. top decile portfolio on the basis of OLS alpha). To calculatethis statistics, we divide the sample of simulated hedge fund returns in two parts: 90 per cent of thesample represents low skill funds with λA = 0.01; the rest 10 per cent of the funds are simulated withλA = 0.6. The rest of the parameters used i teh simulation are as follows: γ = 5, r = 0.05, λB = 0.2,σB = 0.05 and σA = 0.05, where λA denotes true skill, γ denotes constant coefficient of relativerisk aversion, r denotes the risk free rate, λB and σB denote the Sharpe Ratio and volatility of thebenchmark asset. The returns are simulated 10,000 times, with each simulation representing a timeseries of 13-years.

Panel A: Out-of-sample performance metrics

Portfolio Alpha t-stat Mean Ret $1 growth IR SR

(pct/ann.) (pct/ann.)

OLS Alpha 1.96 1.77 7.80 2.08 0.56 0.30

Structural True Skill 6.27 6.81 12.09 3.20 2.16 0.78

Panel B: OLS alpha’s false identification of skilled funds

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

Percentage of

False Discovery66.70% 66.50% 61.50% 57.60% 59.10% 47.40% 55.10% 62.50% 59.10% 59.80%

40

Table 3: Evidence on Risk Shifting - Simulated Economy. This table reports evidence on riskshifting, proxied by standardized betas (Panel A) and residual volatility (Panel B), in a simulatedeconomy. Each panel reports regression coefficients, p-values (in parentheses) and adjusted R-squaredin a simulated economy. 13-year returns for 10,000 hedge funds are simulated using the followingparameters: λA ∈ {0.01, 0.6}, γ = 5, r = 0.05, λB = 0.2, σB = 0.05 and σA = 0.05, where λA denotestrue skill, γ denotes the constant coefficient of relative risk aversion, r denotes risk free rate, λB

and σB denotes the Sharpe Ratio and volatility of the benchnark asset B. We assume that 5,000hedge funds returns are of low skill, with λA ∈ {0.01}; another 5,000 funds are assumed to have highskill with λA ∈ {0.6}. Betas and residual volatilities are computed using 24-months rolling windowregressions. The estimates are then standardized at the fund level with their unconditional meansand standard deviations

Ri,t= u+ c1×1{Dist2HWMi,t>0}×Dist2HWM i,t

+ c2×1{Dist2HWMi,t<0}×Dist2HWM i,t+εi,t,

Intercept c1 c2 Adj.R2

Panel A: Ri,t ≡ Standardized Beta

Low skill funds 1.72 -2.05 5.74 0.330(0.00) (0.00) (0.00)

High skill funds 1.63 -1.58 4.53 0.277(0.00) (0.00) (0.00)

Panel B: Ri,t ≡ Standardized Residual Volatility

Low skill fund 0.27 -0.56 1.08 0.859(0.00) (0.00) (0.00)

High skill fund 0.29 -0.44 0.82 0.760(0.00) (0.00) (0.00)

41

.

Tab

le4:

SummaryStatisticsforHedge

FundsReturns.

This

table

displays,

foreach

invesm

entcatego

ry,thenumber

offunds(in

parentheses)an

dfundcross-sectional

medianas

wellas

firstandthirdquartiles

(inparentheses)ofthean

nualisedexcess

meanreturns

over

risk

free

rate,stan

darddeviation

,skew

nessan

dkurtosis.

Thestatisticsare

computedusingmonthly

net-of-feereturnsoflivean

ddeadhedge

fundsreportedin

theBarclay

Hedge

datab

asebetweenJanuary

1994andDecem

ber

201

0.

InvestmentObjectives

Mean(pct/a

nn.)

Std

(pct/a

nn.)

Skew

ness

Kurtosis

CTA

(165

5)5.80(1.33

,12

.06)

15.28(9.94

,24

.57)

0.34(-0.12,0.86)

4.28(3.41,6.03)

Con

vertible

Arbitrage

(155

)4.97

(2.34

,6.49

)6.98

(4.22

,10

.55)

-0.72(-1.45,-0.08)

6.13(4.37,10

.97)

EmergingMarkets(512

)9.20

(3.98

,16

.04)

19.40(10

.97,29

.33)

-0.30(-0.94,0.25)

5.14(3.84,

7.92)

EquityLon

g/Short(807

)6.57

(2.74

,11

.21)

11.73(8.43

,17

.54)

-0.01(-0.53,0.54)

4.52(3.56,6.24)

EquityMarket

Neutral

(154

)2.64

(-0.11,5.64

)7.27

(5.28

,10

.13)

-0.10(-0.42,0.22)

4.14(3.10,5.58)

EquityShortBias(34)

1.40(-3.76,6.40

)21

.57(13.40

,30

.79)

0.30(-0.08,0.66)

4.32(3.56,5.46)

EventDriven

(211

)7.10

(4.34

,10

.65)

10.84(7.19

,14

.86)

-0.43(-1.19,0.32)

5.91(4.53,9.79)

Fixed

IncomeArbitrage

(93)

3.57

(0.31

,8.53

)9.02

(5.52

,12

.99)

-0.77(-2.45,0.06)

8.53(4.68,15.20)

Global

Macro

(185

)5.79(2.55

,10

.43)

12.35(8.58

,17

.08)

0.09(-0.43,0.49)

4.33(3.56,6.24)

MultiStrategy

(196

)5.23(1.22

,8.96

)9.85

(6.04

,15

.13)

-0.38(-1.21,0.34)

5.35(3.60,9.36)

Others(826

)8.23

(3.66

,14

.15)

13.91(8.44

,23

.29)

-0.03(-0.65,0.59)

4.90(3.60,7.00)

AllFunds(4828

)6.38

(2.26

,11

.99)

13.33(8.29

,21

.68)

0.05(-0.56,0.62)

4.70(3.58,6.94)

42

.

Tab

le5:

SummaryStatisticsforHedge

FundStyle

Index

Returns.

Thistable

displays,byinvestmentcategory,

maxim

um

drawdow

n,risk

premium

over

risk

free

rate,SharpeRatio,mean,median,1st

and3rd

quartiles,standarddeviation,skew

nessandkurtosisof

theindex

returns.

Thestatistics

arecomputedusingmon

thly

hedgefundstyle

index

returnsreportedin

theDow

Jones

Credit

Suisse

between

Jan

uary1994

andDecem

ber

2010.

Index

Max.

Draw-

dow

n

Risk

Pre-

mium

Sharpe

Ratio

Mean

Med

ian

1st

Quar-

tile

3rd

Quar-

tile

Std

Skewness

Kurtosis

(%)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

CTA

17.74

3.70

0.31

7.13

4.74

-21.78

36.18

11.81

0.00

2.98

Con

vertible

Arbitrage

32.86

4.42

0.62

7.86

12.36

1.02

17.94

7.08

-2.74

18.75

EmergingMarkets

45.15

5.66

0.37

9.10

16.26

-16.26

34.50

15.22

-0.78

7.88

EquityLon

g/Short

21.97

6.85

0.69

10.29

10.38

-9.96

28.44

9.97

-0.01

6.35

EquityMarket

Neutral

45.11

2.29

0.22

5.73

8.34

2.64

15.42

10.64

-11.71

156

.21

EquityShortBias

58.64

-5.88

-0.35

-2.45

-8.28

-41.70

35.46

17.05

0.69

4.45

EventDriven

19.15

6.67

1.09

10.11

12.30

1.86

22.68

6.11

-2.42

15.93

Fixed

IncomeArbitrage

29.03

1.88

0.32

5.32

8.88

2.34

14.46

5.93

-4.29

31.27

GlobalMacro

26.78

8.88

0.88

12.31

13.08

-1.68

27.36

10.03

-0.03

6.44

MultiStrategy

24.75

4.63

0.85

8.07

9.96

1.26

19.74

5.45

-1.76

9.06

All

19.67

5.89

0.77

9.33

9.78

-1.98

23.10

7.70

-0.22

5.38

43

.

Tab

le6:

RegressionResultsof

Hedge

FundStyle

Index

ReturnsonFungandHsieh

7-factorBen

chmark.This

table

reports

Fungan

dHsieh

7factorsalpha,

betacoeffi

cients

andtheirNew

ey-W

estt-statistics(inparentheses)of

hedgefundindex

returnsbyinvestment

objective.

Theestimates

arecomputedbyusingmon

thly

hedgefundstyle

index

returnsreportedin

theDow

Jones

Credit

Suisse,Fung

and

Hsieh

(2004)

risk

factorsreported

inDav

idA.Hsieh’s

hedgefund

data

library.

Bon

dfactorsareduration

adjusted,thebond

duration

sdataarefrom

Datastream.AlldataarebetweenJanuary

1994andDecem

ber

201

0.

Hedge

fundindex

Alpha

PTFSBD

PTFSFX

PTFSCOM

SNPMRF

SCMLC

BD10

RET

BAAMTSY

Adj.

R2

(pct/a

nn.)

CTA

4.05

0.03

0.04

0.04

0.01

0.04

0.24

0.01

15.43

(1.63)

(1.92

)(2.67

)(2.52

)(0.14

)(0.66)

(1.97)

(0.08)

Con

vertible

Arbitrage

2.12

-0.02

0.00

0.00

0.04

0.01

0.08

0.60

44.45

(1.06

)(-2.07)

(-0.88)

(-0.43)

(1.39

)(0.40)

(1.52)

(5.54)

EmergingMarkets

1.26

-0.04

-0.01

0.01

0.42

0.24

0.07

0.35

35.38

(0.32

)(-1.42)

(-0.59)

(0.73

)(6.60

)(3.43)

(0.68)

(3.81)

EquityLon

g/S

hort

3.50

-0.02

0.01

0.01

0.39

0.32

0.14

0.09

58.24

(1.94)

(-1.55)

(1.04

)(1.02

)(8.16

)(3.63)

(2.43)

(1.51)

EquityMarket

Neutral

1.68

-0.03

0.02

0.02

0.14

0.02

-0.31

0.12

15.03

(0.66

)(-1.10)

(1.69

)(1.52

)(2.68

)(0.52)

(-1.15)

(1.56)

EquityShortBias

-1.49

0.00

0.00

-0.02

-0.86

-0.50

0.06

0.24

69.62

(-0.52)

(0.27

)(-0.40)

(-1.02)

(-10.14

)(-8.89)

(0.79)

(1.53)

EventDriven

4.57

-0.03

0.00

0.01

0.18

0.10

-0.03

0.23

54.93

(3.21)

(-2.07)

(1.11

)(1.03

)(7.17

)(3.86)

(-0.65)

(5.39)

Fixed

IncomeArbitrage

-0.10

-0.02

-0.01

0.01

0.02

0.00

0.10

0.51

45.02

(-0.06)

(-2.52)

(-1.72)

(0.71

)(0.57

)(0.14)

(2.07)

(4.53)

Global

Macro

6.18

-0.03

0.01

0.02

0.13

0.04

0.34

0.26

12.58

(2.53

)(-1.77)

(0.94

)(0.90

)(1.78

)(0.50)

(4.26)

(2.41)

MultiStrategy

3.33

-0.01

0.01

0.00

0.05

0.02

-0.03

0.37

32.32

(2.42)

(-1.05)

(1.08

)(0.52

)(1.92

)(0.83)

(-0.53)

(5.21)

All

3.10

-0.03

0.01

0.02

0.23

0.14

0.14

0.24

43.78

(2.11

)(-2.50)

(1.48

)(1.51

)(5.72

)(2.33)

(2.16)

(4.56)

44

.

Tab

le7:

SummaryStatisticsforProxiesof

BenchmarkAssets.

This

table

displays,

byinvestmentobjective,

benchmarkassetreturns

max

imum

drawdow

n,risk

premium

over

risk

free

rate,SharpeRatio,mean,median,1stan

d3rd

quartiles,stan

dard

deviation

,skew

ness

andkurtosis.Thebenchmarkassetforeach

investmentob

jectiveishedgefundstyle

index

adjusted

forpotential

Fungan

dHsieh

alpha.

Thestatistics

arecomputedbyusingmon

thly

benchmarkasset

returnswhichare

computedfrom

hed

gefundstyle

index

returnsreported

intheDow

Jon

esCredit

Suisse

andFungan

dHsieh

7risk

factors

reported

inDav

idA.Hsieh

’shedgefunddatalibrary.Bondfactors

areduration

adjusted,thebon

dduration

sdataarefrom

Datastream.Alldata

are

betweenJan

uary199

4andDecem

ber

201

0.

Benchmarkasset

Max.

Draw-

dow

n

Risk

Pre-

mium

Sharpe

Ratio

Mean

Med

ian

1st

Quar-

tile

3rd

Quar-

tile

Std

(Ann.)

Skewness

Kurtosis

(%)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

CTA

12.60

0.23

0.05

3.67

1.75

-7.18

12.14

4.42

0.68

3.75

Con

vertible

Arbitrage

18.29

1.83

0.41

5.26

5.92

-0.08

11.45

4.42

-1.71

16.61

EmergingMarkets

31.24

3.45

0.39

6.88

12.88

-9.18

25.17

8.74

-1.15

7.00

EquityLon

g/Short

20.97

2.88

0.40

6.31

9.17

-11.18

22.90

7.28

-0.67

4.22

EquityMarket

Neutral

7.44

0.80

0.34

4.24

5.23

-0.42

10.02

2.34

-0.76

3.96

EquityShortBias

47.17

-5.25

-0.35

-1.81

-9.67

-37.33

32.11

14.99

0.70

4.10

EventDriven

16.23

2.23

0.48

5.67

8.34

-1.97

15.12

4.63

-1.44

8.29

Fixed

IncomeArbitrage

14.27

1.88

0.51

5.31

6.20

0.78

11.00

3.65

-1.98

19.61

GlobalMacro

5.71

1.56

0.56

5.00

5.79

-0.57

10.93

2.76

-0.84

8.37

MultiStrategy

10.48

0.93

0.35

4.36

4.88

1.71

7.94

2.62

-1.99

19.23

All

17.66

2.96

0.55

6.40

9.05

-3.03

18.41

5.34

-1.39

8.14

45

Table 8: Put Option Strike Estimation. This tables displays, by investment objective, the estimationof put option strike price as a fraction of high water mark, K and their t-statistics in parentheses. Theestimates are computed using montly net-of-fee returns of live and dead hedge funds, by investmentobjectives, reported in the BarclayHedge database between January 1994 and December 2010.

Investment Objectives K

CTA 0.8 (3.81)Convertible Arbitrage 0.6 (6.51)

Emerging Markets 0.6 (0.61)Equity Long/Short 0.8 (0.51)

Equity Market Neutral 0.6 (11.56)Equity Short Bias 0.8 (2.74)

Event Driven 0.8 (3.79)Fixed Income Arbitrage 0.6 (6.02)

Global Macro 0.8 (3.04)Multi Strategy 0.8 (5.86)

Other 0.8 (2.33)

46

.

Tab

le9:

In-sam

ple

SkillEstim

ates.This

table

displays,

byinvestmentobjective,

in-sample

estimationof

trueskill,λA,trad

itional

OLSalpha,

αOLS,an

dstructuralalpha,

αt,OLS,im

plied

from

theoreticalrestrictions.

Each

columnreports

thecross-sectional

med

ian

oftheestimates

aswellas

theirfirstan

dthirdquartilesin

parentheses.Since

thestructuralalphaestimatevaries

over

time,

wereport

themeanan

dstan

darddeviation

ofthemeasure

inthefourthandfifthcolumn,respectively.

Theestimatesarecomputedusingmontly

net-of-feereturnsof

livean

ddeadhedge

fundsreportedin

theBarclayHedgedatabase

betweenJan

uary

1994

andDecem

ber

2010

.


TrueSkill

OLSAlpha(pct/a

nn.)

MeanStructuralAlpha(pct/a

nn.)

Std

StructuralAlpha(pct/an

n.)

λA

αOLS

Mean(α

t,OLS)

Std(α

t,OLS)

CTA

(1655

)0.55

(0.33,

0.93)

5.58

(1.48

,11

.60)

3.14

(1.57

,6.53)

2.16(0.93,4.24)

Convertible

Arbitrage

(155

)0.00(0.00,

0.26

)3.19

(1.03

,5.60

)0.00

(0.00

,4.58

)0.00(0.00,0.56)

EmergingMarkets(512)

0.70(0.37,

1.01

)6.09

(1.60

,12

.07)

3.29

(1.03

,7.46

)3.96(1.18,7.07)

EquityLon

g/S

hort(80

7)0.44(0.31

,0.67

)4.34

(0.82

,8.44

)1.23

(0.59

,2.75

)1.24(0.49,2.54)

EquityMarket

Neutral

(154

)0.12

(0.00,

0.26)

2.00

(-0.99,4.53

)1.06

(0.00

,1.86

)0.40(0.00,1.08)

EquityShort

Bias(34)

0.47(0.25,

1.15

)3.20

(0.02

,9.18

)6.90

(2.95

,15

.77)

2.82(1.46,7.61)

EventDriven

(211

)0.41(0.30

,0.59

)4.11

(0.58

,8.34

)1.11

(0.45

,2.20

)1.31(0.12,2.86)

Fixed

IncomeArbitrage

(93)

0.17(0.10,

0.33

)1.63

(-2.08,5.26

)0.65

(0.27

,1.31)

0.51(0.18,

1.42)

Global

Macro(185

)0.61

(0.46

,0.80)

4.33

(1.13

,8.83

)1.89

(1.14

,2.82)

1.56(0.75,

2.55)

MultiStrategy(196

)0.41

(0.04,

0.74)

5.99

(3.39

,9.20

)6.37

(0.66

,11

.40)

2.22(0.20,

4.82)

Others(826

)0.53(0.34,

0.88

)5.15

(0.74

,10

.44)

1.58

(0.51

,4.87

)1.19(0.12,4.76)

AllFunds(482

8)0.49(0.29

,0.82

)4.85

(1.09

,9.87

)2.08

(0.75,

5.24

)1.62(0.44,3.96)

47

Table 10: Evidence on Risk Shifting - Standardized betas and residual volatility. This table re-ports, for all investment categories, evidence on risk shifting using standardized betas (Panel A)and standardized residual volatility (Panel B). We report regressions’ coefficients, their p-values (inparentheses) and adjusted R squared for all hedge funds, low skill funds (the bottom quartile of hedgefunds ranked by true skill measure) and high skill funds (the top quartile of hedge funds ranked bytrue skill measure). Betas and residual volatilities are computed using 24-months rolling windowregressions and its alpha-adjusted hedge fund style index. The betas and residual volatilities arethen standardized using fund specific unconditional means and standard deviations. The hedge funddata are monthly net-of-fee returns reported in BarclayHedge database between January 1994 andDecember 2010

Ri,t= u+ c1×1{Dist2HWMi,t>0}×Dist2HWM i,t

+ c2×1{Dist2HWMi,t<0}×Dist2HWM i,t+εi,t,

Intercept c1 c2 Adj.R2

Panel A: Ri,t ≡ Standardized Beta

All funds 0.86 -0.18 0.24 0.014(0.00) (0.00) (0.01)

Low skill funds 0.90 -0.30 0.43 0.003(0.00) (0.02) (0.02)

High skill funds 0.89 -0.11 0.24 0.003(0.00) (0.20) (0.03)

Panel B: Ri,t ≡ Standardized Residual Volatility

All funds 0.08 -0.01 0.01 0.001(0.00) (0.13) (0.21)

Low skill funds 0.07 -0.04 0.09 0.02(0.00) (0.00) (0.00)

High skill funds 0.13 -0.03 0.03 0.001(0.00) (0.10) (0.25)

48

.

Tab

le11:Evidence

onRiskShifting-Chan

gein

realized

volatility.Thistable

reports,byinvestmentcategory,

regression

coeffi

cients

and

adjusted

Rsquared.In

Pan

elA

andB,wereport

evidence

onrisk

shiftingforfundsthat

haveexperiencedat

leasta15percentdeviation

ofassets

valuefrom

theirhighwater

mark.In

Pan

elA

weuse

thedifference

betweensubsequent12-month

andprior

12-m

onth

realized

volatilityas

thedep

endentvariab

le;Pan

elB

reports

resultswhen

usingthedifference

betweensubsequent6-m

onth

andprior

6-month

realized

volatilityas

dep

endentvariab

le.Thehedge

funddata

are

monthly

net-of-feereturnsreported

inBarclayHed

gedatabasebetween

Jan

uary1994

andDecem

ber

2010.

σi,[t,t+T]−

σi,[t−T,t]=

ui+

β1×1 {

Dist2HW

Mi,t>0}×

Dist2HW

Mi,t

+β2×1 {

Dist2HW

Mi,t<0}×

Dist2HW

Mi,t+

β3×(A

VGVIX

[t,t+T]−

AVGVIX

[t−lag,t])+ε i

,t,

Pan

elA

:σt+

12−

σt−

12

Pan

elB:σt+

6−

σt−

6


Intercept

Beta1

Beta2

Beta3

Adj.R

2Intercep

tBeta1

Beta2

Beta3

Adj.R

2

AllStrategies

0.009

∗∗∗

-0.063

∗∗∗

0.13

7∗∗∗

0.55

6∗∗∗

0.18

8-0.004

∗∗∗

-0.038

∗∗∗

0.087

∗∗∗

0.528∗∗

∗0.115

CTA

0.00

1-0.096

∗∗∗

0.02

5∗∗∗

0.30

4∗∗∗

0.10

4-0.010∗∗

∗-0.067∗∗

∗-0.057∗∗

∗0.303∗∗

∗0.058

Con

vertible

Arbitrage

0.00

9∗∗∗

-0.067∗

∗∗0.09

2∗∗∗

1.11

6∗∗∗

0.40

4-0.001∗∗

∗-0.078∗∗

∗0.124∗∗

∗0.997

∗∗∗

0.319

EmergingMarkets

0.00

6∗∗∗

-0.015

∗∗0.12

7∗∗∗

0.91

3∗∗∗

0.28

1-0.012

∗∗∗

0.020

∗∗∗

0.098∗∗

∗0.839

∗∗∗

0.198

EquityLon

g/S

hort

0.00

5∗∗∗

-0.019

∗∗∗

0.11

8∗∗∗

0.40

8∗∗∗

0.12

6-0.005

∗∗∗

-0.005

0.085∗∗

∗0.412

∗∗∗

0.088

EquityMarket

Neutral

-0.003

∗∗∗

0.056

∗∗∗

0.02

9∗∗

0.22

4∗∗∗

0.18

1-0.009

∗∗∗

0.067

∗∗∗

-0.009

0.208

∗∗∗

0.111

EquityShortBias

0.05

2∗∗∗

-0.156

∗∗∗

0.19

6∗∗∗

0.41

8∗∗∗

0.06

80.009

-0.120

∗∗0.028

0.620∗∗

∗0.040

EventDriven

0.00

8∗∗∗

-0.101

∗∗∗

0.10

1∗∗∗

0.78

2∗∗∗

0.32

70.001

-0.108

∗∗∗

0.127∗∗

∗0.706

∗∗∗

0.266

Fixed

IncomeArbitrage

0.019

∗∗∗

-0.173

∗∗∗

0.33

9∗∗∗

0.83

3∗∗∗

0.43

40.01

4∗∗

∗-0.195∗∗

∗0.377∗∗

∗0.758

∗∗∗

0.348

Global

Macro

0.004

∗∗-0.054

∗∗∗

0.03

6∗∗

0.45

7∗∗∗

0.13

7-0.011

∗∗∗

0.027

∗∗-0.054∗∗

∗0.411∗∗

∗0.097

MultiStrategy

0.00

9∗∗∗

-0.065∗

∗∗0.16

7∗∗∗

0.66

9∗∗∗

0.43

40.00

3∗∗

∗-0.069∗∗

∗0.206∗∗

∗0.604

∗∗∗

0.338

Others

0.01

4∗∗∗

-0.040

∗∗∗

0.18

5∗∗∗

0.64

2∗∗∗

0.28

40.000

-0.016

∗∗0.149∗∗

∗0.631

∗∗∗

0.159

49

.

Tab

le12:Variation

inhedge

fundalpha.

Thistable

summarizestheresultsofpanel

regression

sofreduced-form

OLSalpha(α

OLS,i,t)on

structuralskill(λ

i,t)

andeff

ectiverisk

aversion

(γi,t):

αOLS,i,t=

const

+b 1

×λi,t+

b 2×

γi,t+

ε i,t,

andin

logs,withlog(1

+αOLS,i,t)=

const

+b 1

×log(λ

i,t)+b 2

×log(γ

i,t)+ε i

,t.Theright-han

dsidevariablesareob

tained

estimatingthe

structuralmodel

using24-m

onth

rollingwindow

.Thehedgefunddata

ismonthly

net-of-feereturnsreported

inBarclayHedgedatabase

betweenJan

uary1994

andDecem

ber

2010.

constan

tλi,t

γi,t

AUM

i,t

log(λ

i,t)

log(γ

i,t)

log(A

UM

i,t)

Adj.R

2

(1)

αi,t

0.03

0.09

5.11%

(11.73)

(11.43)

(2)

αi,t

0.04

0.12

-0.01

7.11%

(22.63)

(13.00)

(-13.94)

(3)

αi,t

0.03

0.06

-0.01

0.00

7.91%

(14.52)

(8.81)

(-13.57)

(-2.62)

(4)

log(1

+αi,t)

0.05

0.00

0.38%

(42.04)

(11.33)

(5)

log(1

+αi,t)

0.09

0.02

-0.01

1.72%

(21.55)

(11.12)

(-10.53)

(6)

log(1

+αi,t)

0.13

0.01

-0.01

-0.01

3.70%

(11.67)

(9.13)

(-8.70)

(-7.44)

50

Table 13: Performance persistence tests - Ranking funds on OLS alphas. This table displays out-of-sample performances of decile portfolios ranked by OLS alphas. The hedge funds are sorted everyyear into deciles on the 1st January (from 2001 until 2010) based on their OLS alpha obtained usingmonthly individual net-of-fee returns for the 36-month rolling window preceding every 1st January.The portfolios are equally weighted on monthly basis, with weights which are readjusted if a funddies. Decile 1 comprises funds with the highest OLS alphas, while decile 10 comprises the lowest.The last row represents spread returns between 1st and 10th decile portfolios, Decile 1 - Decile 10.Column two reports Fung and Hsieh (2004) OLS alphas based on out-of-sample portfolios returnsbetween January 2001 and December 2010. Column three and four report t-statistics and p-valuesof the Fung and Hsieh (2004) OLS alphas. Column five reports the adjusted R2 (Adj R2) of theOLS regression of out-of-sample portfolio returns on Fung and Hseih 7 factors. The rest of thecolumns report annualised percentage mean returns (Mean Ret), the growth of 1 dollar investment($1 growth), the annualised information ratio (IR), tracking error (TE), and Sharpe Ratio (SR) ofout-of-sample portfolios returns.

Portfolio FH Alpha t-stat p-value Adj R2 Mean Ret $1 growth IR TE SR


Decile 1 6.28 2.13 0.04 40.37 11.25 2.90 0.78 8.00 0.85

Decile 2 3.14 1.77 0.08 40.43 7.03 1.96 0.58 5.45 0.66

Decile 3 3.36 2.21 0.03 33.04 7.44 2.07 0.77 4.36 0.95

Decile 4 3.46 3.18 0.00 40.37 6.72 1.93 0.92 3.77 0.90

Decile 5 3.38 2.86 0.01 42.46 7.15 2.02 1.01 3.34 1.09

Decile 6 2.79 3.00 0.00 49.97 6.30 1.86 0.94 2.98 0.95

Decile 7 2.40 2.58 0.01 47.21 6.01 1.81 0.85 2.81 0.95

Decile 8 2.79 3.54 0.00 47.55 6.13 1.83 1.05 2.65 1.05

Decile 9 3.23 2.82 0.01 52.86 6.69 1.93 1.03 3.14 0.96

Decile 10 4.43 2.71 0.01 56.30 8.05 2.18 1.00 4.44 0.85

Spread (Decile1-10) -0.46 -0.14 0.89 12.83 3.20 1.33 -0.06 7.41 0.12

51

Table 14: Performance persistence tests - Ranking funds on True skill measure. This table displaysout-of-sample performances of decile portfolios ranked by true skill measure, λA, relative to empiricalproxies for their benchmark index. The hedge funds are sorted every year into deciles on the 1stJanuary ( from 2001 until 2010 ) based on the true skill measure, λA, obtained using monthlynet-of-fee returns in the 36-month rolling window preceding every 1st January. The portfolios arethen equally weighted on monthly basis, with the weights beng readjusted if a fund dies. Decile1 comprises funds with the highest true skill measure, while decile 10 comprises the lowest. Thelast row represents the spread returns between 1st and 10th decile portfolios, Decile 1 - Decile 10.Column two reports Fung and Hsieh (2004) OLS alphas based on out-of-sample portfolios returnsbetween January 2001 and December 2010. Column three and four report t-statistics and p-valuesof the Fung and Hsieh (2004) OLS alphas. Column five reports the adjusted R2 (Adj R2) of theOLS regression of out-of-sample portfolio returns on Fung and Hseih 7 factors. The rest of thecolumns report annualised percentage mean returns (Mean Ret), the growth of 1 dollar investment($1 growth), the annualised information ratio (IR), tracking error (TE), and Sharpe Ratio (SR) ofout-of-sample portfolios returns.

Portfolio Alpha t-stat p-value Adj R2 Mean Ret $1 growth IR TE SR


Decile 1 11.44 3.55 0.00 41.73 15.50 4.32 1.22 9.41 1.05

Decile 2 5.58 3.15 0.00 52.14 9.36 2.45 0.98 5.69 0.85

Decile 3 5.09 2.89 0.00 42.92 9.34 2.47 0.95 5.38 0.98

Decile 4 2.25 1.36 0.18 42.24 6.85 1.95 0.51 4.37 0.78

Decile 5 2.87 2.35 0.02 43.72 6.56 1.90 0.76 3.79 0.83

Decile 6 2.21 1.89 0.06 41.86 5.96 1.79 0.64 3.43 0.81

Decile 7 2.35 2.50 0.01 47.30 5.76 1.76 0.86 2.75 0.91

Decile 8 1.45 1.63 0.11 48.58 4.91 1.62 0.56 2.60 0.72

Decile 9 1.58 2.32 0.02 27.06 4.65 1.58 0.73 2.16 0.91

Decile 10 0.43 0.38 0.71 39.20 3.87 1.46 0.19 2.25 0.54

Spread (Decile1-10) 8.70 2.43 0.02 35.29 11.64 2.98 0.94 9.26 0.80

52

.

Tab

le15:

Perform

ance

persistence

tests-Con

ditional

oninvestmentobjectives

andskillmeasures.

This

table

displays,

byinvestment

objective,

out-of-sam

ple

Fungan

dHsieh

(2004)

OLSalphasofdecileportfoliosranked

byOLSalphaan

dtrueskillmeasure,λA,relative

totheirem

pirical

benchmark.Thehedge

fundsaresorted

everyyearinto

deciles

onthe1st

January(from

2001

until201

0)based

ontheOLSalphaan

dtrueskillmeasure,λA,ob

tained

usingmonthly

net-of-feereturnsin

the36-month

rollingwindow

precedingevery

1stJan

uary.

Theportfoliosareequally

weigh

tedon

mon

thly

basis,

withtheweights

beingreadjusted

ifafunddies.

Decile1comprises

fundswiththehighesttrueskillmeasure,whiledecile10

comprisesthelowest.

Thelast

columnreports

thedifference

betweenFungand

Hsieh

(2004)

OLSalphas

ofDecile1an

dDecile10

portfolio.


Portfolio

Decile1

23

45

67

89

Decile10

Decile1-

10

CTA

OLSalpha

10.71

7.48

4.84

3.37

4.51

3.39

3.73

4.14

2.91

1.41

9.31

Trueskill

12.52

5.50

6.63

4.88

5.13

4.37

2.62

3.11

0.06

1.55

10.96

Con

vertible

Arbitrage

OLSalpha

-2.39

8.83

4.47

5.56

1.81

-1.67

1.15

2.91

5.43

-2.72

0.33

Trueskill

3.97

5.63

3.65

2.35

0.11

1.62

3.96

3.53

-0.54

1.11

2.86

EmergingMarkets

OLSalpha

11.65

11.13

8.96

6.39

4.16

5.39

7.18

8.69

8.95

10.75

0.89

Trueskill

17.51

12.28

5.77

12.78

8.39

4.41

4.83

6.53

3.26

6.58

10.93

Lon

g/ShortEquity

OLSalpha

2.49

-0.63

2.79

2.30

1.12

2.11

1.85

2.91

4.19

3.97

-1.48

Trueskill

6.09

1.68

3.55

1.86

1.80

2.08

0.89

2.49

0.75

1.66

4.43

EquityMarket

Neutral

OLSalpha

1.26

3.69

4.74

-1.10

1.04

-1.01

-0.82

1.43

0.57

-0.29

1.54

Trueskill

2.17

1.32

-0.82

0.43

2.19

-0.60

0.11

-0.68

2.59

1.21

0.97

EquityShortBias

OLSalpha

-1.26

-7.51

2.02

-1.56

3.05

4.17

-2.00

-2.74

-5.16

-1.71

0.44

Trueskill

0.38

5.34

6.36

4.72

-0.23

3.36

-9.09

-0.94

-0.83

-6.92

7.30

53

.Table

15(continued)


Portfolio/D

ecile

Decile1

23

45

67

89

Decile10

Decile1-10

EventDriven

OLSalpha

9.41

1.98

3.97

1.24

2.86

3.17

0.84

3.48

4.56

3.99

5.42

Trueskill

8.76

6.39

2.93

5.55

2.44

2.37

-2.06

2.56

1.30

5.62

3.14

Fixed

IncomeArbitrage

OLSalpha

1.54

-1.95

-3.17

-0.50

2.64

0.91

-0.10

0.62

-1.91

8.52

-6.98

Trueskill

12.87

-3.21

3.09

0.79

2.81

0.67

-6.93

-1.15

-6.08

4.13

8.74

Global

Macro

OLSalpha

3.70

8.87

3.98

3.87

3.44

0.84

7.12

5.68

2.61

9.06

-5.37

Trueskill

17.34

9.52

2.02

1.87

7.58

2.03

1.37

1.19

5.24

-0.25

17.59

Multi-Strategy

OLSalpha

7.27

2.02

5.32

5.31

5.71

3.08

0.54

3.85

2.21

1.04

6.23

Trueskill

5.78

2.59

9.32

4.98

3.64

1.87

3.77

1.82

2.27

1.63

4.15

Other

(Hedge

FundIndex)

OLSalpha

4.16

2.44

2.67

-0.63

2.63

-0.81

1.15

-0.92

4.58

2.77

1.39

Trueskill

3.75

3.03

2.79

-0.86

2.73

4.06

2.03

-1.77

0.28

1.41

2.34

All

OLSalpha

6.28

3.14

3.36

3.46

3.38

2.79

2.40

2.79

3.23

4.43

1.85

Trueskill

11.44

5.58

5.09

2.25

2.87

2.21

2.35

1.45

1.58

0.43

11.01

54

.

Table1:Reduced-FormandStructuralSkillEstimatesinSimulatedEconomies.Thistabledisplaysmeansandstandarddeviations(in

parentheses)ofskillestimatesfrom

2,500simulationsof36monthlyreturns.PanelAshowstheestimateofreduced-form

alpha,�OLS,

reduced-form

beta,�OLSandinformationratio,IR,whilePanelBshowstheestimatesoftrueskillmeasure,�A,meanandstandard

deviationoftimevaryingreduced-form

alpha,�t;OLS,whichisimpliedfrom

thetheoreticalrestriction:�t;OLS=�2 A=~ tandmeanand

standarddeviationoftimevaryingreduced-form

beta,�t;OLS,whichisimpliedfrom

thetheoreticalrestriction:�t;OLS=�B=(~ t�B).

The36-monthreturnsaresimulatedwiththefollowingparameters;�A2f0:5;1;1:5g; =5;r=0:05;�B=0:2�B=0:05and�A=0:05,

where�Adenotestrueskilll, denotesconstantcoe¢cientofrelativeriskaversion,rdenotesriskfreerate,while�Band�Bdenotethe

SharpeRatioandvolatilityofthebenchmarkassetB.

Trueparameter

PanelA:Reduced-formEstimation

PanelB:StructuralEstimation

�A

b� OLSb � OLS

IR

b � AMean(b� t;OL

S)

Std(b� t;OL

S)

Mean(b � t;OL

S)

Std(b � t;OL

S)

0.5

5.93

1.82

0.29

0.49

5.39

3.12

1.76

1.26

(10.15)

(0.74)

(065)

(0.09)

(2.44)

(1.49)

(0.57)

(0.50)

119.67

2.04

0.85

0.99

19.12

8.05

2.03

1.60

(14.99)

(0.92)

(0.65)

(0.15)

(5.95)

(2.60)

(0.58)

(0.41)

1.5

39.82

2.14

1.37

1.48

38.80

13.68

2.12

1.83

(18.20)

(1.11)

(0.64)

(0.19)

(9.28)

(3.13)

(0.52)

(0.31)

39

Table 2: Out-of-sample analysis - Simulated Economy. Panel A displays the out-of-sample perfor-mance of the top decile portfolio ranked by true skill measure, �A, and OLS alphas in a simulatedeconomy. The out-of-sample performance metrics are annualised alpha relative to simulated bench-mark (Alpha), t-statistic of the alpha (t-stat), mean annualised returns (Mean Ret), growth of 1dollar investment ($1 growth), annualised information ratio (IR) and Sharpe Ratio (SR). The simu-lated hedge funds are sorted at the beginning of every year (10 years in total) into deciles based on thetrue skill measure, �A, and OLS alphas estimated using 36-month rolling window preceding portfoliorebalancing time. The portfolios are equally weighted on a monthly basis, with the weights beingre-adjusted whenever a fund disappears. Panel B document the percentage of unskilled managersbeing identi�ed as skilled managers (i.e. top decile portfolio on the basis of OLS alpha). To calculatethis statistics, we divide the sample of simulated hedge fund returns in two parts: 90 per cent of thesample represents low skill funds with �A = 0:01; the rest 10 per cent of the funds are simulated with�A = 0:6. The rest of the parameters used i teh simulation are as follows: = 5; r = 0:05; �B = 0:2,�B = 0:05 and �A = 0:05, where �A denotes true skill, denotes constant coe¢ cient of relativerisk aversion, r denotes the risk free rate, �B and �B denote the Sharpe Ratio and volatility of thebenchmark asset. The returns are simulated 10,000 times, with each simulation representing a timeseries of 13-years.

Panel A: Out-of-sample performance metrics

Portfolio Alpha t-stat Mean Ret $1 growth IR SR(pct/ann.) (pct/ann.)

OLS Alpha 1.96 1.77 7.80 2.08 0.56 0.30Structural True Skill 6.27 6.81 12.09 3.20 2.16 0.78

Panel B: OLS alpha�s false identi�cation of skilled funds

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

Percentage ofFalse Discovery

66.70% 66.50% 61.50% 57.60% 59.10% 47.40% 55.10% 62.50% 59.10% 59.80%

40

Table 3: Evidence on Risk Shifting - Simulated Economy. This table reports evidence on riskshifting, proxied by standardized betas (Panel A) and residual volatility (Panel B), in a simulatedeconomy. Each panel reports regression coe¢ cients, p-values (in parentheses) and adjusted R-squaredin a simulated economy. 13-year returns for 10,000 hedge funds are simulated using the followingparameters: �A 2 f0:01; 0:6g; = 5; r = 0:05; �B = 0:2, �B = 0:05 and �A = 0:05, where �A denotestrue skill, denotes the constant coe¢ cient of relative risk aversion, r denotes risk free rate, �Band �B denotes the Sharpe Ratio and volatility of the benchnark asset B. We assume that 5,000hedge funds returns are of low skill, with �A 2 f0:01g; another 5,000 funds are assumed to have highskill with �A 2 f0:6g. Betas and residual volatilities are computed using 24-months rolling windowregressions. The estimates are then standardized at the fund level with their unconditional meansand standard deviations

Ri;t= u+ c1�1fDist2HWMi;t>0g�Dist2HWM i;t

+ c2�1fDist2HWMi;t<0g�Dist2HWM i;t+"i;t;

Intercept c1 c2 Adj:R2

Panel A: Ri;t � Standardized Beta

Low skill funds 1.72 -2.05 5.74 0.330(0.00) (0.00) (0.00)

High skill funds 1.63 -1.58 4.53 0.277(0.00) (0.00) (0.00)

Panel B: Ri;t � Standardized Residual Volatility

Low skill fund 0.27 -0.56 1.08 0.859(0.00) (0.00) (0.00)

High skill fund 0.29 -0.44 0.82 0.760(0.00) (0.00) (0.00)

41

.

Table4:SummaryStatisticsforHedgeFundsReturns.Thistabledisplays,foreachinvesmentcategory,thenumberoffunds(in

parentheses)andfundcross-sectionalmedianaswellas�rstandthirdquartiles(inparentheses)oftheannualisedexcessmeanreturns

overriskfreerate,standarddeviation,skewnessandkurtosis.Thestatisticsarecomputedusingmonthlynet-of-feereturnsofliveand

deadhedgefundsreportedintheBarclayHedgedatabasebetweenJanuary1994andDecember2010.


Mean(pct/ann.)

Std(pct/ann.)

Skewness

Kurtosis

CTA(1655)

5.80(1.33,12.06)

15.28(9.94,24.57)

0.34(-0.12,0.86)

4.28(3.41,6.03)

ConvertibleArbitrage(155)

4.97(2.34,6.49)

6.98(4.22,10.55)

-0.72(-1.45,-0.08)

6.13(4.37,10.97)


9.20(3.98,16.04)

19.40(10.97,29.33)

-0.30(-0.94,0.25)

5.14(3.84,7.92)

EquityLong/Short(807)

6.57(2.74,11.21)

11.73(8.43,17.54)

-0.01(-0.53,0.54)

4.52(3.56,6.24)

EquityMarketNeutral(154)

2.64(-0.11,5.64)

7.27(5.28,10.13)

-0.10(-0.42,0.22)

4.14(3.10,5.58)

EquityShortBias(34)

1.40(-3.76,6.40)

21.57(13.40,30.79)

0.30(-0.08,0.66)

4.32(3.56,5.46)

EventDriven(211)

7.10(4.34,10.65)

10.84(7.19,14.86)

-0.43(-1.19,0.32)

5.91(4.53,9.79)

FixedIncomeArbitrage(93)

3.57(0.31,8.53)

9.02(5.52,12.99)

-0.77(-2.45,0.06)

8.53(4.68,15.20)

GlobalMacro(185)

5.79(2.55,10.43)

12.35(8.58,17.08)

0.09(-0.43,0.49)

4.33(3.56,6.24)

MultiStrategy(196)

5.23(1.22,8.96)

9.85(6.04,15.13)

-0.38(-1.21,0.34)

5.35(3.60,9.36)

Others(826)

8.23(3.66,14.15)

13.91(8.44,23.29)

-0.03(-0.65,0.59)

4.90(3.60,7.00)

AllFunds(4828)

6.38(2.26,11.99)

13.33(8.29,21.68)

0.05(-0.56,0.62)

4.70(3.58,6.94)

42

.

Table5:SummaryStatisticsforHedgeFundStyleIndexReturns.Thistabledisplays,byinvestmentcategory,maximum

drawdown,risk

premiumoverriskfreerate,SharpeRatio,mean,median,1stand3rdquartiles,standarddeviation,skewnessandkurtosisoftheindex

returns.Thestatisticsarecomputedusingmonthlyhedgefund

styleindexreturnsreportedintheDow

JonesCreditSuissebetween

January1994andDecember2010.

Index

Max.

Draw-

down

Risk

Pre-

mium

Sharpe

Ratio

Mean

Median

1st

Quartile

3rd

Quartile

Std

Skewness

Kurtosis

(%)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

CTA

17.74

3.70

0.31

7.13

4.74

-21.78

36.18

11.81

0.00

2.98

ConvertibleArbitrage

32.86

4.42

0.62

7.86

12.36

1.02

17.94

7.08

-2.74

18.75

EmergingMarkets

45.15

5.66

0.37

9.10

16.26

-16.26

34.50

15.22

-0.78

7.88

EquityLong/Short

21.97

6.85

0.69

10.29

10.38

-9.96

28.44

9.97

-0.01

6.35

EquityMarketNeutral

45.11

2.29

0.22

5.73

8.34

2.64

15.42

10.64

-11.71

156.21

EquityShortBias

58.64

-5.88

-0.35

-2.45

-8.28

-41.70

35.46

17.05

0.69

4.45

EventDriven

19.15

6.67

1.09

10.11

12.30

1.86

22.68

6.11

-2.42

15.93

FixedIncomeArbitrage

29.03

1.88

0.32

5.32

8.88

2.34

14.46

5.93

-4.29

31.27

GlobalMacro

26.78

8.88

0.88

12.31

13.08

-1.68

27.36

10.03

-0.03

6.44

MultiStrategy

24.75

4.63

0.85

8.07

9.96

1.26

19.74

5.45

-1.76

9.06

All

19.67

5.89

0.77

9.33

9.78

-1.98

23.10

7.70

-0.22

5.38

43

.

Table6:RegressionResultsofHedgeFundStyleIndexReturnsonFungandHsieh7-factorBenchmark.ThistablereportsFungand

Hsieh7factorsalpha,betacoe¢cientsandtheirNewey-Westt-statistics(inparentheses)ofhedgefund

indexreturnsbyinvestment

objective.TheestimatesarecomputedbyusingmonthlyhedgefundstyleindexreturnsreportedintheDowJonesCreditSuisse,Fung

andHsieh(2004)riskfactorsreportedinDavidA.Hsieh�shedgefund

datalibrary.Bondfactorsareduration

adjusted,thebond

durationsdataarefrom

Datastream.AlldataarebetweenJanuary1994andDecember2010.

Hedgefundindex

Alpha

PTFSBD

PTFSFX

PTFSCOM

SNPMRF

SCMLC

BD10RET

BAAMTSY

Adj.R2

(pct/ann.)

CTA

4.05

0.03

0.04

0.04

0.01

0.04

0.24

0.01

15.43

(1.63)

(1.92)

(2.67)

(2.52)

(0.14)

(0.66)

(1.97)

(0.08)


2.12

-0.02

0.00

0.00

0.04

0.01

0.08

0.60

44.45

(1.06)

(-2.07)

(-0.88)

(-0.43)

(1.39)

(0.40)

(1.52)

(5.54)

EmergingMarkets

1.26

-0.04

-0.01

0.01

0.42

0.24

0.07

0.35

35.38

(0.32)

(-1.42)

(-0.59)

(0.73)

(6.60)

(3.43)

(0.68)

(3.81)

EquityLong/Short

3.50

-0.02

0.01

0.01

0.39

0.32

0.14

0.09

58.24

(1.94)

(-1.55)

(1.04)

(1.02)

(8.16)

(3.63)

(2.43)

(1.51)

EquityMarketNeutral

1.68

-0.03

0.02

0.02

0.14

0.02

-0.31

0.12

15.03

(0.66)

(-1.10)

(1.69)

(1.52)

(2.68)

(0.52)

(-1.15)

(1.56)

EquityShortBias

-1.49

0.00

0.00

-0.02

-0.86

-0.50

0.06

0.24

69.62

(-0.52)

(0.27)

(-0.40)

(-1.02)

(-10.14)

(-8.89)

(0.79)

(1.53)

EventDriven

4.57

-0.03

0.00

0.01

0.18

0.10

-0.03

0.23

54.93

(3.21)

(-2.07)

(1.11)

(1.03)

(7.17)

(3.86)

(-0.65)

(5.39)


-0.10

-0.02

-0.01

0.01

0.02

0.00

0.10

0.51

45.02

(-0.06)

(-2.52)

(-1.72)

(0.71)

(0.57)

(0.14)

(2.07)

(4.53)

GlobalMacro

6.18

-0.03

0.01

0.02

0.13

0.04

0.34

0.26

12.58

(2.53)

(-1.77)

(0.94)

(0.90)

(1.78)

(0.50)

(4.26)

(2.41)

MultiStrategy

3.33

-0.01

0.01

0.00

0.05

0.02

-0.03

0.37

32.32

(2.42)

(-1.05)

(1.08)

(0.52)

(1.92)

(0.83)

(-0.53)

(5.21)

All

3.10

-0.03

0.01

0.02

0.23

0.14

0.14

0.24

43.78

(2.11)

(-2.50)

(1.48)

(1.51)

(5.72)

(2.33)

(2.16)

(4.56)

44

.

Table7:SummaryStatisticsforProxiesofBenchmarkAssets.Thistabledisplays,byinvestmentobjective,benchmarkassetreturns

maximum

drawdown,riskpremiumoverriskfreerate,SharpeRatio,mean,median,1stand3rdquartiles,standarddeviation,skewness

andkurtosis.ThebenchmarkassetforeachinvestmentobjectiveishedgefundstyleindexadjustedforpotentialFungandHsiehalpha.

Thestatisticsarecomputedbyusingmonthlybenchmarkassetreturnswhicharecomputedfrom

hedgefundstyleindexreturnsreported

intheDow

JonesCreditSuisseandFungandHsieh7riskfactorsreportedinDavidA.Hsieh�shedgefunddatalibrary.Bondfactors

aredurationadjusted,thebonddurationsdataarefrom

Datastream.AlldataarebetweenJanuary1994andDecember2010.

Benchmarkasset

Max.

Draw-

down

Risk

Pre-

mium

Sharpe

Ratio

Mean

Median

1st

Quartile

3rd

Quartile

Std

(Ann.)

Skewness

Kurtosis

(%)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

(Ann.)

CTA

12.60

0.23

0.05

3.67

1.75

-7.18

12.14

4.42

0.68

3.75


18.29

1.83

0.41

5.26

5.92

-0.08

11.45

4.42

-1.71

16.61

EmergingMarkets

31.24

3.45

0.39

6.88

12.88

-9.18

25.17

8.74

-1.15

7.00

EquityLong/Short

20.97

2.88

0.40

6.31

9.17

-11.18

22.90

7.28

-0.67

4.22

EquityMarketNeutral

7.44

0.80

0.34

4.24

5.23

-0.42

10.02

2.34

-0.76

3.96

EquityShortBias

47.17

-5.25

-0.35

-1.81

-9.67

-37.33

32.11

14.99

0.70

4.10

EventDriven

16.23

2.23

0.48

5.67

8.34

-1.97

15.12

4.63

-1.44

8.29


14.27

1.88

0.51

5.31

6.20

0.78

11.00

3.65

-1.98

19.61

GlobalMacro

5.71

1.56

0.56

5.00

5.79

-0.57

10.93

2.76

-0.84

8.37

MultiStrategy

10.48

0.93

0.35

4.36

4.88

1.71

7.94

2.62

-1.99

19.23

All

17.66

2.96

0.55

6.40

9.05

-3.03

18.41

5.34

-1.39

8.14

45

Table 8: Put Option Strike Estimation. This tables displays, by investment objective, the estimationof put option strike price as a fraction of high water mark, K and their t-statistics in parentheses. Theestimates are computed using montly net-of-fee returns of live and dead hedge funds, by investmentobjectives, reported in the BarclayHedge database between January 1994 and December 2010.

Investment Objectives K

CTA 0.8 (3.81)Convertible Arbitrage 0.6 (6.51)

Emerging Markets 0.6 (0.61)Equity Long/Short 0.8 (0.51)

Equity Market Neutral 0.6 (11.56)Equity Short Bias 0.8 (2.74)

Event Driven 0.8 (3.79)Fixed Income Arbitrage 0.6 (6.02)

Global Macro 0.8 (3.04)Multi Strategy 0.8 (5.86)

Other 0.8 (2.33)

46

.

Table9:In-sampleSkillEstimates.Thistabledisplays,byinvestmentobjective,in-sampleestimationoftrueskill,�A,traditionalOLS

alpha,�OLS,andstructuralalpha,�t;OLS,impliedfrom

theoreticalrestrictions.Eachcolumnreportsthecross-sectionalmedianofthe

estimatesaswellastheir�rstandthirdquartilesinparentheses.Sincethestructuralalphaestimatevariesovertime,wereportthe

meanandstandarddeviationofthemeasureinthefourthand�fthcolumn,respectively.Theestimatesarecomputedusingmontly

net-of-feereturnsofliveanddeadhedgefundsreportedintheBarclayHedgedatabasebetweenJanuary1994andDecember2010.


TrueSkill

OLSAlpha(pct/ann.)

MeanStructuralAlpha(pct/ann.)

StdStructuralAlpha(pct/ann.)

�A

�OLS

Mean(�t;OLS)

Std(�t;OLS)

CTA(1655)

0.55(0.33,0.93)

5.58(1.48,11.60)

3.14(1.57,6.53)

2.16(0.93,4.24)

ConvertibleArbitrage(155)

0.00(0.00,0.26)

3.19(1.03,5.60)

0.00(0.00,4.58)

0.00(0.00,0.56)


0.70(0.37,1.01)

6.09(1.60,12.07)

3.29(1.03,7.46)

3.96(1.18,7.07)

EquityLong/Short(807)

0.44(0.31,0.67)

4.34(0.82,8.44)

1.23(0.59,2.75)

1.24(0.49,2.54)

EquityMarketNeutral(154)

0.12(0.00,0.26)

2.00(-0.99,4.53)

1.06(0.00,1.86)

0.40(0.00,1.08)

EquityShortBias(34)

0.47(0.25,1.15)

3.20(0.02,9.18)

6.90(2.95,15.77)

2.82(1.46,7.61)

EventDriven(211)

0.41(0.30,0.59)

4.11(0.58,8.34)

1.11(0.45,2.20)

1.31(0.12,2.86)

FixedIncomeArbitrage(93)

0.17(0.10,0.33)

1.63(-2.08,5.26)

0.65(0.27,1.31)

0.51(0.18,1.42)

GlobalMacro(185)

0.61(0.46,0.80)

4.33(1.13,8.83)

1.89(1.14,2.82)

1.56(0.75,2.55)

MultiStrategy(196)

0.41(0.04,0.74)

5.99(3.39,9.20)

6.37(0.66,11.40)

2.22(0.20,4.82)

Others(826)

0.53(0.34,0.88)

5.15(0.74,10.44)

1.58(0.51,4.87)

1.19(0.12,4.76)

AllFunds(4828)

0.49(0.29,0.82)

4.85(1.09,9.87)

2.08(0.75,5.24)

1.62(0.44,3.96)

47

Table 10: Evidence on Risk Shifting - Standardized betas and residual volatility. This table re-ports, for all investment categories, evidence on risk shifting using standardized betas (Panel A)and standardized residual volatility (Panel B). We report regressions�coe¢ cients, their p-values (inparentheses) and adjusted R squared for all hedge funds, low skill funds (the bottom quartile of hedgefunds ranked by true skill measure) and high skill funds (the top quartile of hedge funds ranked bytrue skill measure). Betas and residual volatilities are computed using 24-months rolling windowregressions and its alpha-adjusted hedge fund style index. The betas and residual volatilities arethen standardized using fund speci�c unconditional means and standard deviations. The hedge funddata are monthly net-of-fee returns reported in BarclayHedge database between January 1994 andDecember 2010

Ri;t= u+ c1�1fDist2HWMi;t>0g�Dist2HWM i;t

+ c2�1fDist2HWMi;t<0g�Dist2HWM i;t+"i;t;

Intercept c1 c2 Adj:R2

Panel A: Ri;t � Standardized Beta

All funds 0.86 -0.18 0.24 0.014(0.00) (0.00) (0.01)

Low skill funds 0.90 -0.30 0.43 0.003(0.00) (0.02) (0.02)

High skill funds 0.89 -0.11 0.24 0.003(0.00) (0.20) (0.03)

Panel B: Ri;t � Standardized Residual Volatility

All funds 0.08 -0.01 0.01 0.001(0.00) (0.13) (0.21)

Low skill funds 0.07 -0.04 0.09 0.02(0.00) (0.00) (0.00)

High skill funds 0.13 -0.03 0.03 0.001(0.00) (0.10) (0.25)

48

.

Table11:EvidenceonRiskShifting-Changeinrealizedvolatility.Thistablereports,byinvestmentcategory,regressioncoe¢cientsand

adjustedRsquared.InPanelAandB,wereportevidenceonriskshiftingforfundsthathaveexperiencedatleasta15percentdeviation

ofassetsvaluefrom

theirhighwatermark.InPanelAweusethedi¤erencebetweensubsequent12-monthandprior12-monthrealized

volatilityasthedependentvariable;PanelBreportsresultswhenusingthedi¤erencebetweensubsequent6-monthandprior6-month

realizedvolatilityasdependentvariable.Thehedgefunddataaremonthlynet-of-feereturnsreportedinBarclayHedgedatabasebetween

January1994andDecember2010.

�i;[t;t+T]��i;[t�T;t]=ui+�1�1 fDist2HWMi;t>0g�Dist2HWMi;t

+�2�1 fDist2HWMi;t<0g�Dist2HWMi;t+�3�(AVGVIX[t;t+T]�AVGVIX[t�lag;t])+" i;t;

PanelA:�t+12��t�12

PanelB:�t+6��t�6


Intercept

Beta1

Beta2

Beta3

Adj:R2

Intercept

Beta1

Beta2

Beta3

Adj:R2

AllStrategies

0.009��

-0.063��

0.137��

0.556��

0.188

-0.004��

-0.038��

0.087��

0.528��

0.115

CTA

0.001

-0.096��

0.025��

0.304��

0.104

-0.010��

-0.067��

-0.057��

0.303��

0.058


0.009��

-0.067��

0.092��

1.116��

0.404

-0.001��

-0.078��

0.124��

0.997��

0.319

EmergingMarkets

0.006��

-0.015��

0.127��

0.913��

0.281

-0.012��

0.020��

0.098��

0.839��

0.198

EquityLong/Short

0.005��

-0.019��

0.118��

0.408��

0.126

-0.005��

-0.005

0.085��

0.412��

0.088

EquityMarketNeutral

-0.003��

0.056��

0.029��

0.224��

0.181

-0.009��

0.067��

-0.009

0.208��

0.111

EquityShortBias

0.052��

-0.156��

0.196��

0.418��

0.068

0.009

-0.120��

0.028

0.620��

0.040

EventDriven

0.008��

-0.101��

0.101��

0.782��

0.327

0.001

-0.108��

0.127��

0.706��

0.266


0.019��

-0.173��

0.339��

0.833��

0.434

0.014��

-0.195��

0.377��

0.758��

0.348

GlobalMacro

0.004��

-0.054��

0.036��

0.457��

0.137

-0.011��

0.027��

-0.054��

0.411��

0.097

MultiStrategy

0.009��

-0.065��

0.167��

0.669��

0.434

0.003��

-0.069��

0.206��

0.604��

0.338

Others

0.014��

-0.040��

0.185��

0.642��

0.284

0.000

-0.016��

0.149��

0.631��

0.159

49

.

Table12:Variationinhedgefundalpha.Thistablesummarizestheresultsofpanelregressionsofreduced-form

OLSalpha(�OLS;i;t)on

structuralskill(�i;t)ande¤ectiveriskaversion( i;t):

�OLS;i;t=const+b 1��i;t+b 2�e i;t+

" i;t;

andinlogs,withlog(1+�OLS;i;t)=const+b 1�log(�i;t)+b 2�log(e i;t)

+" i;t.Theright-handsidevariablesareobtainedestimatingthe

structuralmodelusing24-monthrollingwindow.Thehedgefunddataismonthlynet-of-feereturnsreportedinBarclayHedgedatabase

betweenJanuary1994andDecember2010.

constant

�i;t

e i;tAUMi;tlog(�i;t)

log(e i;t)

log(AUMi;t)

Adj:R2

(1)

�i;t

0.03

0.09

5.11%

(11.73)(11.43)

(2)

�i;t

0.04

0.12

-0.01

7.11%

(22.63)(13.00)(-13.94)

(3)

�i;t

0.03

0.06

-0.01

0.00

7.91%

(14.52)

(8.81)

(-13.57)

(-2.62)

(4)log(1+�i;t)

0.05

0.00

0.38%

(42.04)

(11.33)

(5)log(1+�i;t)

0.09

0.02

-0.01

1.72%

(21.55)

(11.12)

(-10.53)

(6)log(1+�i;t)

0.13

0.01

-0.01

-0.01

3.70%

(11.67)

(9.13)

(-8.70)

(-7.44)

50

Table 13: Performance persistence tests - Ranking funds on OLS alphas. This table displays out-of-sample performances of decile portfolios ranked by OLS alphas. The hedge funds are sorted everyyear into deciles on the 1st January (from 2001 until 2010) based on their OLS alpha obtained usingmonthly individual net-of-fee returns for the 36-month rolling window preceding every 1st January.The portfolios are equally weighted on monthly basis, with weights which are readjusted if a funddies. Decile 1 comprises funds with the highest OLS alphas, while decile 10 comprises the lowest.The last row represents spread returns between 1st and 10th decile portfolios, Decile 1 - Decile 10.Column two reports Fung and Hsieh (2004) OLS alphas based on out-of-sample portfolios returnsbetween January 2001 and December 2010. Column three and four report t-statistics and p-valuesof the Fung and Hsieh (2004) OLS alphas. Column �ve reports the adjusted R2 (Adj R2) of theOLS regression of out-of-sample portfolio returns on Fung and Hseih 7 factors. The rest of thecolumns report annualised percentage mean returns (Mean Ret), the growth of 1 dollar investment($1 growth), the annualised information ratio (IR), tracking error (TE), and Sharpe Ratio (SR) ofout-of-sample portfolios returns.

Portfolio FH Alpha t-stat p-value Adj R2 Mean Ret $1 growth IR TE SR(pct/ann.) (pct/ann.)

Decile 1 6.28 2.13 0.04 40.37 11.25 2.90 0.78 8.00 0.85Decile 2 3.14 1.77 0.08 40.43 7.03 1.96 0.58 5.45 0.66Decile 3 3.36 2.21 0.03 33.04 7.44 2.07 0.77 4.36 0.95Decile 4 3.46 3.18 0.00 40.37 6.72 1.93 0.92 3.77 0.90Decile 5 3.38 2.86 0.01 42.46 7.15 2.02 1.01 3.34 1.09Decile 6 2.79 3.00 0.00 49.97 6.30 1.86 0.94 2.98 0.95Decile 7 2.40 2.58 0.01 47.21 6.01 1.81 0.85 2.81 0.95Decile 8 2.79 3.54 0.00 47.55 6.13 1.83 1.05 2.65 1.05Decile 9 3.23 2.82 0.01 52.86 6.69 1.93 1.03 3.14 0.96Decile 10 4.43 2.71 0.01 56.30 8.05 2.18 1.00 4.44 0.85

Spread (Decile1-10) -0.46 -0.14 0.89 12.83 3.20 1.33 -0.06 7.41 0.12

51

Table 14: Performance persistence tests - Ranking funds on True skill measure. This table displaysout-of-sample performances of decile portfolios ranked by true skill measure, �A, relative to empiricalproxies for their benchmark index. The hedge funds are sorted every year into deciles on the 1stJanuary ( from 2001 until 2010 ) based on the true skill measure, �A, obtained using monthlynet-of-fee returns in the 36-month rolling window preceding every 1st January. The portfolios arethen equally weighted on monthly basis, with the weights beng readjusted if a fund dies. Decile1 comprises funds with the highest true skill measure, while decile 10 comprises the lowest. Thelast row represents the spread returns between 1st and 10th decile portfolios, Decile 1 - Decile 10.Column two reports Fung and Hsieh (2004) OLS alphas based on out-of-sample portfolios returnsbetween January 2001 and December 2010. Column three and four report t-statistics and p-valuesof the Fung and Hsieh (2004) OLS alphas. Column �ve reports the adjusted R2 (Adj R2) of theOLS regression of out-of-sample portfolio returns on Fung and Hseih 7 factors. The rest of thecolumns report annualised percentage mean returns (Mean Ret), the growth of 1 dollar investment($1 growth), the annualised information ratio (IR), tracking error (TE), and Sharpe Ratio (SR) ofout-of-sample portfolios returns.

Portfolio Alpha t-stat p-value Adj R2 Mean Ret $1 growth IR TE SR(pct/ann.) (pct/ann.)

Decile 1 11.44 3.55 0.00 41.73 15.50 4.32 1.22 9.41 1.05Decile 2 5.58 3.15 0.00 52.14 9.36 2.45 0.98 5.69 0.85Decile 3 5.09 2.89 0.00 42.92 9.34 2.47 0.95 5.38 0.98Decile 4 2.25 1.36 0.18 42.24 6.85 1.95 0.51 4.37 0.78Decile 5 2.87 2.35 0.02 43.72 6.56 1.90 0.76 3.79 0.83Decile 6 2.21 1.89 0.06 41.86 5.96 1.79 0.64 3.43 0.81Decile 7 2.35 2.50 0.01 47.30 5.76 1.76 0.86 2.75 0.91Decile 8 1.45 1.63 0.11 48.58 4.91 1.62 0.56 2.60 0.72Decile 9 1.58 2.32 0.02 27.06 4.65 1.58 0.73 2.16 0.91Decile 10 0.43 0.38 0.71 39.20 3.87 1.46 0.19 2.25 0.54

Spread (Decile1-10) 8.70 2.43 0.02 35.29 11.64 2.98 0.94 9.26 0.80

52

.

Table15:Performancepersistencetests-Conditionaloninvestmentobjectivesandskillmeasures.Thistabledisplays,byinvestment

objective,out-of-sampleFungandHsieh(2004)OLSalphasofdecileportfoliosrankedbyOLSalphaandtrueskillmeasure,�A,relative

totheirempiricalbenchmark.Thehedgefundsaresortedeveryyearintodecilesonthe1stJanuary(from

2001until2010)basedon

theOLSalphaandtrueskillmeasure,�A,obtainedusingmonthlynet-of-feereturnsinthe36-monthrollingwindowprecedingevery

1stJanuary.Theportfoliosareequallyweightedonmonthlybasis,withtheweightsbeingreadjustedifafunddies.Decile1comprises

fundswiththehighesttrueskillmeasure,whiledecile10comprisesthelowest.Thelastcolumnreportsthedi¤erencebetweenFungand

Hsieh(2004)OLSalphasofDecile1andDecile10portfolio.


Portfolio

Decile1

23

45

67

89

Decile10

Decile1-10

CTA

OLSalpha

10.71

7.48

4.84

3.37

4.51

3.39

3.73

4.14

2.91

1.41

9.31

Trueskill

12.52

5.50

6.63

4.88

5.13

4.37

2.62

3.11

0.06

1.55

10.96


OLSalpha

-2.39

8.83

4.47

5.56

1.81

-1.67

1.15

2.91

5.43

-2.72

0.33

Trueskill

3.97

5.63

3.65

2.35

0.11

1.62

3.96

3.53

-0.54

1.11

2.86

EmergingMarkets

OLSalpha

11.65

11.13

8.96

6.39

4.16

5.39

7.18

8.69

8.95

10.75

0.89

Trueskill

17.51

12.28

5.77

12.78

8.39

4.41

4.83

6.53

3.26

6.58

10.93

Long/ShortEquity

OLSalpha

2.49

-0.63

2.79

2.30

1.12

2.11

1.85

2.91

4.19

3.97

-1.48

Trueskill

6.09

1.68

3.55

1.86

1.80

2.08

0.89

2.49

0.75

1.66

4.43

EquityMarketNeutral

OLSalpha

1.26

3.69

4.74

-1.10

1.04

-1.01

-0.82

1.43

0.57

-0.29

1.54

Trueskill

2.17

1.32

-0.82

0.43

2.19

-0.60

0.11

-0.68

2.59

1.21

0.97

EquityShortBias

OLSalpha

-1.26

-7.51

2.02

-1.56

3.05

4.17

-2.00

-2.74

-5.16

-1.71

0.44

Trueskill

0.38

5.34

6.36

4.72

-0.23

3.36

-9.09

-0.94

-0.83

-6.92

7.30

53

.Table15(continued)


Portfolio/Decile

Decile1

23

45

67

89

Decile10

Decile1-10

EventDriven

OLSalpha

9.41

1.98

3.97

1.24

2.86

3.17

0.84

3.48

4.56

3.99

5.42

Trueskill

8.76

6.39

2.93

5.55

2.44

2.37

-2.06

2.56

1.30

5.62

3.14


OLSalpha

1.54

-1.95

-3.17

-0.50

2.64

0.91

-0.10

0.62

-1.91

8.52

-6.98

Trueskill

12.87

-3.21

3.09

0.79

2.81

0.67

-6.93

-1.15

-6.08

4.13

8.74

GlobalMacro

OLSalpha

3.70

8.87

3.98

3.87

3.44

0.84

7.12

5.68

2.61

9.06

-5.37

Trueskill

17.34

9.52

2.02

1.87

7.58

2.03

1.37

1.19

5.24

-0.25

17.59

Multi-Strategy

OLSalpha

7.27

2.02

5.32

5.31

5.71

3.08

0.54

3.85

2.21

1.04

6.23

Trueskill

5.78

2.59

9.32

4.98

3.64

1.87

3.77

1.82

2.27

1.63

4.15

Other(HedgeFundIndex)

OLSalpha

4.16

2.44

2.67

-0.63

2.63

-0.81

1.15

-0.92

4.58

2.77

1.39

Trueskill

3.75

3.03

2.79

-0.86

2.73

4.06

2.03

-1.77

0.28

1.41

2.34

All

OLSalpha

6.28

3.14

3.36

3.46

3.38

2.79

2.40

2.79

3.23

4.43

1.85

Trueskill

11.44

5.58

5.09

2.25

2.87

2.21

2.35

1.45

1.58

0.43

11.01

54

Figure 1: Payo¤ Function. This �gure displays the payo¤ to a hedge fund manager plotted againstterminal fund value. The payo¤ is derived from performance fee, management fee and concern aboutshort put positions. The underlying parameters are as follows p = 0:2;m = 0:02 and c = 0:02.

0 1 2 K 4 HWM 6 7 8 9 100.2

0

0.2

0.4

0.6

0.8

1

1.2

Terminal fund value

Pay

off

b

55

Figure 2: Concavi�ed Utility Function. This �gure displays the concavi�ed utility function at di¤er-ent levels of terminal fund value. The solid line is the original utility function which is superimposedwith a dotted line between XK and XHWM :

K HWM

Terminal portfolio value

Util

ity

b XK XHWM

56

Figure 3: Optimal Allocation. This �gure displays the optimal allocation to the risky asset againstcurrent fund value. The manager considers an incentive fee, management fee and the presenceof put options when making her investment decision. The underlying parameters are as follows:p = 0:2;m = 0:02; c = 0:02; = 1:5; � = 0:4; r = 0 and t = 0; where p denotes the performance fee;mdenotes the management fee; c denotes the concern level on short put option positions; denotes thelevel of risk aversion; � denotes the Sharpe Ratio and r denotes the risk free rate:

2 K 4 HWM 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Current fund value

Opt

imal

allo

catio

n to

ris

ky a

sset

p = 0.2 m = 0.02 c= 0.02

Merton constant

57

Figure 4: Sensitity Anlaysis of Optimal Allocation. This �gure displays the optimal allocation to arisky asset against current fund value. The dashed line represents the decision made by a managerwho considers only an incentive fee, while the solid line represents a manager who considers bothincentive fee and management fee when making her investment decision. The underlying parametersare as follows: p = 0:2;m = [0; 0:02]; c = 0; = 1:5; � = 0:4; r = 0 and t = 0; where p denotes the per-formance fee;m denotes the management fee; c denotes the concern level on short put oppositions; denotes level of the risk aversion; � denotes the Sharpe Ratio and r denotes the risk free rate:

2 K 4 HWM 6 7 8 9 100

0.5

1

1.5

Merton constant

2.5

3

3.5

4

4.5

5

Current fund value

Opt

imal

allo

catio

n to

risk

y as

set

alpha = 0.2, m=0, c=0alpha = 0.2, m=0.02, c=0

58

Figure 5: Scatter Plot of Fung-Hsieh Alpha. This �gure displays the scatter plot of OLS Fung andHsieh alpha on the y-axis and the ratio of current fund value to fund high water mark on the x-axis.Each dot corresponds to the estimated alpha of a fund over 36-month rolling estimation window.The hedge fund data is monthly net-of-fee returns of live and dead hedge funds reported in theBarclayHedge database. We exclude funds with less than 36 monthly observations from our funduniverse.

59

Figure 6: Di¤erence between Reduced Form and True Alpha. This �gure displays the di¤erencebetween typical OLS alpha and true alpha against current fund value. The underlying parametersare as follows: �� = 0:05; � = 0:1; c = 0:02; = 1:5; � = 0:5; and r = 0:05; where �� denotes truealpha, � denotes volatility of alpha generating process; c denotes concern level about short put optionpositions; denotes level of risk aversion; � denotes Sharpe Ratio and r denotes the risk free rate:

0.4 K 0.8 1 1.2 1.4 1.6 1.8 20.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Fund Value

αr

α*

60

Figure 7: Risk Taking Plots - Sensitivity with respect to structural skill. Panel A displays the exactsolution of 1

tas function of current fund value (Xt) normalised by current level of high water mark

(HWMt) at di¤erent value of �. Panel B displays the approximated solution of the plots. Otherparameters are calibrated to the following values: K = 0:6 �HWMt; = 1:5; � = 20%;m = 2%; c =2%:

0.4 K/HWM 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Current Fund Value/Current HWM

1/γ t

λ = 0.2λ = 0.4λ = 0.6λ = 0.8

(a) Panel A: Exact solution plots

0.4 K/HWM 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5


1/γ t

λ =0.2λ =0.4λ=0.6λ =0.8

(b) Panel B: Approximated solution plots

61

Figure 8: Risk Taking Plots - Sensitivity with respect to put option�s strike. Panel A displays theexact solution of 1

tas function of current fund value (Xt) normalised by current level of high water

mark (HWMt) at di¤erent value of put strike K which is measured as a fraction of current high watermark. Panel B displays the approximated solution of the plots. Other parameters are calibrated tothe following values: � = 0:4; = 1:5; � = 20%;m = 2%; c = 2%:

0.4 K/HWM 0.8 1 1.2 1.4 1.6

0.5

1

1.5

2

2.5


K=0.6HWMK=0.7HWMK=0.8HWM

(a) Panel A: Exact solution plots

0.4 K/HWM 0.8 1 1.2 1.4 1.6

0.5

1

1.5

2

2.5


K=0.6HWMK=0.7HWMK=0.8HWM

(b) Panel B: Approximated solution plots

62

Figure 9: Performance persistence test - Cumulative returns of 1 dollar investment. This �gureplots, for all hedge funds and by investment objectives, cumulative returns of 1 dollar investment intop decile portfolios sorted by true skill measure and typical OLS alpha between January 2001 andDecember 2010.

Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100

1

2

3

4

5 ALL


1

2

3

4

CTA


0.5

1

1.5

2

2.5

3 Convertible Arbitrage


2

4

6

8

10

12

Emerging Markets

OLS alphaTrue skill

63

Figure 10: Performance persistence test - Cumulative returns of 1 dollar (cont) investment. This�gure plots, for all hedge funds and by investment objectives, cumulative returns of 1 dollar invest-ment in top decile portfolios sorted by true skill measure and typical OLS alpha between January2001 and December 2010.

Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100.5

1

1.5

2

2.5

3 Long/Short Equity


0.5

1

1.5

2

2.5 Equity Market Neutral


0.5

1

1.5

2 Equity Short Bias


1

2

3

4

5 Event Driven

OLS alphaTrue Skill

64

Figure 11: Performance persistence test - Cumulative returns of 1 dollar investment (cont). This�gure plots, for all hedge funds and by investment objectives, cumulative returns of 1 dollar invest-ment in top decile portfolios sorted by true skill measure and typical OLS alpha between January2001 and December 2010.


1

2

3

4

Fixed Income Arbitrage


1

2

3

4

5

6 Global Macro


0.5

1

1.5

2

2.5

3

3.5 MultiStrategy


0.5

1

1.5

2

2.5

3 Others

OLS alphaTrue skill

65

Figure 12: Nonconcave Utility Function. This �gure displays a nonconcave utility function. Panel(a) displays the nonconcave utility function with a tangent line which satis�es Condtion 1. Panel(b) displays the nonconcave utility function with a tangent line which violates Condtion 1.

K HWM(b) Condition 1 violated

Util

ity

K HWM(a) Condition 1 satisfied

Util

ity

XK=K XHWM

K< XKXHWM

66

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