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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.
Electronic copy available at: http://ssrn.com/abstract=1785995
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)
1
Electronic copy available at: http://ssrn.com/abstract=1785995
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 non- linear
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
[[ INSERT Figure 2 about Here ]]
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.
10
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)
φt ∥λ∥√T − 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.
[[ INSERT Figure 3 about Here ]]
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
[[ INSERT Figure 4 about Here ]]
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.
[[ INSERT Figure 5 about Here ]]
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.
[[ INSERT Figure 6 about Here ]]
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.
[[ INSERT Table 2 about Here ]]
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.
[[ INSERT Table 3 about Here ]]
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
[[ INSERT Table 4 about Here ]]
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.
[[ INSERT Table 5 about Here ]]
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.
[[ INSERT Table 8 about Here ]]
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.
[[ INSERT Table 9 about Here ]]
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.
[[ INSERT Table 10 about Here ]]
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”.
[[ INSERT Table 11 about Here ]]
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
[[ INSERT Table 12 about Here ]]
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.
[[ INSERT Table 13 about Here ]]
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.
[[ INSERT Table 14 about Here ]]
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.
[[ INSERT Table 15 about Here ]]
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.
[[ INSERT Figure 9 about Here ]]
[[ INSERT Figure 10 about Here ]]
[[ INSERT Figure 11 about Here ]]
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
φt ∥λ∥√T − t
[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)
φt ∥λ∥√T − 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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.
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
.
InvestmentObjectives
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
InvestmentObjectives
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
(pct/ann.) (pct/ann.)
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
(pct/ann.) (pct/ann.)
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.
InvestmentObjectives
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)
InvestmentObjectives
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.
InvestmentObjectives
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)
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)
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)
ConvertibleArbitrage
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)
FixedIncomeArbitrage
-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
ConvertibleArbitrage
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
FixedIncomeArbitrage
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.
InvestmentObjectives
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)
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)
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
InvestmentObjectives
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
ConvertibleArbitrage
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
FixedIncomeArbitrage
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.
InvestmentObjectives
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
ConvertibleArbitrage
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)
InvestmentObjectives
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
FixedIncomeArbitrage
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
Current Fund Value/Current HWM
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
Current Fund Value/Current HWM
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
Current Fund Value/Current HWM
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
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
1
2
3
4
CTA
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
0.5
1
1.5
2
2.5
3 Convertible Arbitrage
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
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
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
0.5
1
1.5
2
2.5 Equity Market Neutral
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
0.5
1
1.5
2 Equity Short Bias
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
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.
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
1
2
3
4
Fixed Income Arbitrage
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
1
2
3
4
5
6 Global Macro
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
0.5
1
1.5
2
2.5
3
3.5 MultiStrategy
Jan 2001 Jul 2003 Jan 2006 Jun 2008 Dec 20100
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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