Optimal Hedge Fund Contract Parameters: Theory and
Empirical Evidence
Karan Puria , Andreas Krause∗,b
a Department
b Department
of Economics, University of Bath, Bath BA2 7AY, Great Britain
of Economics, University of Bath, Bath BA2 7AY, Great Britain
Abstract
We model the evolution of the value of a hedge fund from the perspective of a
hedge fund manager and investor. We use these values to assess the optimal
contract specification between the manager and investor, considering the fee
schedule that the manager receives, the investment characteristics represented
by the distribution of returns that are generated as well as managerial ownership in the fund. We find that our results are consistent with managers having
low bargaining power and low ability to generate excess returns. We then derive predictions for the relationship between contract parameters and the actual
fund performance. These predictions are confirmed empirically using an extensive database.
Keywords: hedge funds, incentive contract, optimal investment strategy, fund
performance
1. Introduction
Hedge funds have become increasingly popular as investment vehicles not
only for high net worth individuals but also for institutional investors such as
pension funds, insurance companies and investment banks. Reaching a high of
over 2 trillion USD assets under management in 2007, this industry has attracted
∗ Corresponding
author
Email addresses: [email protected] (Karan Puri), [email protected] (Andreas Krause)
more interest in recent years by regulators due to their supposed involvement
in the credit crisis 2007/8 and continuing concerns about shadow banks causing
instability to the financial sector. One key feature of hedge funds is that very
few, if any, regulatory constraints exist that limit the scope of their investments
or the way the hedge fund manager is rewarded for managing the fund. Any
such contracts have to be negotiated individually between the investors and the
hedge fund manager.
This paper seeks to explore the optimal contract specification between an
investor and a hedge fund manager. Thus far little research has extensively explored this key feature of hedge funds with most contributions focussing on the
implications of given compensation contracts of hedge fund managers on their
investment behavior. In contrast we seek to find a unifying framework that addresses the investment decision of hedge fund managers and the compensation
they receive in a single step. For those contributions that actually seek to find
the optimal contract specification, they focus mainly on a single aspect of the
contract itself rather than all possible aspects that require negotiation. We aim
at closing this gap in the literature by providing a much more general context
that allows for a large degree of freedom in setting hedge fund contract parameters. This model will then be tested empirically using an extensive database
to establish how far market participants are actually behaving optimally.
The coming section reviews the literature on the contract specification in the
hedge fund industry and how they relate to performance. We will then in section
3 introduce our model and derive its key properties. After that section 4 derives
predictions on the empirical relationship between the contract parameters and
performance and test this empirically. Finally section 5 concludes our findings
and suggests some further work.
2. Characteristics of hedge funds
Hedge funds are a private arrangement between an investor and a hedge
fund manager that invests the money on his behalf. This private arrangement
does include an agreement on the way the money is invested, the investment
2
strategy to be followed, and how the hedge fund manager is compensated for his
efforts. Unlike in mutual funds the considerable size of the investments made by
investors allows many investors to individually negotiate with the hedge fund
manager the terms of conditions of their investment.
Hedge funds have a wide range of investment strategies that can be classified
according to many criteria, often based on the type of investment instruments
used and the way this investment is conducted. Distinguishing hedge funds from
many other forms of investments is their ability to invest into instruments that
are not commonly used in other asset management arrangements such as private banking or mutual funds, e. g. commodities, arbitrage trading strategies,
derivatives markets or property. Hedge funds also make extensive use of leverage, which in is also uncommon in other forms of asset management. McCrary
(2002), Jaeger (2003), Lhabitant (2006), and Jaeger (2008) provide an overview
of the different strategies employed by hedge funds and how they operate. Table
7 shows the main investment strategies we consider in this paper.
Apart from the investment strategy itself, what distinguishes hedge funds
in particular from other forms of investments is the way compensation for the
manager is determined. Most such arrangements in mutual funds and other
forms of asset management charge a fee to the investor that is proportional
to the amount invested, i. e. assets under management, often supplemented
by a fraction of the return generated above a benchmark return. In contrast,
the typical hedge fund contract has, besides a fee comprising of a fraction of
the assets under management, also a performance component that is based
around a highwater mark. This highwater mark is the highest value of the
fund in its lifetime and the hedge fund manager receives a fraction of the value
of the fund that exceeds this value, calculated periodically. It is common to
find a contract stipulating that the hedge fund manager receives 2% of the
assets under management per year and 20% of the returns generated above the
highwater mark, paid and calculated quarterly. Sometimes a minimum return
per calculation period, the hurdle rate, needs to be exceeded to trigger the
payment of this performance-based fee.
3
In addition, and in contrast to other forms of asset management, the hedge
fund manager retains a stake in the fund himself, i. e. he invests himself into
the hedge fund and is thus exposed to his own investments. This is uncommon
in other forms of asset management, like mutual funds, where managers do not
have any such stakes. Finally, investors are required to invest their money for
a given period of time before they can withdraw any money, the lock-in period,
and after that investors have to give notice before being able to withdraw their
money, the redemption period. If investors want to withdraw money without
observing those periods, this might be possible on payment of a penalty.
Most of the literature on hedge funds is concerned with explaining the performance of hedge funds, using the contract specifications as given and then
establishing how these aspects affect their performance, with a special focus
often on specific elements of this contract. In many cases they focus in particular on the risks a hedge fund manager takes, such as Kouwenberg and Ziemba
(2007) finding that his stake in the fund changes his behavior depending on
whether he is making losses or gains. Extending their model to multiple time
period settings Hodder and Jackwerth (2007) and Drechsler (2013) show that
these misalignments vanish. Kritzman (1987), Grinblatt and Titman (1989) and
Carpenter (2000) discuss the incentives to increase risks with performance fees
in a general setting. The impact the inclusion of a highwater mark has on this
incentive is explored in Drechsler (2013), Panageas and Westerfield (2009) and
Drechsler (2014), extended by the inclusion of requiring a hurdle rate in Patrick
and Motaze (2013). Aragon and Qian (2009) investigate the impact of the highwater mark on manager behavior. Similarly Buraschi, Kosowski, and Sritrakul
(2014) investigate the effects on hedge funds performance of performance fees,
early redemption by investors and leverage. Ou-Yang (2003) considers a portfolio optimization problem assuming normally distributed returns and does not
take into account the investment of the fund manager, thus missing two important aspects that distinguish hedge funds from other forms of asset management.
The resulting optimal contract specification does not include a performance fee
component.
4
One line of research in asset management explores the importance of the
performance fee as an incentive for managers to maximize their efforts in managing the hedge fund. Li and Tiwari (2009) show in a mutual fund setting
that performance fees induce higher levels of efforts. This view is confirmed
when including the impact of performance on capital inflows and outflows by
Das and Sundaram (1998) and Stracca (2006) shows that the limited downside
of performance fee components attracts lower ability managers to the industry.
However, these model are mainly applicable to mutual funds and similar asset
management constructs, but not necessarily suitable to modelling hedge funds
as here the manager retains a stake in the fund and different compensation
schemes are employed, potentially altering his incentives considerably.
Goetzmann, Ingersoll, and Ross (2003) and Lan, Wang, and Yang (2013)
investigate the fees generated from the contract and how they depend on various
contract parameters, most importantly finding that managers make the most
profits from fees, in particular management fees, rather than their stake in the
fund itself. Other works investigate the implications of differences in the risk
aversions between managers and investors, e. g. Agarwal, Gomez, and Priestley
(2012). In a reversal of our analysis, Koijen (2014) extracts the ability, risk
aversion and incentives from performance data of mutual funds.
Overall what is missing is a general model of hedge fund behavior such as
Berk and Green (2004) is for mutual funds, although its focus on competition is
different from the setting to be employed by us. It is common to explore individual aspects of a contract and investigate how a hedge fund manager adjusts
his behavior to maximize fee income. Very little attention has been paid to the
characteristics of the optimal hedge fund contract itself along all of its major
aspects simultaneously, including its investment strategy and considerations for
the incentives to investors in agreeing such a contract.
We seek to determine in this paper the optimal contract parameters for a
hedge fund - management and performance fees, managerial ownership, reporting frequencies, investment lengths, and investment strategies. Hence rather
than taking contract parameters as given, we let those parameters all be deter5
mined endogenously.
3. The optimal hedge fund contract
In order to assess what properties a hedge fund contract should have, this
section introduces a model of hedge funds that includes a wide range of parameters governing the fees and investment characteristics of hedge funds. These
parameters are then optimized assuming a risk averse hedge fund manager and
investor and the properties of the resulting optimal hedge fund contract are
discussed.
3.1. The contract parameters
Let us consider a situation in which we have two market participants, one
being a hedge fund manager (”manager”) and the other a potential investor
(”investor”), both negotiating the terms and conditions of an investment into a
hedge fund (”fund”). The manager has the ability to generate a Sharpe ratio
s when making investments, which is common knowledge for the manager and
investor, interpreted here as his ”ability”. While it may seem a strong assumption that the ability of a manager is known with certainty, it has to be noted
that in many cases investors have some idea about their past performance from
managing other hedge funds or their track record while employed within the industry and can thus make inferences about their ability, making this assumption
not too unrealistic. We assume that the manager will perform according to his
ability, thus the manager does not engage in shirking and no decision on effort
by the manager needs to be considered. We thus abstract from the existence
of a principal-agency problem in the management of the fund to allow us to
focus on the contract specification itself. While this might go against common
assumptions in the modelling of such situations, it has to be noted that typically
hedge fund managers are working long hours and it is not perceived to be an
issue that they do not employ sufficient effort.
Those two participants will negotiate the hedge fund contract along a wide
range of specifications that include the characteristics of its investments through
6
a selection of the optimal moments of the distribution of returns it generates and
a choice of the compensation structure to the manager by optimally choosing
the parameters of the performance contract, as well as the optimal investment
of the manager itself into the fund. This contract will be enforced at no costs
and the manager will not be able to deviate from its specification, thus again
we abstract from the possibility of a principal-agency problem and incomplete
contract enforcement. Finally, we assume that investor and manager have to
come to an agreement on the contract, i. e. neither side can walk away from
the negotiations.1
In our analysis we consider a generalization of the contract specification
that is commonly used in actual hedge funds and optimize for the parameters of
such a contract. A typical contract consists of a management fee that charges a
fraction γ of the value of the fund for each time period and a performance fee. In
order to obtain the performance fee we need to determine the high water mark of
the fund, which is the highest value that the fund at time t has ever achieved and
b t = maxi=0,...,t−1 Fi , where Fi denotes the value of the fund at
is defined as H
time i, assuming no capital inflows or capital outflows and hence no redemption
bt as the return required to reach the
period is modelled. We define H = ln H
t
Ft−1
highwater mark. Let us now define a benchmark return Rt = ψHt , which will
be the return required to trigger payment of the performance fee. For ψ = 1 this
constitutes the standard highwater mark widely employed by hedge funds. The
manager will then obtain a fraction δ of the return generated in excess of this
benchmark, i.e. δ max{rt − Rt , 0} where rt denotes the actual return obtained
in time period t. We assume that any fees are paid out periodically from the
fund as cash payments to the manager, who then retains these funds until the
end of the lock-in period under consideration without investing them into any
other assets.
Fees are calculated periodically in fixed time intervals. We assume that the
frequency of these intervals are also to be negotiated between the manager and
1 We
will make clear later that this assumption does not affect the results of our model.
7
investor. The investor is committed to invest his money with the manager for T
time periods (the ”lock-in period”) before the conditions can be re-negotiated or
the money be withdrawn. During this lock-in period fees are calculated every τ
time periods, giving a total of
T
τ
fee payments. We do not allow for any money
to be withdrawn or repaid until the time period T has expired, i.e. we impose a
hard lock-in that also cannot be broken with a redemption penalty. We also do
not allow for the liquidation of the fund if its value reaches a lower boundary,
as some actual hedge funds allow for. For simplicity we do not allow for any
flows of funds, e.g. additional money being invested into the fund.
Furthermore, the manager will retain a fraction λ of the fund and thus be
exposed to the risks of his investments. We assume that the initial amount
invested by the investor is determined exogenously in his portfolio decision and
not affected by the choice of contract. The manager does not face a constraint
on his wealth to obtain the desired investment into the hedge fund. We do
not consider portfolio considerations by investors nor managers for the total
investments they are holding, such considerations are outside of the remit of
this paper.2
With the fees being paid in cash from the fund itself to the manager, any fee
payment reduces the value of the fund3 and using running costs of a fraction c
of the fund value to cover administration and trading costs, we can determine
the value of the fund in time period t easily as
t
Ft = F0
τ
Y
(1 + ri )(1 − c)(1 − γ)(1 − δ max{ri − Ri , 0}),
(1)
i=1
where F0 denotes the combined initial investment of the investor and manager
and c the running costs of the funds that might include administrative costs
2 The outside options of investors can be interpreted to include such considerations in their
value.
3 We also assume that a fee is charged to the fraction of the fund that is owned by the
manager. Although this implies that the manager is paying himself a a fee, it is done in order
to treat all investors into the fund equally. Obviously at the time, managers do not suffer a
loss as the fees is paid to them in cash and its amount equals exactly the amount by which
the value of their share of the fund reduces.
8
and trading costs such as brokerage fees, bank fees or other direct costs of the
fund. As we only need to consider the value of the fund at the time its fees are
determined, we can use a discrete time approximation, where one time period
corresponds to a length of τ . Consequently, rt has to be interpreted as the
return for a time period of that length.
Typically in hedge funds, the manager retains a certain amount of their own
money in the fund. If we denote the fraction of the fund held by the manager
with λ, the value of the investment for the investor at the end of the lock-in
period, i. e. at time T , will simply be
VTI = (1 − λ)FT .
(2)
The value to the manager will consist of the value of his investment as well
as the value of the fees generated, PT . This gives a total value to the manager
of
VTM = λFT + PT .
(3)
The final aspect that a manager and investor need to agree on is the investment strategy. We here assume that this negotiation centres around the moments of the distribution of returns that the manager generates as a proxy for
the negotiation on the investment strategy. Empirically we know that different
strategies exhibit different return distributions by the very nature of the investments the fund makes. As an investment strategy cannot directly be modelled,
the negotiation of the distribution of returns is the best proxy for this aspect of
the contract. If we let the negotiation centre on the first four moments of the
distribution, we cannot freely choose the mean as we assumed above that the
manager’s ability is determined by his ability to generate a certain Sharpe ratio,
hence for a given volatility σ, we have the mean µ of the return distribution
given by
µ = rf + sσ,
9
(4)
where rf denotes the risk-free rate and s the Sharpe ratio the manager is able
to produce. Thus we assume the negotiation centres on volatility σ, skewness
κ3 and kurtosis κ4 . In our model we assume that returns follow a Generalized Hyperbolic distribution and through our assumption of the contract being
completely enforceable, it will not vary with the realization of returns.4 This
distribution, introduced in Barndorff-Nielson (1977), has the benefit of allowing
for a wide range of properties, including thin and fat tails as well as skewed distributions. While this distribution is not commonly used in finance, it has been
used on occasions where fat-tailed distributions were required, e. g. in Eberlein
(2001) and Fajardo and Farais (2010). A Generalized Hyperbolic distribution
is characterized by five parameters, a shape parameter ι, an asymmetry parameter %, a scale parameter ς, a tail parameter ϕ, and a location parameter
ζ. Given that we assumed that expected returns and variance are linked via
the Sharpe ratio that indicated the manager’s ability, we can neglect the location parameter as a free parameter and may freely choose the remaining four
parameters. A more detailed discussion of the properties of this distribution
can be found in Emberlin and Hammerstein (2003) and Fischer (2011). Using
a distribution that allows a wide range of shapes allows us to capture the full
range of actual return distributions observed when analysing hedge funds. We
do use these different distributions of the returns as an approximation for an
agreement between the investor and manager on an investment strategy. Each
specific investment strategy will have its own characteristic distribution, e. g.
high kurtosis and skewness, that is modelled here and it is assumed for simplicity that investment strategies can be thus characterized.5 Thus using the
4 We do not consider the possibility that managers vary their distributions in response to
realized returns, e.g. increase risks after losses. However, as we only consider the negotiations
for the length of the investment horizon, we might interpret those as the ”average” distribution and could allow for short-term deviations from this agreed distribution as long as the
distribution over the investment length is observed. This simplification allows us to abstract
from the details of the investment process and focus on the main aspects of the contract
specifications.
5 There is evidence in our database that variations in the variance, skewness and kurtosis
of funds following the same investment strategy are lower than those following different investment strategies. However, there is still significant heterogeneity between funds in each
10
moments of the return distribution is the closest approximation to modelling
the investment strategy.
Thus overall a hedge fund contract consists of of a set of parameters Θ =
(γ, δ, ψ, τ, T, λ, σ, ι, %, ς, ϕ), which needs to be negotiated between the investor
and manager. The next section will detail how the optimal contract specification
is determined.
3.2. Deriving the contract curve
We assume that both investors and managers have a CRRA utility function
with risk aversions zI and zM , respectively, and they seek to maximize their
expected utilities over final wealth, adjusted for the time horizon by normalizing
i T1
V
to a single time period: VbTi = VTi
for i ∈ {I, M }, where V0I = λF0 and
0
V0M = (1 − λ)F0 . Conducting this optimization simultaneously for investor
and manager will give rise to a multi-objective optimization problem. In line
with standard microeconomic theory the joint optimal solution will be all those
where the corresponding marginal rates of substitutions are equal for all contract
parameters Θi , i = 1, . . . , 11:
∂E[U (VbTM )]
∂E[U (VbTI )]
=
.
∂Θi
∂Θi
(5)
Solving this equation will lead not to a single optimal contract but to a a
set of solutions that are Pareto-optimal, commonly called a contract curve. The
actual point on this contract curve that the investor and manager select will be
the result of a bargaining process. In our model we assume that an agreement
on a contract specification has to be reached, hence no outside options exist
that could limit the choices available in the negotiation process. Adding such
outside options would only reduce the contract curve on its margins, thus the
possible contracts become a subset of those contracts presented here. We are
here not interested in the aspects of this bargaining process, but will instead
derive the entire contract curve, hence ignoring outside options will only provide
investment strategy, reflecting the significantly different investments that are conducted.
11
us with a larger set of possible contracts. Thereby we will interpret a situation
in which the investor achieves his highest expected utility while the manager has
the lowest expected utility as the investor having all the bargaining power and
in the opposite case the manager having all the bargaining power. Any point in
between will have partial bargaining power on each side; in the analysis below
we will order all contracts accordingly using a 100 point scale for the relative
position of the investor and manager in their bargaining power.6
Although the set-up of the final value seems relatively straightforward, conducting the optimization analytically is very difficult as the highwater mark
does not only introduce an element of non-linearity but also path-dependence
into the final value of the wealth of the investor and manager. Furthermore,
we cannot simply separate the decision of choosing the distribution of returns
and then, taking this decision as given, optimize the contract parameters as a
rule for the optimal sharing of returns. Firstly, the way returns are distributed
will affect how these returns will be shared, e. g. heavy tailed distributions or
skewed distributions might be shared differently to those that are thin tailed.
Secondly, the way returns are shared will affect the optimal distribution of returns, e. g. if performance fees are more prominent, this might induce more (or
less) risky distributions. Hence return distributions and the sharing of these
returns have to be determined simultaneously. This complication lead us to
simulate the expected utilities and in that way determine the Pareto-optimal
points to construct the contract curve. For a given level of expected utility of
the investor we sought the maximum level of utility achievable for the manager
and record the corresponding contract specification. This is repeated for the
entire range of utilities.
In order to obtain the contract curve we consider 10,000 random contract
specifications and for each generate 500 time series to calculate the final value
of the fund to the investor and manager and thus determine their expected
6 Please note that this scale is not equivalent to a proportional division of surplus between
investor and manager but merely used to indicate an order.
12
Variable Definition
Ranges
Fee parameters
λ
Managerial equity ownership
[0 1]
γ
Management fee
[0 1]
δ
Performance fee
[0 1]
ψ
High-water mark
[0 2]
τ
Frequency of reporting (years)
[0.00001 1]
T
Minimum investment length (years) [0.001 5]
Return distribution for annual returns
ι
Shape parameter
[0 10]
%
Asymmetry parameter
[0 10]
ς
Scale parameter
[0 10]
ϕ
Tail parameter
[-2 2]
Other variables (fixed at the level indicated)
rf
Risk free rate p.a.
0.05
c
Fund costs p.a.
0.01
s
Sharpe ratio
[-0.25 1.25]
zI
Investor risk aversion
1.5
zM
Manager risk aversion
1.5
Table 1: Ranges of contract specifications considered
utilities. We sort these contracts such that only those that are Pareto-optimal
are retained, providing us with a a single contract curve. This procedure is
repeated 1,000 times such that we obtain multiple contract curves whose contract specifications are then averaged to obtain more robust results in line with
the resampling method used in portfolio construction, see Scherer (2002) and
Michaud and Michaud (2008). Using this methodology we select 100 equally
spaced points on each of the 1,000 contract curves and then determine the average for each of the contract parameters across the contract curves for each of
the selected points. We need to select a fixed number of points on the calculated
contract curves as the number of actual points that are Pareto-optimal will be
different for each simulation.
The range of contract specifications we used are shown in table 1. Each of
these ranges covers either the entire possible range or a range far in excess of
that found empirically and in no instance did we observe the contract curve to
contain a solution on the edge of the range. A random contract specification is
13
selected by drawing a random uniformly distributed number over the range of
each parameter, independent across parameters.
In addition, table 1 also shows the other variables we have to select in our
analysis. In order to obtain the expected returns from the Sharpe ratio, we need
to specify the risk free rate, which we set at 5% p.a. We find that changing this
rate does not affect our results in a meaningful way. Similarly we set the costs
of running the fund (to include trading costs as well as other administrative
costs of running the fund) at 1% p.a., the level of which again does not affect
results. The risk aversion of investors and managers are set at 1.5 and we have
checked that lower and higher as well as different risk aversions for investors
and managers do not affect the results we present here as differences are very
small.7 Finally, in our analysis below we will consider Sharpe ratios from -0.25
to 1.25, capturing most of the reasonable range found in the performance of
hedge funds, which we find in our database.
3.3. Equilibrium contract specifications
Conducting the optimization as described above, we can observe how the
different contract parameters are determined with managers of different abilities and bargaining power. All other elements of our model are determined
endogenously as part of the optimization process. Table 2 summarizes the results on the evolution of the optimal contract parameters as we change those
two variables, the detailed results of our simulations are visualized in figure 1.
Most of the results are intuitively obvious. A manager that has higher
bargaining power in the negotiation process will naturally be able to extract
higher fees from the investor, both management and performance fees, as we
can observe in our results. The performance fee is only paid for relatively
large (positive) returns, with no penalty for lower performance, this provides
7 We conducted detailed investigations with risk aversions of 1.5, 3, and 10 in any combination between investor and manager. We found most results to be indistinguishable between
those cases, for obvious reasons only the optimal volatility and kurtosis were found to be
significantly lower for higher risk aversions, but the general properties presented here are
unchanged. We therefore focus our attention on the case of a risk aversion of 1.5.
14
This table shows the partial derivatives of the contract parameters with respect to the bargaining power and Sharpe ratio. ”+” indicates a positive sign of this partial derivative, ”–” a
negative sign and ”o” a slope very close to zero.
Variable
Definition
Fee parameters
λ
Managerial equity ownership
γ
Management fee
δ
Performance fee
ψ
High-water mark
τ
Frequency of reporting (years)
T
Lock-in (years)
Return distribution
σ
Volatility
κ3
Skewness
κ4
Kurtosis
Manager
bargaining power
Sharpe ratio
–
+
+
+
+
–
o
o
–
–
o/o
+
o
–
+
o
–
Table 2: Properties of the optimal contract specifications
incentives for the manager to seek more risky positions, equivalent to higher
volatility. The kurtosis reduces slightly to partly compensate for this higher
risk arising from higher volatility. With such investments, the manager is more
likely to receive large payments from the performance fee, while in the case of
large negative returns no losses will be accrue to the manager. Given the low
exposure of the manager to the investment itself through a low co-investment,
the higher risks have only very limited downside potential for managers. As
he has increasing bargaining power, the manager will reduce his investment
into the fund as this reduces his risk exposure to these risks while at the same
time benefitting from any large gains through the performance fee. In order
to minimize the risk of the fees eroding after a positive return, the period for
calculating fees is reduced if the manager has bargaining power. The impact on
the high-water mark benchmark is very small but increasing in the bargaining
power, this is to offset some of the higher fees such that the fee income is
not too high for the investor to agree the higher risks taken. Similarly the
manager seeks a shorter lock-in in order to reduce his risk from longer-term
investments as with shorter time periods the total risk is reduced, but also the
chances of high returns, which for risk averse managers is optimal. The impact
15
of the bargaining power on skewness is negligible and the distribution remains
symmetric for all practical purposes. This is because an asymmetry would result
in higher volatility and kurtosis that are already captured directly.
From the perspective of an investor, as he gains bargaining power, these
aspects reverse. Obviously such an investor would negotiate lower fees and in
order to reduce the amount of performance fees to be paid he seeks to extend the
time length of reporting periods to ensure that short term high returns do not
lead to the payment of performance fees if they are followed by negative returns
causing him losses. Similarly, requiring the manager to hold larger investments
in the fund will reduce his appetite for higher risks, i. e. volatility and kurtosis,
that the investors seeks to limit given his risk aversion.
Looking at the impact the ability of the manager, measured by his Sharpe
ratio, has on the contract specification we notice firstly that as his ability increases, the performance fee reduces. This is due to the fact that as the result
of his increased ability, the amount charged will nevertheless increase and the
reduced charge partially offsets this increase in the amount paid out, ensuring
that it does not increase beyond bounds. The small reduction in the highwater mark partially offsets this lower performance fee for but cases where the
manager has a very high bargaining power. Furthermore, a higher Sharpe ratio
means that the rewards for taking on additional risk in the form of volatility are
higher and consequently the volatility increases with managerial ability. Low
ability managers will seek to compensate for their low performance by employing an investment strategy with a high kurtosis to maintain a chance of making
large positive returns. The other contract parameters are not affected by the
managerial ability as here the ability has no direct effect on the performance for
either manager or investor.
If we compare the typical contract specifications actually found in hedge
funds with a low management fee of 1.5%, a performance fee of 20% and substantial co-investment by the manager, see Ackermann, McEnally, and Ravenscraft
(1999), Edwards and Caglayan (2001), Agarwal and Naik (2004), and Aragon
and Nanda (2012) for empirical evidence to that effect, we see that such a con16
17
(a) Managerial co-investment
(b) Management fee
(c) Performance fee
Figure 1: Optimal contract specifications
18
(d) Highwater mark
(e) Lock-in period
(f) Reporting period
Figure 1: Optimal contract specifications (ctd.)
19
(g) Volatility
(h) Skewness
(i) Kurtosis
Figure 1: Optimal contract specifications (ctd.)
tract is consistent with a situation in which the manager has a low bargaining
power. This will also result in relatively long lock-ins and long fee calculation
periods, consistent with actual funds. At the same time the volatility chosen
will be reasonably low unless the manager has a high ability; similarly the kurtosis will be close to 3, indicating no fat tails. Only if the ability of managers is
low will a moderately higher kurtosis be chosen as can be found in many funds.
Similarly we find that with an increasing ability of the fund manager the
volatility chosen increases as the marginal benefits in terms of expected returns
are increasing. As the risks from increased volatility increase, this is compensated for by reducing the risk arising from kurtosis. As the higher expected
returns will increase the performance fees paid, their fraction will be reduced to
offset this effect partially. There is no effect on the management fee component,
lock-in provision, fee calculation period or co-investment as these are unaffected
by the ability of the manager.
Another observation we can make from our model is that management fees
account for the majority of fee income for the manager, and not the performance
fee, as shown in figure 2 and consistent with results found in Lan, Wang, and
Yang (2013). This is particularly dominant when managers have a low ability
and high bargaining power. In this case it is intuitively understandable that
the manager will negotiate a contract that will guarantee him a steady income
stream rather than being exposed to the risks of accepting a higher performance component. As his bargaining position deteriorates he has to accept a
larger performance component and as his ability increases the better prospects
of receiving performance fees increases their fraction in the optimal contract.
However, a low bargaining position combined with high ability increases the
fraction of fees earned because the investor can insist on a contract that gives
the manager a smaller part of the high performance while allowing them to
retain more of these benefits. Overall we observe that management fees are the
main source of income for managers.
Also in line with actual hedge funds, the fee income forms the majority
of the final value to managers and is thus their main concern. Only when
20
21
(a) Fees as fraction of investment performance
(b) Fees as fraction of final value by manager
(c) Investor return (p.a.)
Figure 2: Return composition
22
(d) Management fees as fraction of total fees
Figure 2: Return composition (ctd.)
managers have a very low bargaining position and are thus required to hold
larger positions in the fund itself, do concerns about the final value of the
fund become relevant for decision making. The total fees, seen relative to the
performance of the investment, i. e. prior to deducting fees, are also substantial.
The amount of fees are actually largely increasing in the ability of the manager.
This reflects the increased performance fee such high ability will generate, even
if the fraction charged is reduced. As the bargaining power of the manager
increases, the amount of fees is largely reduced because the risk aversion of the
manager drives the fee towards management fees who are smaller but less risky
than performance fees.
A final observation that is confirmed empirically is that the return to investors is low. Even if the bargaining power of managers is low, the investor
barely obtains a positive return, with most of the performance eroded by fees.
Having thus established the properties of the optimal contract specification
we can now continue to assess whether the contracts actually chosen by investors
and managers are consistent with the model proposed here.
4. The empirical evidence
The last section showed how the optimal contract parameters are determined
and how they relate to the bargaining power and ability of the manager. This
section now seeks to evaluate the validity of these results empirically by investigating the relationship between the performance of funds and their contract
parameters. As we do not have any information on the bargaining power, we
can only infer this from the level of contract parameters as outlined above to
be low and therefore derive observable properties from our model that does not
include the bargaining power as a variable.
4.1. The relationships between performance net of fees and contract parameters
Any data on hedge fund returns are the value of the fund to the investor, i. e.
after the deduction of fees. Hence we need to derive the appropriate net fund
23
This table shows the partial derivatives of the net Sharpe ratio with respect to the contract
parameters for low and high ability managers. ”+” indicates a positive sign of this partial
derivative, ”–” a negative sign and ”o” a slope very close to zero.
Variable Definition
Fee parameters
λ
Managerial equity ownership
γ
Management fee
δ
Performance fee
ψ
High-water mark
T
Minimum investment length (years)
Return distribution
κ3
Skewness
κ4
Kurtosis
Low ability
High ability
o
–
–
o
o
–
–
–
–
o
o
o
o
o
Table 3: Relationship between net Sharpe ratios and contract specifications
returns from our model and then determine the performance for these returns.8
We will use the Sharpe ratio again as our performance measure and call this
the ”net Sharpe ratio” to distinguish it from the Sharpe ratio that determines
the investment returns of the funds and used as a measure of the ability of
the manager. This Sharpe ratio we call for clarity ”gross Sharpe ratio” in this
section if required. Applying this measure to the model described above we can
easily derive the main properties between the net Sharpe ratio and the contract
parameters.
Based on our simulations to obtain the contract curve, we generate time series of the returns after fees for each contract specification with returns recorded
at the end of every reporting period until the end of the lock-in period, i.e. for
t = 1, . . . , Tτ we determine the returns net of fees as
rbt = rt − cτ − γτ − δ max{rt − Rt }
(6)
and calculate the Sharpe ratio based on these returns, where the Sharpe ratio
is averaged over 500 individual simulations.
The results of this analysis are summarized in table 3, where results some8 As our database does not contain time series information on assets under management,
rather only the assets under management at one point of time, we cannot recontsruct the
gross Sharpe ratio as in Agarwal, Daniel, and Naik (2009) or Aragon and Nanda (2012).
24
times differ for managers with high or low ability. As we use the Sharpe ratio as
our performance measure, the use of volatility as one of the independent variables is not possible, so we excluded this variable from our analysis, as much
as we excluded the reporting period as no data were available in our database.
The full results are visualized in figure 3.
Our results on the net Sharpe ratio are obviously closely related to those of
the model presented above. As we have no information in our dataset on the
bargaining power of the manager, we do not consider this variable in our analysis
as a relevant explanatory variable. We find not surprisingly that as management and performance fee are increasing, the performance of the fund reduces.
No effect is found for the skewness and kurtosis as we have argued above the
additional risks are balanced against other aspects of the optimal contract, such
as the level of fees, making their influence on the fund performance irrelevant.
We find that for low ability managers the managerial ownership does not affect
performance while for higher abilities higher manager ownership is detrimental
to the fund performance, although the impact is very small. The reason for
this observation is that with low ability the manager does not obtain significant
returns from his investment and thus effects of the other parameters are approximately off setting each other. Higher ability managers seek to extract higher
fees from the fund as he has a larger stake because this would reduce the risks
associated with their final wealth, who would depend otherwise too much on the
risky returns of the investment itself rather than the more steady fee income.
As we see, the highwater mark does not affect the net Sharpe ratio for low ability managers, mainly because in this case the likelihood of paying performance
fee in the first place is very much reduced, thus its impact is negligible, while
for higher ability manager an increase in the highwater mark is associated with
lower returns due to the fact that the performance fee parameter does increase
at the same time and the increase of the highwater does not fully compensate
for this effect.
As mentioned above the returns to investors are negative in many cases, resulting in a negative net Sharpe ratio for a wide range of contract specifications,
25
26
(a) Managerial co-investment
(b) Management fee
(c) Performance fee
Figure 3: Relationship between net Sharpe ratio and contract specifications
27
(d) Highwater mark
(e) Lock-in period
(f) Skewness
Figure 3: Relationship between net Sharpe ratio and contract specifications (ctd.)
28
(g) Kurtosis
Figure 3: Relationship between net Sharpe ratio and contract specifications (ctd.)
generally for most scenarios in which the ability of the manager is low and his
bargaining power high. Empirically we also find a significant fraction of funds
exhibiting negative Sharpe ratios. We can now use the above properties on the
partial derivatives obtained here to assess whether actual hedge fund contract
exhibit properties consistent with our model.
4.2. Data and variable description
The data for our analysis are obtained from Hedge Fund Intelligence (HFI)
using the time period from 1969 to the end of 2012. The database consists of
12699 funds for which a range of information is available; 3438 funds were found
to be duplicates that were removed, leaving 9261 funds for analysis. Duplicate
funds were identified by manually checking all those funds that had a return correlation exceeding 0.95 after converting any returns into USD for those funds
that were quoted in different currencies. Most duplicate funds were merely
quoted in different currencies but were otherwise identical; in these cases the
fund using the most commonly used currency was retained and later transferred
into USD. Of the remaining funds 5041 are ”dead”, i.e. do not provide information to the database at the end of 2012, 3960 are ”live”, i.e. are continuing
to report information and operating normally, 236 funds are ”closed”, i.e. have
restricted capital flows and 24 funds have an unknown status.
As hedge funds provide information to database providers voluntary, the
available dataset can be biased due to the self-selection of reporting. The survivorship bias we find is in line with those of other studies, like Schuhmacher
and Eling (2012) or Bali, Brown, and Caglayan (2012) amongst others, and
will address this issue in the empirical analysis through the use of appropriate
dummy variables on the fund status. In addition, hedge funds often decide to
provide data once they have a history of good performance, with data being
backfilled to times prior to this data. This instant history bias, as discussed in
Agarwal, Daniel, and Naik (2009), is not present in our database as HFI does
only use data after they have been provided by the hedge fund and does not
accept data from prior periods retrospectively.
29
The HFI database provides information on the management and performance fees, whether a high watermark and or hurdle rate exists, lock-in periods,
redemption terms and penalties, size of the fund, its currency, and domicile. Table 4 provides an overview of these variables with their definitions, limited to
those variables we use in our analysis, either as contract parameters or additional control variables.
Compared to other databases that have been used in previous research, the
descriptive statistics of the key variables of interest are in line with those. Our
database includes, however, larger funds than in other databases used previously. Table 5 shows the descriptive statistics of the key quantitative variables.
Throughout this table and the remaining analysis, we only considered funds
that have at least 24 months of continuous return information to be used for
calculating net Sharpe ratios. For comparability all returns have been calculated in US dollars, using monthly exchange rates from Datastream to convert
non-US Dollar denominated funds.9
In terms of composition of funds, our database consists of approximately
three quarter funds denominated in US dollars, with the EURO the second
most common currency and other currencies having only few funds, see table
7. Similarly the domiciles of funds are spread widely. However, most funds are
located in off-shore centres in 83% of the cases as is common for hedge funds
and therefore a breakdown by fund location would not enhance the quality of
the analysis in a meaningful way.
In addition to the variables above, we also have information on the investment strategy of the funds. Rather than relying on the provided coding of the
investment strategy provided by HFI, which seemed rather inconsistent, we use
the description of funds which is also provided and manually code each fund into
a strategy based on the classifications used by McCrary (2002), Jaeger (2003),
Lhabitant (2004), and Jaeger (2008). The description of these strategies is pro9 A sensitivity analysis of requiring 36 and 60 months of continuous return information has
been conducted. We found no meaningful differences to the results obtained here.
30
31
Redemption terms
Lock-in
Assets under management
Currency
Fund location
Fund status
Fund age
High-water mark
Variable
Returns
Management fee
Performance fee
Hurdle rate
Table 4: Data definitions
Description
Monthly net of fee return
Fees as a percentage of assets under management
Fees as a percentage of the value of the fund exceeding the highwater mark
Dummy variable which is 1 if a minimum performance per reporting period is
required to trigger payment of the performance fee, and 0 otherwise
Dummy variable which is 1 if a high-water mark is used in the calculation of
performance fees and 0 otherwise
The time lag between the decision to divest and the actual withdrawal of funds
Time until the investment can be withdrawn
The net asset value of the fund at a specified date
Currency in which the fund reports
Dummy variable which is 1 if the fund is offshore and 0 otherwise
An indicator whether the fund is open, closed or dead
The year in which the fund started reporting. We change this variable to a
dummy variable indicating a start in a particular decade
32
1.50%
20%
35
1
0
52
12
40
0.63%
3.73%
-0.12
1.29
0.13
2%
20%
121.9
1
0
88
12
90
1%
5.34%
0.37
3.28
0.23
6%
50%
56,015
1
1
429
84
1800
11.31%
118%
10.93
152
9.06
1.60%
19.43%
213.3
0.8633
0.1663
66
12.5
57
0.68%
4.30%
-0.15
3
0.15
Descriptive Statistics
Median
75%
Maximum Mean
Table 5: Descriptive statistics
1.50%
20%
11
1
0
28
12
30
0.28&
2.39%
-0.58
0.27
0.03
6655
6563
5508
5508
5508
6831
5576
6254
5497
5497
5497
5497
5497
Management fee
Performance fee
Fund size (m$)
High-water mark
Hurdle rate
Fund age (months)
Lock-in (months)
Redemption (days)
Mean return
Volatility
Skewness
Kurtosis
Sharpe ratio
0
0
0.096
0
0
1
1
1
-5.36%
0.08%
-12.1
-1.65
-0.78
Observations
Minimum 25%
Variable
Standard
Deviation
0.48%
3.44%
1191
0.1180
0.1386
53
7.1
62
0.80%
3.28%
1.16
6.17
0.3
33
Fund Reporting Currency
Australian Dollar
Brazilian Real
Canadian Dollar
Swiss Franc
Chinese Yuan
Czech Koruna
Danish Krone
Euro
British Pound
Hong Kong Dollar
Indian Rupee
Japanese Yen
Norwegian Krone
New Zealand Dollar
Swedish Krona
Singapore Dollar
United States Dollar
South African Rand
Total
Symbol
AUD
BRL
CAD
CHF
CNY
CZK
DKK
EUR
GBP
HKD
INR
JPY
NOK
NZD
SEK
SGD
USD
ZAR
Number of Funds
106
52
87
29
1
1
4
991
173
4
1
125
11
3
68
5
5067
103
6831
Table 6: Hedge Fund Reporting Currencies
Percent of database
1.55%
0.76%
1.27%
0.42%
0.01%
0.01%
0.06%
14.51%
2.53%
0.06%
0.01%
1.83%
0.16%
0.04%
1.00%
0.07%
74.18%
1.51%
100.00%
vided in table 7. We have excluded from our analysis so called ”funds of funds”,
i. e. those hedge funds that invest into other funds only. They make 2430 hedge
funds in total, reducing our sample to a total 6831, of which 5497 have return
data of at least 24 continuous months. Table 8 provides an overview of the
frequency of the different main strategies in our database. We will use these
main strategies as control variables in our further analysis.10
We measure performance of a hedge fund using the Sharpe ratio. While this
is clearly consistent with the model developed in the previous section, we have
also considered a wide range of other performance measures such as Sortino
ratio (Sortino and van der Meer (1991)), Omega ratio (Shadwick and Keating
(2002)), Upside potential (Schuhmacher and Eling (2011)), Kappa (Kaplan and
Knowles (2004)), Excess Return VaR (Agarwal and Naik (2004)), Treynor ratio,
Generalized Treynor ratio (Hübner (2005)), Calmar ratio (Young (1991)), Sterling ratio (Kestner (1996)) and Burke ratio (Burke (1991)). We found generally
that when ranking funds using these measures, results were highly consistent.
The Spearman rank correlation coefficient was always found to be in excess of
0.83, with most values being above 0.9. Only when using Jensen’s α do we find
correlations below 0.7. Using the procedure in (Rees, 1987, p. 383), we find that
when excluding Jensen’s α we cannot reject the hypothesis that the correlations
are equal to 0.99 at the 1% significance level. This result is consistent with
Schuhmacher and Eling (2007), Ding, Shawky, and Tian (2009) and Abugri and
Dutta (2009) and suggests that the Sharpe ratio will provide a robust measure
of performance and that results using alternative measures will provide similar
outcomes.
4.3. Variable definitions
We will be using a range of contract parameters and control variables in our
regressions of the net Sharpe ratio. Starting with the contract parameters, we
have no information on the amount of the funds held by the manager in our
10 We also considered alternative strategy specifications according to Agarwal, Daniel, and
Naik (2009) as well as using substrategies without changing the results in a meaningful manner.
34
35
Fixed Income
Exotic
Emerging Markets
Multi Strategy
Global Macro
Systematic
Event Driven
Main Strategy
Equity Hedge
Table 7: Hedge Fund Strategy Coding
Strategy Description
A portfolio of listed stocks consisting of long and short positions, including derivatives based on them
A focus of investment into stocks around particular events such as distress or mergers
A fund that invests to a wider theme in a wide range of asset classes
An algorithmically constructed portfolio investing into a wide range of
assets, mainly based on futures
Investments into fixed income securities taking long and short positions
Investments into less liquid markets, such as volatility trading
Investments into listed assets from emerging markets
A fund that combines two or more of the above main strategies
Strategy
Equity Hedge
Event Driven
Global Macro
Systematic
Fixed Income
Exotic
Emerging Markets
Multi Strategy
Total
Number of Funds
2705
456
473
655
681
334
1136
391
6831
% of Database
39.60%
6.68%
6.92%
9.59%
9.97%
4.89%
16.63%
5.72%
100.00%
Table 8: Database Composition by Strategy
database. In line with Hodder and Jackwerth (2007) we assume here that large
funds will only have a small amount of manager holdings while smaller funds
will have more. We therefore propose to use
1
ln A0
as a proxy for the amount
of managerial ownership. Here A0 denotes the assets under management at the
time the fund starts reporting in our database. As the assets under management
are generally not given at this starting date but at some time t in our database,
Qt−1
we approximate the assets under management by using A0 = At τ =0 e−rt−τ ,
assuming no fund in and outflows. In addition, while the inception date will in
general not be the date at which it has first provided data, we treat this date
as the best available proxy.
The management fee is directly given in our database as a fraction of the
assets under management and the performance fee is also given as the fraction
of profits that are retained by the manager if the value of the fund exceeds
the highwater mark. The highwater mark in our database is a dummy variable
taking the value of 1 if a highwater mark exists and 0 otherwise. Hence it
is not a direct equivalence of the highwater mark parameter ψ in our model,
although it is nearly unheard of to observe other variables than ψ = 1 and
the absence of a highwater mark can then be interpreted as ψ = 0, thus this
variable can serve as a useful proxy for our contract parameter. The lock-in
in our database refers to the time until the investor is able to withdraw his
investment, measured in months. We can treat this information as a proxy for
36
the length of the investment in our model. Skewness and kurtosis are estimated
from the monthly return data in a standard way.
A wide variety of control variables have been used in the literature so far.
We employ similar sets of control variables that we briefly describe here. As
for control variables we have information on the fund location, which we set 1 if
it is classified as an offshore location and 0 otherwise. Some funds might have
a hurdle rate in which in addition to the highwater mark the return in a time
period must exceed a certain threshold to trigger payment of the performance
fee, e.g. the riskfree rate. This variable is set to 1 if a hurdle exists and 0
otherwise. The redemption terms measures how many days notice an investor
needs to give to withdraw money from the fund, after the lock-in period. As
the redemption period is often related to the liquidity of the investments of the
hedge fund, we can use this variable as a proxy to control for the liquidity of
their investment strategy.
The fund status takes a value of 1 if the fund is dead and 0 otherwise. This
allows us to account for any effect the survivorship bias has on the performance
of hedge funds. The Currency includes dummies for the fund being quoted in
EUROs or other currencies, with the US dollar serving as the benchmark currency. This allows us to account for any effects arising from different currency
exposures. The fund age is a series of dummy variables that takes into account
the time in which a fund first reports its returns by introducing a range of
dummy variables for different decades. That way we can account for any effects
on performance arising from the initiation of funds during different market conditions.11 Finally we account for the investment strategy of the fund as specified
in table 7 in a series of dummy variables for each strategy. This allows to adjust
for any effects of the strategy that have not been captured by the moments of
the distribution.
To assess the possibility of multi-collinearity of the contract parameters as
11 Different specifications of this dummy variable have been considered without affecting the
main results.
37
Variable
Fund Location
Hurdle Rate
Redemption terms
Fund Status
Currency Exposure
Fund Age
Investment Style
Control variable Set
Set 1 Set 2 Set 3 Set 4
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
Table 9: Sets of control variables
well as control variables, we considered the correlation matrix of all these variables and found no concerns in this respect.
4.4. Regression results
Using these data we can now conduct a regression analysis to assess the
contract specifications of hedge funds. When conducting standard OLS regessions we find the error terms to be highly non-normal. To stay consistent with
the model used, we instead assume the error terms to follow a Generalized Hyperbolic distribution. In similar circumstances Kouwenberg and Ziemba (2007)
used a skewed t-distribution. Following Nelder and Mead (1965) we conduct
a two-stage maximum likelihood estimation of our regression coefficients with
bootstrapping of the standard errors. Similar to Kouwenberg and Ziemba (2007)
we find that our results are largely consistent with those obtained from OLS
regressions, supporting the stability of our results. In order to account for the
influence of any outliers in our database, we winsorize our data at the 5% and
95% level to avoid any bias in our results. Again, conducting an analysis without
winsorizing we found the results to be consistent with those presented here.
We conducted regressions with a wide variety of control variables and present
some of these here. Table 9 shows the control variables used in each of those
regressions presented here, while results did not change in a meaningful way
for other combinations attempted, again providing evidence for the stability of
our results. Furthermore, conducting regressions with data being available for
at least 36 and 60 months, respectively, yielded results that are consistent with
38
those presented here.
We start by conducting a regression on the whole sample, whose results
are shown in table 10, neglecting the coefficient values of our control variables.
Firstly we observe stability of results as we include different sets of control
variables in terms of significant contract parameters as well as coefficient values.
Secondly, we observe the relevance of the higher moments skewness and kurtosis
as well as managerial ownership, with the management and performance fees
becoming significant in the final set of control variables. Comparing these results
with those arising from our model in table 3 we observe that the result on
managerial ownership is broadly consistent, while the sign for the management
and performance fees are opposite of those predicted by theory. The insignificant
coefficient for the highwater mark is consistent with the predicted results. While
the model predicted no significant relationship between the performance of the
fund and its skewness and kurtosis, we empirically find this to be positive.
The coefficient size associated with the kurtosis is, however, so small, that its
economic significance is negligible. The insignificance of the lock-in is consistent
with the model.
When discussing the results from our model, we discovered that in some
instances the results were different for high and low Sharpe ratios. Hence we
have split our sample into quartiles according to the Sharpe ratio observed
and conduct the same regressions as previously, reporting only the top and
bottom quartile, see table 11.12 Using different splits, e.g. into deciles lead
to comparable results. Once again, results are stable across specifications with
different sets of control variables.
The first observation is that with this split of the sample the results for the
management fee is now consistent with the predicted signs of the theory, while
the performance fee remains statistically insignificant rather than negative. The
12 While we argued above for differences between low ability and high ability managers to
be taken into account, i. e. the gross Sharpe ratio, we here split our sample according to
the net Sharpe ratio. Inspection of figure 3 shows that in the relevant cases for managerial
co-investment and highwater mark gross and net Sharpe ratio are consistent with each other,
thus allowing us to make this distinction in our regressions.
39
40
Controls
Adjusted R2
Sample Size
Lock-in
Managerial ownership
Kurtosis
Skewness
Highwater mark
Performance fee
Management fee
Intercept
Set 1
0.1490
3364
2
0.0401**
(2.1820)
0.1171
(0.2846)
0.1227*
(1.9024)
0.0104
(1.1189)
0.0430***
(16.2495)
0.0052***
(8.0618)
-0.0279***
(-4.2445)
-0.0012
(-0.6206)
Set 2
0.1600
3364
3
0.0521***
(3.0589)
0.2585
(0.5649)
0.1078*
(1.8643)
0.0039
(0.3859)
0.0421***
(16.1516)
0.0048***
(7.8484)
-0.0343***
(-4.7905)
-0.0014
(-0.7788)
Table 10: Regression Results with all funds
–
0.0918
3489
1
0.0994***
(5.7373)
-0.0178
(-0.0415)
0.1115*
(1.6864)
0.0207**
(2.2493)
0.0491***
(18.8867)
0.0062***
(9.5014)
-0.0467***
(-7.2857)
0.0118***
(6.3957)
Set 3
0.1583
3364
4
0.0474***
(2.6211)
0.2823
(0.6243)
0.0950
(1.5101)
0.0063
(0.7142)
0.0394***
(14.9290)
0.0040***
(6.6614)
-0.0387***
(-5.6553)
-0.0008
(-0.3969)
Set 4
0.2045
3364
5
0.0369*
(1.9533)
0.8376**
(1.9764)
0.1346**
(2.0909)
0.0042
(0.4938)
0.0445***
(17.4175)
0.0019***
(3.0881)
-0.0303***
(-4.231)
-0.0015
(-0.7743)
This table presents results of the cross-sectional regression using bootstrapped standard errors assuming error terms are follow a generalized
Hyperbolic distribution. Coefficient estimates are followed by bootstrapped t-statistic values in parentheses in the row below. *** shows statistical
significance at the 1% level, ** at the 5% level and * at the 10% level.
41
Controls
Adjusted R2
Sample Size
Lock-in
Managerial ownership
Kurtosis
Skewness
Highwater mark
Performance fee
Management fee
Intercept
–
1
-0.0326
872
Bottom
0.3140***
(18.2704)
-0.6527
(-1.5294)
0.0767
(1.3134)
0.0428***
(4.1375)
0.0158***
(6.6611)
-0.0020***
(-3.5219)
-0.0622***
(-9.3754)
0.0107***
(6.2910)
0.1026
841
4
0.0610
841
Bottom
0.3041***
(15.2894)
0.0193
(0.0397)
0.1263**
(2.0286)
0.0000
(0.000)
0.0052**
(1.9772)
0.0001
(0.2003)
-0.0144**
(-2.0573)
-0.0035*
(-1.9223)
Set 3
Top
-0.0507***
(-4.0938)
-0.6302**
(-2.1888)
-0.0062
(-0.1420)
0.0031
(0.5341)
0.0081***
(3.8276)
0.0015***
(3.4559)
-0.0038
(-0.7739)
0.0015
(1.1116)
0.0990
841
5
0.1540
841
Bottom
0.2899***
(12.5937)
-0.2062
(-0.3885)
0.1229*
(1.6514)
0.0008
(0.0618)
0.0147***
(5.2178)
-0.0017**
(-2.4339)
-0.0072
(-0.8622)
-0.0054***
(-2.7802)
Set 4
Top
-0.0510***
(-4.5874)
-0.7533**
(-2.4505)
0.0007
(0.0170)
0.0022
(0.3920)
0.0074***
(3.9794)
0.0015***
(2.9297)
-0.0046
(-0.8710)
0.0013
(0.8593)
Table 11: Regression Results of top and bottom quartile of funds
0.0438
872
Top
-0.0303**
(-2.5504)
-0.9326***
(-2.9441)
0.0061
(0.1335)
-0.0030
(-0.5377)
0.0127***
(6.1253)
0.0019***
(4.0159)
-0.0026
(-0.5015)
-0.0012
(-0.8500)
This table presents the results of the cross-sectional regressions with the sample split into the top and bottom quartiles by performance. Coefficient
estimates are followed by bootstrapped t-statistic values in parentheses in the row below. *** shows statistical significance at the 1% level, ** at
the 5% level and * at the 10% level.
high water mark remains statistically insignificant, which is consistent with our
model for low Sharpe ratios, but not high. We find some evidence for the
managerial ownership to be negative for high Sharpe ratios and insignificant
for lower ratios. the lock-in shows results consistent again for low Sharpe ratios
while for high Sharpe ratios the coefficients are statistically significant, although
of different signs across specifications. The small coefficient values combined
with the changing sign leads us to conclude that our empirical results are broadly
consistent with theory. Here the economic impact of changing this contract
parameter is negligible.
We continue to observe a significant positive coefficient for skewness, while
theory predicts no significant relationship. The coefficient value is much reduced, however, leading us to conclude that the relationship is weak and thus
broadly consistent with our results. The kurtosis, for which we predicted no
significant result, shows a positive relationship for high Sharpe ratios and a
negative for low. But again the coefficients are so low that the economic significance is very low, suggesting broad consistently with our results.
Hence we overall conclude that our results are consistent with the theoretical model developed above, with the exception of the relationship with the
performance fee and to an extend the high water mark. This difference can
be attributed to a number of features that we have not included in our model.
Most prominently we have assumed that contracts, once agreed, are executed
to the letter of the agreement and the ability of managers are known. In reality,
however, such contracts will be subject to principal-agent problems where firstly
the ability of the manager will have to be subject to effort, which is costly for
managers. Li and Tiwari (2009) show that effort, and hence performance, is
increasing if the performance fee is increasing, implying a positive relationship
between the performance fee and actual performance. This positive relationship
superimposed on the projected negative relationship arising from our model,
might well result in the insignificant relationship we found empirically. Thus
including this missing aspect into our model might well explain this difference
in results.
42
5. Conclusions
Hedge funds are investment vehicles that enjoy not only flexibility in their
investment strategies but also in the compensation of the manager and any restrictions on withdrawing these investments by the investor or manager. We
considered a model of hedge fund returns by explicitly taking into a account a
wide range of parameters concerning the management contract and investment
strategy, including the fee structure, lock-in period, managerial investments and
properties of the return distribution and determined the optimal contract specification a risk averse manager and investor could agree on. We showed how these
contract parameters will depend on the ability of the manager and his bargaining power in the contract negotiations. We found that the results of our model
are consistent with managers having relatively little bargaining power as well as
low ability. We then continued to use our model to derive empirical predictions
on the relationship between contract specifications and investor performance of
hedge funds. Comparing these with actual observations we found our model to
predict correctly most of the relationships between these variables, providing
evidence that hedge funds contracts are negotiated optimally. We only found
the relationship between the performance fee and the actual performance to be
inconsistent and were able to suggest that this might be due to the lack of a
principal-agency problem in our model as suggested in other research.
Our model has some obvious extensions by including a principal-agent problem to allow the manager to decide his own effort level. We could also include
additional aspects in the contract specification like a hurdle rate, allow for capital inflows and outflows during the lifetime of the fund rather than a fixed
lock-in period. In addition we might want to consider a liquidation boundary,
i. e. a lower limit that once crossed triggers the dissolution of the hedge fund.
Such extensions should not change significantly the results obtained here, but
might most importantly add to the understanding of such additional provisions.
43
References
Abugri, B. A., and Sandip Dutta, 2009, Emerging market hedge funds: Do
they perform like regular hedge funds?, Journal of International Financial
Markets, Institutions and Money 19, 834–849.
Ackermann, Carl, Richard McEnally, and David Ravenscraft, 1999, The performance of hedge funds: risk, return and incentives, The Journal of Finance
54, 833–874.
Agarwal, Vikas, Naveen Daniel, and Narayan Naik, 2009, Role of managerial
incentives and discretion in hedge fund performance, Journal of Finance 64,
2221–2256.
Agarwal, Vikas, Juan-Pedro Gomez, and Richard Priestley, 2012, Management
compensation and market timing under portfolio constraints, Journal of Economic Dynamics and Control 36, 1600–1625.
Agarwal, Vikas, and Narayan Naik, 2004, Risk and portfolio decisions involving
hedge funds, Review of Financial Studies 17, 63–98.
Aragon, George, and Vikram Nanda, 2012, Tournament behavior in hedge
funds: High-water marks, fund liquidation, and managerial stake, Review
of Financial Studies 25, 936–974.
Aragon, George O., and Jun ”QJ” Qian, 2009, High-water marks and hedge
fund compensation*, Unpublished Working Paper Arizona State University.
Bali, Tarun, Stephen Brown, and Mustafa Caglayan, 2012, Systematic risk and
the cross section of hedge fund returns, Journal of Financial Economics 106,
114–131.
Barndorff-Nielson, Ol, 1977, Exponentially decreasing distributions for the logarithm of particle size, Proceedings of the Royal Society London Series A,
Mathematical and Physical Sciences 353, 401–419.
44
Berk, Jonathan, and Richard Green, 2004, Mutual fund flows and performance
in rational markets, Journal of Political Economy 112, 1269–1295.
Buraschi, Andrea, Robert Kosowski, and Worrawat Sritrakul, 2014, Incentives
and endogenous risk taking: A structureal view on hedge fund alphas, Journal
of Finance forthcoming.
Burke, Gray, 1991, A sharper sharp ratio, Futures 23, 56.
Carpenter, Jennifer, 2000, Does option compensation increase managerial risk
appetite?, Journal of Finance 55, 2311–2331.
Das, Sanjeev R., and Rangarajan K. Sundaram, 1998, On the regulation of fee
structures in mutual funds, New York University, Center for Law and Business
Working Paper No.98-002.
Ding, Bill, Henry Shawky, and Jianbo Tian, 2009, Liquidity shocks, size and
the relative performance of hedge fund strategies, Journal of Banking and
Finance 33, 883–891.
Drechsler, Itamar, 2013, Risk choice under high-water marks, Unpublished
Working Paper New York University.
, 2014, Risk choice under high-water marks, Review of Financial Studies
27, 2052–2096.
Eberlein, Ernst, 2001, Application of generalized hyperbolic lévy motions to
finance, in Ole E. Barndorff-Nielson, Thomas Miskosch, and Sidney I. Resnick,
ed.: Lévy processes: Theory and applications . pp. 319–336 (Springer Verlag).
Edwards, Franklin, and Mustafa Caglayan, 2001, Hedge fund performance and
manager skill, Journal of Futures Markets 21, 1003–1028.
Emberlin, Ernst, and Ernst-August Hammerstein, 2003, Generalized hyperbolic
and inverse gaussian distributions: limiting cases and approximation of processes, GDM Reprint Nr.80, University of Freiburg.
45
Fajardo, Jośe, and Anquiles Farais, 2010, Derivative pricing using multivariate
affine generalized hyperbolic distributions, Journal of Banking and Finance
34, 1607–1617.
Fischer, Matthias, 2011, Generalized hyperbolic distributions, in Miodrag
Lovric, ed.: International Encyclopedia of Statistical Science . pp. 589–590
(Springer Berlin Heidelberg).
Goetzmann, William, Jonathan Ingersoll, and Stephen Ross, 2003, High-water
marks and hedge fund management contracts, Journal of Finance 58, 1685–
1718.
Grinblatt, Mark, and Sheridan Titman, 1989, Adverse risk incentives and the
design of performance-based contracts, Management Science 35, 807–822.
Hodder, J. E., and J. C. Jackwerth, 2007, Incentive contracts and hedge fund
management, Journal of Financial and Quantitative Analysis 42, 811–826.
Hübner, Georges, 2005, The generalized treynor ratio, Review of Finance 9,
415–435.
Jaeger, Lars, 2008, Alternative Beta Strategies and Hedge Fund Replication
(John Wiley & Sons) 1st edn.
Jaeger, R. A., 2003, All About Hedge Funds: The Easy Way to Get Started
(McGraw Hill) 1st edn.
Kaplan, P. D., and J. A. Knowles, 2004, Kappa: A generalised downside riskadjusted performance measure, Journal of Performance Measurement 8, 42–
54.
Kestner, Larry, 1996, Getting a handle on true performance, Futures 25, 44–46.
Koijen, Ralph S. J., 2014, The cross-section of managerial ability, incentives,
and risk preferences, Journal of Finance 69, 1051–1098.
46
Kouwenberg, Roy, and W. T. Ziemba, 2007, Incentives and risk taking in hedge
funds, Journal of Banking and Finance 31, 3291–3310.
Kritzman, Mark, 1987, Incentive fees: Some problems some solutions, Financial
Analysts Journal 43, 21–26.
Lan, Yingcong, Neng Wang, and Jinqiang Yang, 2013, The economics of hedge
funds, Journal of Financial Economics 109, 734–758.
Lhabitant, Francois, 2004, Hedge Funds: Quantitative Insights (John Wiley &
Sons).
, 2006, Handbook of Hedge Funds (John Wiley & Sons).
Li, Wei, and Ashish Tiwari, 2009, Incentive contracts in delegated portfolio
management, The Review of Financial Studies 22, 4681–4714.
McCrary, Stuart, 2002, How to Create and Manage a Hedge Fund: A Professional’s Guide (John Wiley & Sons) 1st edn.
Michaud, R.O, and R.O Michaud, 2008, Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation (Oxford University Press) 2nd edn.
Nelder, J. A., and R. Mead, 1965, A simplex algorithm for function minimization, Computer Journal 7, 308–313.
Ou-Yang, Hui, 2003, Optimal contracts in a continuous-time delegated portfolio
management problem, The Review of Financial Studies 16, 173–208.
Panageas, Stavros, and Mark Westerfield, 2009, High-water marks: High risk
appetites? convex compensation, long horizons, and portfolio choice, The
Journal of Finance 64, 1–36.
Patrick, Serge, and Amvella Motaze, 2013, Managerial incentives and the risktaking behavior of hedge fund managers, International Journal of Economics
and Finance 5, 36–48.
47
Rees, Daniel, 1987, Foundations of Statistics (Chapman & Hall).
Scherer, B, 2002, Portfolio resampling: Review and critique, Financial Analysts
Journal 58, 98–109.
Schuhmacher, Frank, and Martin Eling, 2007, Does the choice of performance
measure influence the evaluation of hedge funds, Journal of Banking and
Finance 31, 2632–2647.
, 2011, Sufficient conditions for expected utility to imply drawdownbased performance rankings, Journal of Banking and Finance 35, 2311–2318.
, 2012, A decision-theoretic foundation for reward-to-risk performance
measures, Journal of Banking and Finance 36, 2077–2082.
Shadwick, William, and Caron Keating, 2002, A universal performance measure,
Journal of Performance Measurement 6, 59–84.
Sortino, Frank, and Robert van der Meer, 1991, Downside risk, Journal of
Portfolio Management 17, 27–31.
Stracca, Livio, 2006, Delegated portfolio management: A survey of the theoretical literature, Journal of Economic Surveys 20, 823–850.
Young, Terry, 1991, Calmar ratio: A smoother tool, Futures 20.
48