Incomplete markets, Knightian uncertainty and high-water marks

Operations Research Letters 48 (2020) 195–201

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

Incomplete markets, Knightian uncertainty and high-water marks Fengjun Liu a,b , Yingjie Niu a , Zhentao Zou c , a b c

∗

School of Finance, Shanghai University of Finance and Economics, Shanghai, China Honghe University, Yunnan, China Economics and Management School, Wuhan University, Hubei, China

article

info

Article history: Received 12 April 2019 Received in revised form 18 February 2020 Accepted 18 February 2020 Available online 28 February 2020 Keywords: High-water mark Hedge fund Non-diversifiable risk Ambiguity

a b s t r a c t This paper extends the pricing of the hedge fund compensation contracts to the case of ambiguity over the appropriate valuation approach originating from market incompleteness. It predicts that an increase in the level of Knightian uncertainty causes the erosion of the values of the fees and the claim, while an increase in the degree of market incompleteness (specified by the correlation between market asset and the fund’s asset or the volatility of non-diversifiable risk) has non-monotonic effects. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Hedge funds, as an alternative investment instruments, have grown faster over the past decades and play a vital role in financial intermediaries. One of the key property of hedge funds is the complicated compensation structure. Specifically, the hedge fund management contract contains regular management fees and performance-based incentive fees. The management fees are paid as a fixed proportion 2% of the net asset value (NAV) of the fund under management, and the performance fees are typically charged as 20% of the fund’s profits. In the hedge fund industry, performance fees are generally associated with high-water mark (HWM) provision. Precisely, performance fees are not calculated unless the NAV of the fund is in excess of the maximum cumulative value, the high-water mark. This feature thus motives the manager to let the NAV move above the HWM. Any previous losses must be recovered before further performance fees apply. This very convex payoffs and asymmetric management compensation structure characteristics make the hedge fund differ remarkably from mutual funds. [7] applies the equilibrium valuation approach to evaluate the present value of fees of the hedge fund management contracts. In their model, the markets are complete and the distribution of shock is known to the investors. However, there are several reasons for us to think about departures from this traditional approach. First, it is not reasonable to assume that markets are complete. In reality, shares of hedge funds are not traded in ∗ Corresponding author. E-mail addresses: [email protected] (F. Liu), [email protected] (Y. Niu), [email protected] (Z. Zou). https://doi.org/10.1016/j.orl.2020.02.005 0167-6377/© 2020 Elsevier B.V. All rights reserved.

the market and positions of the fund asset are complicated so that outside investors cannot simply replicate its returns. Second, the presence of market incompleteness implies that a unique martingale measure is invalid to price all risk in the hedge fund. Instead, there are infinitely many discount factors that price the market. Finally, the idiosyncratic risk is not traded on the market and hence there is no observable price for it, implying that the investors should be ambiguous over the price of non-tradeable risk. Ambiguity is a complex concept that may not be limited to risk. The Ellsberg [5] paradox and related experimental evidence demonstrate that people respond to risk and ambiguity in different ways. Moreover, as [2] and [8] note, economic agents believe that the observed economic data come from a set of unspecified models. The individual may consider other probability measures and be uncertain of the plausibility of these measures. Roughly speaking, risk copes with uncertainty that follows from a given probability measure on a measurable space. Ambiguity, on the other hand, refers to uncertainty that arises if one does not know what the probability measure is. Concerns about ambiguity induce the individual to make robust decisions. The objective of our paper is to investigate the effects of market incompleteness and ambiguity on valuation for a hedge fund management company. For featuring ambiguity that is solely about the proper rate at which any payment should be discounted, we adopt the multiple-priors model which is based on the maxmin criterion and indicates optimization in a worst-case scenario. Concretely, we assume that other probability measures considered are not far from the original measure. That is, the density generator moves only in the range [−κ, κ], where κ can be viewed as the degree of Knightian uncertainty. This method in continuous time is called κ -ignorance by [3] and [15].

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By incorporating market incompleteness and ambiguity, the evaluation of all fees and the claim is affected differently. The main results of the model are as follows. First, the present values of the performance fee, regular annual fee, total fee, and the investor’s claim have unambiguous relations with ambiguity parameter. An increase in ambiguity decreases the values of the fees and the claim believed by a fund manager due to the additional discount for Knightian uncertainty. Second, when market incompleteness is represented by the correlation between the partial spanning asset and the fund’s net asset value, the values of the fees and the claim are non-monotonic with the correlation. The straightforward interpretation is to clarify the equilibrium-pricing effect. Finally, the other way to the measure market incompleteness is the employment of nondiversifiable risk. As the degree of market incomplete increases, the value of performance fee rises, the annual fee and total fee fall, whereas the investor’s claim presents an ambiguous relationship. Our paper is related to a rapidly growing literature that analyses the economics of hedge funds. Among them, our paper is most closely connected to [7], which firstly provides a framework for valuation of a hedge fund manager by quantitatively valuing management and performance incentive fees. [12] states that hedge fund managers trade off the benefits of leveraging on the alpha-generating strategy against the costs of inefficient fund liquidation. [17] extends the model in [7] to the case of partial information. [4] solves in closed form for the optimal dynamic risk choice of a fund manager who is compensated with a high-water mark contract. [1] considers the optimal investment problem for a time-inconsistent manager compensated with a HWM contract. [13] extends the continuous hedge fund framework to model the dynamic leverage choice of a hedge fund manager with time-inconsistent preferences. [6] introduces a new instrument in the context of hedge fund seeding, called fees-for-guarantee swap. One essential difference between this paper and aforementioned paper is that our model economy is incomplete whereas they focus on the valuation in complete markets. Another important point of departure is that we consider the effects of ambiguity while they develop a valuation equation in the absence of ambiguity. The strand of literature associated with exploring other discount factors for valuing the claim is also relevant to the present paper. [11] proposes the subjective and objective evaluation of incentive stock options. [19] argues that market incompleteness leads to ambiguity over the volatility of the discount factor. [10] considers a hedge fund manager who operates the hedge fund asset and her private portfolio simultaneously under high-water mark compensation in an incomplete market. Different from them, we discuss the consequences of market incompleteness and ambiguity model via κ -ignorance by deriving the martingale pricing operator. Furthermore, our work is linked with the strand of literature that embeds ambiguity into contracting problem. [18] introduces ambiguity into dynamic contracting by assuming that the principal is ambiguous about the agent’s effort cost. [14] studies a contracting problem in which the principal has ambiguous beliefs about the project cash flows. [16] analyses the influences of model uncertainty on dynamic agency and investment theory. However, we investigate the role of market incompleteness played in the framework of hedge fund management contract. The remainder of the paper is organized as follows: Section 2 describes the model setup including the baseline model of highwater mark contract and ambiguity. Section 3 provides the model solutions. In Section 4, we present parameter choices and illustrate the quantitative results and economic implications based on the optimal contract.

2. Model Consider the following management contract cost model in a continuous-time framework. Following [7], we assume that, in the absence of payouts, the net asset value (NAV) of the fund denoted by S follows a lognormal diffusion process given by: dSt

= µdt + σs dZt + σw dWt ,

St

where µ, σs and σw are positive constants. For simplicity as [7], we take dynamics of the net asset value (NAV) of the fund as exogenously given. In other words, we do not allow the manager to pick drift/volatilities. Z and W are two-dimensional independent Brownian motion defined on a probability space (Ω , F , P ). The instantaneous volatility of S is denoted by σ , that is, σ 2 = σs2 +σw2 . In addition to the risk characteristics of the asset value, assume that there are two assets that can be traded in the market with price processes (Bt )t ≥0 and (Mt )t ≥0 , following the diffusions: dBt

dMt

= rdt ,

Bt

Mt

= µm dt + σm dZt ,

respectively, where µm > 0 and σm > 0 are constants. Mt is allowed to be correlated with St with correlated coefficient ρ , in that

[ E

dMt dSt Mt St

]

= ρ dt .

As a matter of fact, ρ =

σs . σ

According to [7], we calculate the valuation of hedge fund from the point of view of an investor. This assumes that the manager has to discount future cash flows in the same way as the investor does, which is able to diversify their shares in the hedge fund by only allocating a limited portion of stake in the hedge fund and thus they use a stochastic discount factor ascertained from observable market prices to evaluate their claim. This is the martingale pricing operator given by equilibrium derivation: dΛt

Λt

= −µΛ dt − hs dZt − hw Wt .

The equilibrium derivation indicates that the representative investor in the market must have preferences that imply that they discount the future according to Λt . An implication of Ito’s lemma yields

µΛ = r ,

hs =

µm − r , σm

where h is equal to the Sharp ratio of the spanning asset or equivalently, the market price of Z -risk. As in [19], we also presume that the market price of W -risk equals a constant η. Then the discount factor of the form dΛt

Λt

= −µΛ dt − hs dZt − ηdWt ,

can be used to price the risk-free asset and the risky asset due to market incompleteness. Fixing η, the value of any asset now equals

EP [d(Λf ) + Dt dt] = 0,

(1)

where f represents the value of a generic claim and Dt is any payment made to the claim being valued. Problem (1) can be reformulated into a more amenable form by a change of measure. This change is what our model focuses on throughout the whole paper. Due to the change of measure, there exists a probability measure Q on (Ω , F ) such that

EP [d(Λf ) + Dt dt] = EQ d(e−r τ f ) + Dt dt

[

]

F. Liu, Y. Niu and Z. Zou / Operations Research Letters 48 (2020) 195–201

Further, under Q, the process St follows the stochastic differential equation: dSt St

ˆt + σw dW ˆt = [µ − σs hs − σw η] dt + σs dZ

ˆ and W ˆ are independent Q-Brownian motion. where Z In the typical high-water mark contract, managers are paid via both management and incentive fees. Assume that the management fee is specified as a constant fraction c of the value of the fund. As for the incentive fee, it often links compensation to the fund manger’s performance via high-water mark (HWM). Notably, the HWM, denoted by H, is the highest level the net assets of the fund have reached. When the fund value moves above the HWM, the manager collects the performance fee that equals the fraction k of this return. For some incentive contracts, the HWM grows at the rate of interest or other contractually stated rate, g. The growth rate g may reflect investor’s opportunity costs of not investing elsewhere. In a word, when S is below the HWM H, the HWM H evolves deterministically and grows exponentially at the rate g, dHt = gHt dt ,

St < Ht .

When S touches HWM, we have dHt − gHt dt > 0 and the manager collects an extra or performance fee equal to the fraction k of this difference dHt − gHt dt. Following [7], we hypothesize that the fund can be terminated exogenously. This termination is assumed to be a Poisson problem with a mean arrival rate λ > 0. This can represent the possibility of a liquidity shock for investors that induces liquidation of the fund. Let τ1 be the stochastic moment at which the exogenous liquidation occurs. Alternatively, the fund is liquidated when the asset value drops to some low level S(Ht ) because large losses may cause the investor to loss confidence in the manager, triggering liquidation. As in [7], we specify S(Ht ) = bHt . Notice that 1 − b is the maximum drawdown that investor allows the fund manager before liquidating the fund. Let τ2 denote the endogenous liquidation time. That is, the fund can be liquidated either exogenously at time τ1 or endogenously at τ2 . Upon liquidation time τ = min{τ1 , τ2 }, the manager receives nothing and the investor collects the fund’s value S. When the manager runs the fund prior to liquidation, the fund’s value evolves according to

ˆt + σw dW ˆt dSt = St (µ − σs hs − σw η) dt + σs dZ [

− cSt dt − k [dHt − gHt dt] .

] (2)

The first term on the right side of (2) describes the change of S due to the manager’s investment skill and the change of measure. The second and the third term represent the continuous payouts to the investor and the flow of management fee. The last term gives the incentive fees which are paid if and only if the fund wealth S exceeds the HWM. Ambiguity. Distinct from the above-mentioned literature, we assume that there is no observable price for W -risk since it is not traded on the market. Therefore, η is not uniquely determined and the manager’s optimization problem cannot be solved. The reason behind is that market incompleteness implies that the measure Q cannot be pinned down uniquely. In our model, as soon as the manger participates in the hedge fund management contracts, he operates the hedge fund in an environment of incomplete information. Distinguished from [15] which considers ambiguity with respect to P , our article emphasizes ambiguity with respect to Q arising from market incompleteness. Suppose that the fund’s manager does not have perfect confidence in the probability measure Q and considers alternative probability measures as

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possible. Then the manager faces so-called Knightian uncertainty, in which the manager is confronted with a set of probability measures rather than a single probability measure. However, we assume that the manager does not consider a wild deviations from Q but only small perturbations from Q, that is, the density generator is constrained to a small range. To model this type of Knightian uncertainty, following [3], firstly we assume that the manager considers only a set of probability measures that perfectly agree with Q with zero probability events. That is, if a specified event’s probability is zero with Q, then it is also zero with these measures. Secondly, since the manager considers only small deviations from Q, thus we focus on κ -ignorance. Specifically, κ -ignorance assumes that the density generator moves only in a range [−κ, κ], where κ ≥ 0 measures the degree of Knightian uncertainty or ignorance. It is obvious that if κ = 0, Knightian uncertainty vanishes. If κ increases, it means that the manager is less certain than before that alternative probability measures are close to Q. Suppose that probability measures are described by mutually absolutely continuous with respect to Q and one another. This assumption allows us to use the Girsanov’s theorem for changing measures equivalent to Q so that modelling ambiguity is different only in the drift term. Thus, to introduce Knightian uncertainty into our model, we follow the methodology adopted by [15]. Define a density generator associated with the profit process as follows: dξt

ξt

ˆt , = θt dW ∫t

where θt is a real-valued process satisfying 0 θs2 ds < ∞ for all t ≥ 0, and where ξ0 = 1. By the Girsanov theorem, there is a measure Qθ corresponding to θ defined on (Ω , F ) such that ξt is the Radon–Nikodym derivative of Qθ with respect to Q when restricted to Ft , dQθ /dQ = θt , and the process defined by

ˆt − θt dt , dWtθ = dW is a standard Brownian motion with respect to Qθ . Under the new measure Qθ , the evolution of the fund’s value is given by

ˆt + σw dWtθ − cSt dt − k [dHt − gHt dt] , (3) dSt = St µθ dt + σs dZ [

]

where

µθ = µ − σs hs − σw (η − θ ).

(4)

Thus, uncertainty characterized by Θ , a set of density generators, the decision-maker takes into account all stochastic differential equation (3) with different θ ∈ Θ . We further are interested in determining the present values of the performance fee, P(St , Ht ), the regular annual fee, A(St , Ht ), and the investor’s claim, G(St , Ht ) with ambiguity from the incomplete market. After considering incomplete markets and ambiguity, we obtain the present values of the incentive fee, annual fee and the investor’s claim respectively as follows: P(St , Ht ) = min EW

θ

θ

A(St , Ht ) = min E

Wθ

[∫

[∫t τ

θ

G(St , Ht ) = min E θ

τ

]

e−r(v−t) k(dHv − gHv )dv ,

]

e−r(v−t) cSv dv ,

t

Wθ

[

e

−r(τ −t)

Sτ ,

]

where τ is the stochastic liquidation time. In addition, the present value of the total fee is naturally given by F (St , Ht ) = P(St , Ht ) + A(St , Ht ).

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3. Model solution

PH = p(s) − sp′ (s), PS = p′ (s) and PSS = p′′ (s)/H into (5), the scaled present value of incentive fee p(s) solves the following ODE:

We solve the manager’s problem by using stochastic dynamic programming. Actually, we get that all the present values of the fees and the investor’s claim must be the functions of S and H. In the interior domain St < Ht , the standard dynamic programming argument implies that the present value of incentive fee P(S , H) satisfies the following Hamilton–Jacobi–Bellman (HJB) equation:

1 2 2 ′′ σ s p (s), 2 subject to the following boundary conditions:

(r + λ)P(S , H) = min (µθ − c ) SPS (S , H) θ

+

1 2

σ 2 S 2 PSS (S , H) + gHPH (S , H),

(5)

where µθ is given by (4). The HJB equation (5) implies that the value function is computed by considering the worst case in the set of the probability measures. The left side of (5) elevates the discount rate from the interest rate r to r + λ to reflect the stochastic liquidation of the fund. The first and second terms on the right side of (5) give the drift and the volatility effects of the net asset value S on P(S , H), respectively. Finally, the last term on the right side of (5) describes the effect of the HWM H change on P(S , H). Since the right-hand side is monotonic in θ , there exists a unique solution for minimization as θ = −κ . Substituting it back into (5) yields (r +λ)P(S , H) =

(

µ∗θ

1

− c SPS (S , H)+ σ S PSS (S , H)+gHPH (S , H),

)

2 2

2

where

[

µ∗θ = µ − σs hs − σw (η + κ ) = µ − σ ρ hs +

√

]

1 − ρ 2 (η + κ ) .

[7] finds that all the functions of the present values have the homogeneity property of degree one in fund value S and HWM H. That is, if we double AUM W and the high-water mark H at the same time, the present value of total fees F (W , H) will also double. Additionally, the functions are independent of time. Thus we are able to reduce the manager’s problem to one dimension. The effective state variable is therefore the ratio between the fund value S and the HWM H, s = S /H. Denote by lower cases the variables of the corresponding upper cases scaled by H and express all control variables per unit of HWM. For example, p(S , H) = P(S , H)/H denotes the scaled present value of incentive fee. Except for the scaling property, we also need some boundary conditions. The first condition indicates that when the asset value falls to the liquidation level, S(Ht ), then the investor will withdraw all his money and there is no further costs or fees, i.e. P(S(Ht ), Ht ) = 0. The second is the upper boundary (S = H). When S ≥ H, a positive return shock increases the asset value from S = H to H + ∆H. The present value of incentive fee for the manager is then given by P(H + ∆H , H) before the HWM adjusts. Immediately after the positive shock, the HWM adjusts to H + ∆H. The manager then collects the incentive fee in flow terms k∆H. Consequently the asset value is changed from H + ∆H to H + ∆H − k∆H. The present value of incentive fee is then equal to P(H + ∆H − k∆H , H + ∆H). Using the continuity of P(·, ·) before and after the adjustment of the HWM, we have P(H + ∆H , H) = k∆H + P(H + ∆H − k∆H , H + ∆H). By taking the limit as ∆H approaches zero and using Taylor’s expansion rule, we obtain

(r + λ − g)p(s) = (µ∗θ − c − g)sp′ (s) +

p(b) = 0,

(6)

p(1) = (k + 1)p (1) − k. ′

(7)

Eq. (6) states that the manager’s value is zero at the liquidation boundary b and (7) gives the condition at the upper boundary s = 1 when the manager is close to collecting incentive fees. Using a similar economic analysis and mathematical derivation, we could characterize the solutions for the annual fee, total fee and the investor’s claim respectively in the following proposition. Proposition 1. The value functions F (S , H) and G(S , H) are all homogeneous with degree one in the asset value S and HWM H, i.e., f (s) = F (S , H)/H, G(s) = G(S , H)/H, where f (s) and g(s) solve the following ODEs respectively (r + λ − g)f (s) = cs + (µ∗θ − c − g)sf ′ (s) + (r + λ − g)g(s) = λs + (µ∗θ − c − g)sg ′ (s) +

1

σ 2 s2 f ′′ (s),

2 1 2

σ 2 s2 g ′′ (s),

with the following boundary conditions: f (b) = 0,

f (1) = (k + 1)f ′ (1) − k,

g(b) = b,

g(1) = (k + 1)g ′ (1).

The present value of annual fee is A(S , H) = a(s)H, where a(s) = f (s) − p(s). As documented in [7], the hedge fund management contract has option-like characteristics. With ambiguity resulting from market incompleteness, we have the following closed-form solutions for the values of the incentive fee, total fee and the investor’s claim scaled by the present HWM H: k sγ − bγ −φ sφ

(

p(s) = f (s) =

)

γ (1 + k) − 1 − bγ −φ [φ (1 + k) − 1] c c + λ − µ∗θ − r

(

.

)s

( )] λ − µ∗ − r k + [φ (1 + k) − 1] cb1−φ ( ∗ θ)] [ ] sγ c + λ − µθ − r γ (1 + k) − 1 − bγ −φ (φ (1 + k) − 1) [ ( )] bγ −φ λ − µ∗θ − r k + [γ (1 + k) − 1]cb1−φ ( ∗ )] [ ] sφ . −[ c + λ − µθ − r γ (1 + k) − 1 − bγ −φ (φ (1 + k) − 1) [

+[

g(s) =

λ (

c + λ − µ∗θ − r

)s

[ ( )] λk + [φ (1 + k) − 1] c − µ∗θ − r b1−φ ( ∗ )] [ ] sγ c + λ − µθ − r γ (1 + k) − 1 − bγ −φ (φ (1 + k) − 1) [ ( )] bγ −φ λk + [γ (1 + k) − 1] c − µ∗θ − r b1−φ ( ∗ )] [ ] sφ +[ c + λ − µθ − r γ (1 + k) − 1 − bγ −φ (φ (1 + k) − 1) −[

where 1

µ∗ − c − g γ = − θ 2 + 2 σ

√ (

1

µ∗θ − c − g − σ2

√ (

µ∗ − c − g − θ 2 2 σ

)2

µ∗θ − c − g σ2

)2

1

1

+

2(r + λ − g)

+

2(r + λ − g)

σ2

> 1,

kPS (H , H) = k + PH (H , H).

φ=

The above is the value-matching condition for the manager on the boundary S = H. Substituting P(S , H) = p(s)H and the derivatives

In addition, the present value of the scaled annual fee is a(s) = f (s) − p(s).

2

−

2

−

σ2

< 0.

F. Liu, Y. Niu and Z. Zou / Operations Research Letters 48 (2020) 195–201

199

Fig. 1. Effect produced by a change in NAV-HWM ratio s on the values of the fee and the claim for different levels of ambiguity.

4. Model analysis In this section, we discuss the effects of market incompleteness and ambiguity on the present values of the performance fee, annual fee and the investor’s claim. Following the previous literature (e.g., [4,7,12]), we choose the risk-free rate r = 5%, a management fee rate c = 2%, an incentive fee rate k = 20%, an indexed growth rate of HWM g = 5%, the liquidation boundary b = 0.685, the expected return on the risky asset µ = 10%, the volatility σ = 15%, and a probability of liquidation λ = 10%. All the rates are annualized and continuously compounded whenever applicable. On the other hand, with respect to market incompleteness and ambiguity, following [19], we set the market price of Z risk hs = 0.5, the market price of W -risk η = 0.5, a level of κ -ignorance κ = 0.1, and the instantaneous correlation ρ = 0.5. Table 1 reports the symbols and the parameter values for the baseline quantitative exercises. Fig. 1 shows the present values of the performance fee, regular annual fee, total fee, and the investor’s claim as a function of the NAV-HWM ratio for κ = 0, 0.1, 0.2 and 0.3. As one would expect, this figure tells that the fees and the claim increase with a growth of the NAV-HWM ratio. In the meanwhile, we find from the figure that an increase in ambiguity (via an increase in κ ) decreases the values of the fees and the claim believed by a fund manager. The reason behind is related to ambiguity aversion. If the fund manager is more ambiguous over the market price of W -risk, then an ambiguous-averse firm will, in a certain sense, be more careful about the additional discount for Knightian uncertainty. Consequently, the values of the fees and the claim will be underestimated. One could think of an increase in κ as an increase in the safety margin around the estimate η. The interval [−κ, κ] might even be considered to be a confidence interval around η. Ambiguity aversion could then be interpreted as a ‘‘safety-first’’ procedure, which leads the manager to adopt the worst possible case in the confidence interval. Such an explanation would be

Table 1 Parameter values. Parameter

Symbol

Value

Risk-free rate Indexed growth rate of H Probability of liquidation Management fee parameter Incentive fee parameter Lower liquidation boundary Ambiguity parameter Expected return on risky asset The price of Z -risk The price of W -risk Correlation between market and NAV

r g

5% 5% 10% 2% 20% 0.685 0.1 10% 0.5 0.5 0.5

λ c k b

κ µ hs

η ρ

close to the approach taken by [9]. Whatever way one chooses to look at it, ambiguity aversion explicitly lowers the values of the fees and the claim. As documented in [19], there are two ways to characterize market incompleteness. First, the correlation between the partial spanning asset and the fund’s net asset value, ρ , can be parameterized as a measure of market incompleteness. Fig. 2 depicts the values of fees and the claim as a function of s for ρ = 0.1, 0.5, 0.9 and 1. We see that the effect of a change in ρ (keeping total volatility constant) on the values of the fees and the claim is non-monotonic. The intuition is due to the so-called ‘‘equilibrium-pricing’’ effect. In essence, the fund manager behaves as if he might be able to trade an asset on the market whose payoffs are uncorrelated with the partial spanning asset but own a Sharpe ratio equal to η + κ . To price these two assets forces the manager to construct a proper martingale pricing operator. Since the correlation between the assets is positive, then the variance of a combination of the two is higher than their sum. Hence, the manager should discount at a higher rate to compensate for this higher volatility. However, this volatility is non-monotonic in the correlation, leading to a non-monotonic relationship between the correlation ρ and the value of fees and the claim.

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Fig. 2. Effect produced by a change in NAV-HWM ratio s on the values of the fee and the claim for different levels of correlation.

Fig. 3. Effect produced by a change in NAV-HWM ratio s on the values of the fee and the claim for different levels of σw .

Second, one can measure market incompleteness by the undiversifiable risk σw . Fig. 3 displays the effects produced by a change in σw on p(s), a(s), f (s) and g(s), respectively. Interestingly, an change in σw has various impacts on the performance

fee, regular annual fee, total fee and the investor’s claim. The upper-left panel says that the performance fee increases in σw . First, since the value of the performance fee has option-like characteristics, an increase in the undiversifiable risk induces

F. Liu, Y. Niu and Z. Zou / Operations Research Letters 48 (2020) 195–201

‘‘option effect’’: the larger the σw , the larger the undiversifiable risk involved in the hedge fund contract, and thus the higher the ‘‘option value’’, which is in line with the conventional realoption literature. Second, an increase in σw may be understood from the perspective of a decrease in the correlation between marketed and non-marketed risks, ρ . Same as above, a change in ρ generates the non-monotonic equilibrium-pricing effect. Thus, for the performance fee, the option effect is dominant. As for the annual fee, this relationship is opposite. As the volatility increases, the average time before the liquidation barrier or the high-water mark is hit decreases, which causes a reduction in future annual fees. Although there exists equilibrium-pricing effect, the quantitative analysis indicates that this hitting effect is still dominant, implying that annual fee is decreasing in σw . Additionally, the effect of a change in σw on the investor’s claim is non-monotonic as can be seen in the bottom-right panel of Fig. 3. On the one hand, the numerical analysis in [7] shows that an increase in volatility lowers the investor’s claim. On the other hand, there is a second effect of an increase in σw from the standpoint of a decrease in ρ . In this case, the equilibriumpricing effect is dominant, which indicates that the value of the claim has an ambiguous relationship with the undiversifiable risk σw . Therefore, an increase in market completeness through an increase in the volatility of non-marketed risk has different implications on the performance fee, regular annual fee, total fee and the investor’s claim, which is in contrast to the case that an increase in market completeness is captured by the correlation between the marketed and non-marketed risk. Acknowledgements The authors thank the editor and the anonymous referee for their constructive comments and suggestions. Fengjun Liu acknowledges the support from the Yunnan Philosophy and Social Science Research Base Foundation, China (No. JD2019YB22). Zhentao Zou acknowledges the support from the Fundamental Research Funds for the Central Universities, China (No. 413000172).

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References [1] C.X. A, Z.F. Li, F. Wang, Optimal investment strategy under timeinconsistent preferences and high-water mark contract, Oper. Res. Lett. 44 (2016) 212–218. [2] E.W. Anderson, L.P. Hansen, T.J. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, J. Eur. Econom. Assoc. 1 (2003) 68–123. [3] Z.J. Chen, L. Epstein, Ambiguity, risk, and asset returns in continuous time, Econometrica 70 (2002) 1403–1443. [4] I. Drechsler, Risk choice under high-water marks, Rev. Financ. Stud. 27 (7) (2014) 2052–2096. [5] D. Ellsberg, Risk, Ambiguity and the savage axiom, Quart. J. Econ. 75 (1961) 643–669. [6] Y. Feng, B.H. Huang, H. Zhang, Hedge fund seeding with fees-for-guarantee swaps, Eur. J. Finance 25 (2019) 16–34. [7] William N. Goetzmann, Jonathan Ingersoll, Stephen A. Ross, High-water marks and hedge fund management contracts, J. Finance 58 (2003) 1685–1717. [8] L.P. Hansen, T.J. Sargent, Robust control and model uncertainty, Am. Econ. Rev. 91 (2001) 60–66. [9] L.P. Hansen, T.J. Sargent, Robustness, Princeton University Press, Princeton, NJ, 2008. [10] B.H. Huang, Y. Huang, N.J. Ju, Locked wealth, subjective valuation and managerial hedging under high-water marks: A structural model, Working paper, 2016. [11] Jonathan Ingersoll, The subjective and objective evaluation of incentive stock options, J. Bus. 79 (2006) 453–487. [12] Y.C. Lan, N. Wang, J.Q. Yang, The economics of hedge funds, J. Financ. Econ. 110 (2013) 300–323. [13] J.Y. Li, B. Liu, J.Q. Yang, Z.T. Zou, Hedge fund’s dynamic leverage decisions under time-inconsistent preferences, European J. Oper. Res. 284 (2020) 779–791. [14] J.J. Miao, A. Rivera, Robust contracts in continuous time, Econometrica 84 (2016) 1405–1440. [15] K. Nishimura, H. Ozaki, Irreversible investment and Knightian uncertainty, J. Econ. Theory 136 (2007) 668–694. [16] Y.J. Niu, J.Q. Yang, Z.T. Zou, Dynamic agency and investment theory under model uncertainty, Int. Rev. Financ. 19 (2019) 447–458. [17] D.D. Song, J.Q. Yang, Z.J. Yang, High-water marks and hedge fund management contracts with partial information, Comput. Econ. 42 (2013) 327–350. [18] M. Szydlowski, Ambiguity in dynamic contracts, Working paper, Northwestern University, 2012. [19] J.J. Thijssen, Incomplete markets ambiguity and irreversible investment, J. Econ. Dyn. Control 35 (2011) 909–921.