Optimal investment with high-watermark performance fee Mihai S - PowerPoint PPT Presentation

Optimal investment with high-watermark performance fee Mihai Sˆ ırbu, University of Texas at Austin based on joint work with Karel Janeˇ cek RSJ Algorithmic Trading and Charles University Analysis, Stochastics and Applications A Conference in Honor of Walter Schachermayer Vienna, July 12-16, 2010

Outline Objective The Model Dynamic Programming Solution of the HJB and Verification Impact of the fees on the investor Conclusions

Objective ◮ build and analyze a model of optimal investment and consumption where the investment opportunity is represented by a hedge-fund using the ”two-and-twenty rule” ◮ analyze the impact of the high-watermark fee on the investor

Previous work on hedge-funds and high-watermarks All existing work analyzes the impact/incentive of the high-watermark fees on fund managers ◮ extensive finance literature ◮ Goetzmann, Ingersoll and Ross, Journal of Finance 2003 ◮ Panagea and Westerfield, Journal of Finance 2009 ◮ Agarwal, Daniel and Naik Journal of Finance, forthcoming ◮ Aragon and Qian, preprint 2007 ◮ recently studied in mathematical finance ◮ Guasoni and Obloj, preprint 2009

The model: investment opportunities An investor with investment opportunities ◮ the hedge fund with fund share value process F , given exogenously ◮ the money market paying interest rate zero Observation: since the money market pays zero rate, the investor leaves all the wealth X t with the hedge-fund manager (so we call it one investment opportunity) ◮ some is invested in the fund: θ t at time t ◮ the rest (the money market) sits with the manager: X t − θ t

The model: dynamic trading strategies The investor makes continuous-time investments and withdrawals from the fund, which amounts to choosing the predictable θ t Evolution of the total wealth for a trading strategy ◮ without high-watemark fee dF t dX t = θ t , X 0 = x F t ◮ with high-watermark proportional fee λ > 0 � dX t = θ t dF t F t − λ dM t , X 0 = x M t = max 0 ≤ s ≤ t ( X s ∨ m ) High-watermark of the investor M t = max 0 ≤ s ≤ t ( X s ∨ m ) . Observation: can be also interpreted as taxes on gains, paid right when gains are realized (pointed out by Paolo Guasoni)

Path-wise solutions of the state equation (same as Guasoni and Obloj) Denote by I t the paper profits from investing in the fund � t dF u I t = θ u F u 0 Then λ 0 ≤ s ≤ t [ I s − ( m − x )] + X t = x + I t − λ + 1 max The high-watermark of the investor is 1 0 ≤ s ≤ t [ I s − ( m − x )] + M t = m + λ + 1 max Observations: ◮ the fee λ can exceed 100% and the investor can still make a profit ◮ the high-watemark is measured before the fee is paid

Connection to the Skorohod map Denote by Y = M − X the distance from paying fees. Then Y satisfies the equation: � dY t = − θ t dF t F t + (1 + λ ) dM t Y 0 = m , where Y ≥ 0 and � t I { Y s � =0 } dM s = 0 , ( ∀ ) t ≥ 0 . 0 Skorohod map � · dF u I · = θ u → ( Y , M ) ≈ ( X , M ) . F u 0 Remark: Y will be chosen as state in more general models.

The model: investment and consumption The investor chooses ◮ have θ t in the fund at time t ◮ consume at a rate γ t Observation: the high-watermark of the investor should take into account his accumulated consumption � s �� � � M t = max X s + γ u du ∨ m 0 ≤ s ≤ t 0 The evolution of the wealth is � dX t = θ t dF t F t − γ t dt − λ dM t , X 0 = x � s �� � � M t = max 0 ≤ s ≤ t X s + 0 γ u du ∨ m

Model: cont’d ◮ consumption is a part of the running-max, as opposed to the literature on draw-dawn constraints ◮ Grossman and Zhou ◮ Cvitanic and Karatzas ◮ Elie and Touzi ◮ Roche ◮ we still have a similar path-wise representation for the wealth in terms of the ”paper profit” I t and the accumulated consumption � t C t = γ u du . 0

Optimal investment and consumption Admissible strategies A ( x , m ) = { ( θ, γ ) : X > 0 } . Can represent investment and consumption strategies in terms of proportions c = γ/ X , π = θ. Obervation: ◮ no closed form path-wise solutions for X in terms of ( π, c ) (unless c = 0)

Optimal investment and consumption:cont’d Maximize discounted utility from consumption on infinite horizon �� ∞ � e − β t U ( γ t ) dt A ( x , m ) ∋ ( θ, γ ) → argmax E . 0 Where U : (0 , ∞ ) → R is the CRRA utility U ( x ) = x 1 − p 1 − p , p > 0 . Finally, choose a geometric Brownian-Motion model for the fund share price dF t = α dt + σ dW t . F t

Dynamic programming: state processes First guess: state ( X , M , C ). High-watermark fees are paid only when X + C = M so we can actually choose as only states X and N = M − C . The state process ( X , N ) is a two-dimensional controlled diffusion with reflection on { X = N } . � dX t = � � θ t α − γ t dt + θ t σ dW t − λ dM t , X 0 = x dN t = − γ t dt + dM t , N 0 = m Recall we have path-wise solutions in terms of ( θ, γ ). Objective: expect to find the value function v ( x , m ) as a solution of the HJB, and find the (feed-back) optimal controls.

Dynamic programming equation Use Itˆ o and write formally the HJB � � − β v + U ( γ ) + ( αθ − γ ) v x + 1 2 σ 2 θ 2 v xx − γ v m sup = 0 γ ≥ 0 ,θ for m > x > 0 and the boundary condition − λ v x ( x , x ) + v m ( x , x ) = 0 . (Formal) optimal controls θ ( x , m ) = − α v x ( x , m ) ˆ σ 2 v xx ( x , m ) ˆ γ ( x , m ) = I ( v x ( x , m ) + v m ( x , m ))

HJB cont’d p − 1 Denote by ˜ p p , y > 0 the dual function of the utility. U ( y ) = 1 − p y The HJB becomes α 2 v 2 U ( v x + v m ) − 1 − β v + ˜ x = 0 , m > x > 0 σ 2 2 v xx plus the boundary condition − λ v x ( x , x ) + v m ( x , x ) = 0 . Observation: ◮ if there were no v m term in the HJB, we could solve it closed-form as in Roche or Elie-Touzi using the (dual) change of variable y = v x ( x , m ) ◮ no closed-from solutions in our case (even for power utility)

Reduction to one-dimension Since we are using power utility U ( x ) = x 1 − p 1 − p , p > 0 we can reduce to one-dimension v ( x , m ) = x 1 − p v (1 , m x ) and v ( x , m ) = m 1 − p v ( x m , 1) ◮ first is nicer economically (since for λ = 0 we get a constant function v (1 , m x )) ◮ the second gives a nicer ODE (works very well if there is a closed form solution, see Roche) There is no closed form solution, so we can choose either one-dimensional reduction.

Reduction to one-dimension cont’d We decide to denote z = m x ≥ 1 and v ( x , m ) = x 1 − p u ( z ) . Use v m ( x , m ) = u ′ ( z ) · x − p , � � (1 − p ) u ( z ) − zu ′ ( z ) · x − p , v x ( x , m ) = � � − p (1 − p ) u ( z ) + 2 pzu ′ ( z ) + z 2 u ′′ ( z ) · x − 1 − p , v xx ( x , m ) = to get the reduced HJB (1 − p ) u − zu ′ � 2 � α 2 − 1 − β u +˜ (1 − p ) u − ( z − 1) u ′ ) � � U − p (1 − p ) u + 2 pzu ′ + z 2 u ′′ = 0 2 σ 2 for z > 1 with boundary condition − λ (1 − p ) u (1) + (1 + λ ) u ′ (1) = 0

(Formal) optimal proportions (1 − p ) u − zu ′ α π ( z ) = ˆ p σ 2 · p z 2 u ′′ , (1 − p ) u − 2 zu ′ − 1 c ( z ) = ( v x + v m ) − 1 p (1 − p ) u − ( z − 1) u ′ � − 1 � ˆ = p x Optimal controls ˆ θ ( x , m ) = x ˆ π ( z ) , ˆ γ ( x , m ) = x ˆ c ( x , m ) Objective: solve the HJB analytically and then do verification

Solution of the HJB for λ = 0 This is the classical Merton problem. The optimal investment proportion is given by α π 0 � p σ 2 , while the value function equals 1 1 − p c − p 0 x 1 − p , v 0 ( x , m ) = 0 < x ≤ m , where · α 2 c 0 � β p − 1 1 − p p 2 σ 2 2 is the optimal consumption proportion. It follows that the one-dimensional value function is constant 1 1 − p c − p u 0 ( z ) = 0 , z ≥ 1 .

Solution of the HJB for λ > 0 If λ > 0 we expect that (additional boundary condition) z →∞ u ( z ) = u 0 . lim (For very large high-watermark, the investor gets almost the Merton expected utility) Theorem 1 The HJB has a smooth solution. Idea of solving the HJB: ◮ find a viscosity solution using Perron’s method ◮ show that the viscosity solution is C 2 Avoid the Dynamic Programming Principle.

Verification Theorem 2 The closed loop equation � dX t = ˆ θ ( X t , N t ) dF t F t − ˆ γ ( X t , N t ) dt − λ dM t , X 0 = x � s �� � � M t = max 0 ≤ s ≤ t X s + 0 ˆ γ ( X t , N t ) du ∨ m where � s N t = M t − ˆ γ ( X t , N t ) du 0 has a unique strong solution 0 < ˆ X ≤ ˆ N . Ideea of proof: use the path-wise representation together with the Itˆ o-Picard theory. Theorem 3 The controls ˆ θ ( ˆ X t , ˆ γ ( ˆ X t , ˆ N t ) and ˆ N t ) are optimal. Idea of proof: uniform integrability. Has to be done separately for p < 1 and p > 1.

The impact of fees Certainty equivalent return defined by � � ˜ u 0 α ( z ) ˜ = u ( z ) all other parameters being equal. Can be solved as α 2 ( z ) = 2 σ 2 p 2 � β � � − 1 � ˜ p − (1 − p ) u ( z ) , z ≥ 1 . p 1 − p The relative size of the certainty equivalent excess return is therefore 1 √ � − 1   β 2 � p − (1 − p ) u ( z ) p α ( z ) ˜ 2 σ p = , z ≥ 1 .   α α 1 − p