Optimal investment with high-watermark performance fee Mihai S - - PowerPoint PPT Presentation
Optimal investment with high-watermark performance fee Mihai S rbu, University of Texas at Austin based on joint work with Karel Jane cek RSJ Algorithmic Trading and Charles University Analysis, Stochastics and Applications A
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
Objective The Model Dynamic Programming Solution of the HJB and Verification Impact of the fees on the investor Conclusions
◮ 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
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
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 Xt with the hedge-fund manager (so we call it
◮ some is invested in the fund: θt at time t ◮ the rest (the money market) sits with the manager: Xt − θt
The investor makes continuous-time investments and withdrawals from the fund, which amounts to choosing the predictable θt Evolution of the total wealth for a trading strategy
◮ without high-watemark fee
dXt = θt dFt Ft , X0 = x
◮ with high-watermark proportional fee λ > 0
dXt = θt dFt
Ft − λdMt,
X0 = x Mt = max0≤s≤t(Xs ∨ m) High-watermark of the investor Mt = max
0≤s≤t(Xs ∨ m).
Observation: can be also interpreted as taxes on gains, paid right when gains are realized (pointed out by Paolo Guasoni)
(same as Guasoni and Obloj) Denote by It the paper profits from investing in the fund It = t θu dFu Fu Then Xt = x + It − λ λ + 1 max
0≤s≤t[Is − (m − x)]+
The high-watermark of the investor is Mt = m + 1 λ + 1 max
0≤s≤t[Is − (m − x)]+
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
Denote by Y = M − X the distance from paying fees. Then Y satisfies the equation: dYt = −θt dFt
Ft + (1 + λ)dMt
Y0 = m, where Y ≥ 0 and t I{Ys=0}dMs = 0, (∀) t ≥ 0. Skorohod map I· = · θu dFu Fu → (Y , M) ≈ (X, M). Remark: Y will be chosen as state in more general models.
The investor chooses
◮ have θt in the fund at time t ◮ consume at a rate γt
Observation: the high-watermark of the investor should take into account his accumulated consumption Mt = max
0≤s≤t
s γudu
dXt = θt dFt
Ft − γtdt − λdMt,
X0 = x Mt = max0≤s≤t
s
0 γudu
◮ 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” It and the accumulated consumption Ct = t γudu.
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)
Maximize discounted utility from consumption on infinite horizon A (x, m) ∋ (θ, γ) → argmax E ∞ e−βtU(γt)dt
Where U : (0, ∞) → R is the CRRA utility U(x) = x1−p 1 − p, p > 0. Finally, choose a geometric Brownian-Motion model for the fund share price dFt Ft = αdt + σdWt.
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}. dXt =
dNt = −γtdt + dMt, N0 = 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
Use Itˆ
sup
γ≥0,θ
2σ2θ2vxx−γvm
for m > x > 0 and the boundary condition −λvx(x, x) + vm(x, x) = 0. (Formal) optimal controls ˆ θ(x, m) = − α σ2 vx(x, m) vxx(x, m) ˆ γ(x, m) = I(vx(x, m) + vm(x, m))
Denote by ˜ U(y) =
p 1−py
p−1 p , y > 0 the dual function of the utility.
The HJB becomes −βv + ˜ U(vx + vm) − 1 2 α2 σ2 v2
x
vxx = 0, m > x > 0 plus the boundary condition −λvx(x, x) + vm(x, x) = 0. Observation:
◮ if there were no vm term in the HJB, we could solve it
closed-form as in Roche or Elie-Touzi using the (dual) change
◮ no closed-from solutions in our case (even for power utility)
Since we are using power utility U(x) = x1−p 1 − p, p > 0 we can reduce to one-dimension v(x, m) = x1−pv(1, m x ) and v(x, m) = m1−pv( 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
We decide to denote z = m
x ≥ 1 and
v(x, m) = x1−pu(z). Use vm(x, m) = u′(z) · x−p, vx(x, m) =
vxx(x, m) =
to get the reduced HJB −βu+˜ U
2 α2 σ2
−p(1 − p)u + 2pzu′ + z2u′′ = 0 for z > 1 with boundary condition −λ(1 − p)u(1) + (1 + λ)u′(1) = 0
ˆ π(z) = α pσ2 · (1 − p)u − zu′ (1 − p)u − 2zu′ − 1
pz2u′′ ,
ˆ c(z) = (vx + vm)− 1
p
x =
p
Optimal controls ˆ θ(x, m) = xˆ π(z), ˆ γ(x, m) = xˆ c(x, m) Objective: solve the HJB analytically and then do verification
This is the classical Merton problem. The optimal investment proportion is given by π0 α pσ2 , while the value function equals v0(x, m) = 1 1 − p c−p
0 x1−p,
0 < x ≤ m, where c0 β p − 1 2 1 − p p2 · α2 σ2 is the optimal consumption proportion. It follows that the
u0(z) = 1 1 − p c−p
0 ,
z ≥ 1.
If λ > 0 we expect that (additional boundary condition) lim
z→∞ u(z) = u0.
(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.
Theorem 2 The closed loop equation dXt = ˆ θ(Xt, Nt)dFt
Ft − ˆ
γ(Xt, Nt)dt − λdMt, X0 = x Mt = max0≤s≤t
s
0 ˆ
γ(Xt, Nt)du
Nt = Mt − s ˆ γ(Xt, Nt)du has a unique strong solution 0 < ˆ X ≤ ˆ N. Ideea of proof: use the path-wise representation together with the Itˆ
Theorem 3 The controls ˆ θ( ˆ Xt, ˆ Nt) and ˆ γ( ˆ Xt, ˆ Nt) are optimal. Idea of proof: uniform integrability. Has to be done separately for p < 1 and p > 1.
Certainty equivalent return defined by ˜ u0
α(z)
all other parameters being equal. Can be solved as ˜ α2(z) = 2σ2 p2 1 − p β p −
− 1
p
z ≥ 1. The relative size of the certainty equivalent excess return is therefore ˜ α(z) α = √ 2σp α
β p −
− 1
p
1 − p
1 2
, z ≥ 1.
Certainty equivalent initial wealth: v0(˜ x) = v(x, n) = x1−pu(z) all other parameters being equal. Can be solved as. ˜ x(z) = x · u(z) u0
1−p
= x ·
0 u(z)
1−p ,
z ≥ 1. The quantity ˜ x(z) x = u(z) u0
1−p
=
0 u(z)
1−p ,
z ≥ 1, is the relative certainty equivalent wealth.
1 1.5 2 2.5 3 0.92 0.94 0.96 0.98 1
Equivelant relat. zero fee return
Figure: Parameters: p = 3, β = 5%, α = 10%, σ = 30%, λ = 20%
1 1.5 2 2.5 3 0.9 0.92 0.94 0.96 0.98 1
Loss in value
Figure: Parameters: p = 3, β = 5%, α = 10%, σ = 30%, λ = 20%
1 1.5 2 2.5 3 0.36 0.37 0.38 0.39 0.4
Investment proportion
Figure: Parameters: p = 3, β = 5%, α = 10%, σ = 30%, λ = 20%
1 1.5 2 2.5 3 0.027 0.0275 0.028 0.0285 0.029 0.0295
Consumption proportion
Figure: Parameters: p = 3, β = 5%, α = 10%, σ = 30%, λ = 20%
1 1.5 2 2.5 3 0.94 0.96 0.98 1
Equivalent rel. zero fee return
Figure: Parameters: p = 3, β = 5%, α = 30%, σ = 30%, λ = 20%
1 1.5 2 2.5 3 0.85 0.9 0.95 1
Loss in value
Figure: Parameters: p = 3, β = 5%, α = 30%, σ = 30%, λ = 20%
1 1.5 2 2.5 3 1.1 1.15 1.2 1.25
Investment proportion
Figure: Parameters: p = 3, β = 5%, α = 30%, σ = 30%, λ = 20%
1 1.5 2 2.5 3 0.115 0.12 0.125 0.13
Consumption proportion
Figure: Parameters: p = 3, β = 5%, α = 30%, σ = 30%, λ = 20%
1 1.5 2 2.5 3 0.7 0.8 0.9 1
Figure: Parameters: p = 3, β = 5%, α = 10%, σ = 30%, λ = 200%
1 1.5 2 2.5 3 0.7 0.8 0.9 1
Loss in value
Figure: Parameters: p = 3, β = 5%, α = 10%, σ = 30%, λ = 200%
1 1.5 2 2.5 3 0.35 0.4 0.45 0.5
Investment proportion
Figure: Parameters: p = 3, β = 5%, α = 10%, σ = 30%, λ = 200%
1 1.5 2 2.5 3 0.02 0.022 0.024 0.026 0.028 0.03
Consumption proportion
Figure: Parameters: p = 3, β = 5%, α = 10%, σ = 30%, λ = 200%
Point of view of Finance:
◮ model optimal investment with high-watermark fees from the
point of view of the investor
◮ analyze the impact of the fees
Point of Mathematics:
◮ we are controlling a two-dimensional diffusion ◮ solve the problem using direct dynamic programming: first
find a smooth solution of the HJB and then do verification
with Gerard Brunick and Karel Janeˇ cek
◮ presence of (multiple and correlated) traded stocks, interest
rates and hurdles: can still be modeled as a two-dimensional diffusion problem using X and Y = M − X as state processes (reduced to one-dimension by scaling)
◮ analytic approximations when λ is small ◮ more than one fund: genuinely multi-dimensional problem
with reflection
◮ stochastic volatility, jumps, etc