Optimal consumption-investment strategy under drawdown constraint - PowerPoint PPT Presentation

Optimal consumption-investment strategy under drawdown constraint Romuald ELIE University Paris Dauphine Joint work with Nizar Touzi Optimal consumption-investment strategy under drawdown constraint – p.1/22

Problem A fund manager detains an initial capital x and can Invest θ in a risky asset: dS t = σS t ( dW t + λdt ) Consume C : give dividends to investors Optimal consumption-investment strategy under drawdown constraint – p.2/22

Problem A fund manager detains an initial capital x and can Invest θ in a risky asset: dS t = σS t ( dW t + λdt ) Consume C : give dividends to investors � t � t X x,C,θ Wealth: = x − 0 C r dr + 0 σθ r ( dW r + λdr ) t Optimal consumption-investment strategy under drawdown constraint – p.2/22

Problem A fund manager detains an initial capital x and can Invest θ in a risky asset: dS t = σS t ( dW t + λdt ) Consume C : give dividends to investors � t � t X x,C,θ Wealth: = x − 0 C r dr + 0 σθ r ( dW r + λdr ) t To convince the investors, he imposes a � X x,C,θ � ∗ X x,C,θ Drawdown constraint : ≥ α t t Optimal consumption-investment strategy under drawdown constraint – p.2/22

Problem A fund manager detains an initial capital x and can Invest θ in a risky asset: dS t = σS t ( dW t + λdt ) Consume C : give dividends to investors � t � t X x,C,θ Wealth: = x − 0 C r dr + 0 σθ r ( dW r + λdr ) t To convince the investors, he imposes a � X x,C,θ � ∗ X x,C,θ Drawdown constraint : ≥ α t t What is the optimal strategy ( C, θ ) ? Optimal consumption-investment strategy under drawdown constraint – p.2/22

Problem A fund manager detains an initial capital x and can Invest θ in a risky asset: dS t = σS t ( dW t + λdt ) Consume C : give dividends to investors � t � t X x,C,θ Wealth: = x − 0 C r dr + 0 σθ r ( dW r + λdr ) t To convince the investors, he imposes a � X x,C,θ � ∗ X x,C,θ Drawdown constraint : ≥ α t t What is the optimal strategy ( C, θ ) ? Is there any admissible strategy ( C, θ ) ? Optimal consumption-investment strategy under drawdown constraint – p.2/22

Drawdown Constraint � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t Optimal consumption-investment strategy under drawdown constraint – p.3/22

Admissible strategies � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t � � � X x,C,θ � ∗ X x,C,θ ( C t , θ t ) = ( c t , π t ) − α Strategy: t t � T 0 ( c t + π t 2 ) dt < ∞ with Optimal consumption-investment strategy under drawdown constraint – p.4/22

Admissible strategies � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t � � � X x,C,θ � ∗ X x,C,θ ( C t , θ t ) = ( c t , π t ) − α Strategy: t t � T 0 ( c t + π t 2 ) dt < ∞ with � � �� � α/ (1 − α ) � X x,C,θ � ∗ X x,C,θ � ∗ X x,C,θ Key process: M t := − α t t t Optimal consumption-investment strategy under drawdown constraint – p.4/22

Admissible strategies � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t � � � X x,C,θ � ∗ X x,C,θ ( C t , θ t ) = ( c t , π t ) − α Strategy: t t � T 0 ( c t + π t 2 ) dt < ∞ with � � �� � α/ (1 − α ) � X x,C,θ � ∗ X x,C,θ � ∗ X x,C,θ Key process: M t := − α t t t dM t = M t [( λσπ t − c t ) dt + σπ t dW t ] Optimal consumption-investment strategy under drawdown constraint – p.4/22

Admissible strategies � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t � � � X x,C,θ � ∗ X x,C,θ ( C t , θ t ) = ( c t , π t ) − α Strategy: t t � T 0 ( c t + π t 2 ) dt < ∞ with � � �� � α/ (1 − α ) � X x,C,θ � ∗ X x,C,θ � ∗ X x,C,θ Key process: M t := − α t t t dM t = M t [( λσπ t − c t ) dt + σπ t dW t ] ⇒ M exists and is positive Optimal consumption-investment strategy under drawdown constraint – p.4/22

Admissible strategies � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t � � � X x,C,θ � ∗ X x,C,θ ( C t , θ t ) = ( c t , π t ) − α Strategy: t t � T 0 ( c t + π t 2 ) dt < ∞ with � � �� � α/ (1 − α ) � X x,C,θ � ∗ X x,C,θ � ∗ X x,C,θ Key process: M t := − α t t t dM t = M t [( λσπ t − c t ) dt + σπ t dW t ] ⇒ M exists and is positive �� � 1 / (1 − α ) X x,C,θ � ∗ M ∗ t = (1 − α ) t Optimal consumption-investment strategy under drawdown constraint – p.4/22

Admissible strategies � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t � � � X x,C,θ � ∗ X x,C,θ ( C t , θ t ) = ( c t , π t ) − α Strategy: t t � T 0 ( c t + π t 2 ) dt < ∞ with � � �� � α/ (1 − α ) � X x,C,θ � ∗ X x,C,θ � ∗ X x,C,θ Key process: M t := − α t t t dM t = M t [( λσπ t − c t ) dt + σπ t dW t ] ⇒ M exists and is positive �� � 1 / (1 − α ) X x,C,θ � ∗ ⇒ X x,C,θ exists M ∗ t = (1 − α ) t Optimal consumption-investment strategy under drawdown constraint – p.4/22

Literature on Drawdown � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t Optimal consumption-investment strategy under drawdown constraint – p.5/22

Literature on Drawdown � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t Long term growth rate [GZ93] [CK95] �� � p � 1 X x,C,θ u ( x ) = sup lim sup t log E t t →∞ θ ∈A D Optimal consumption-investment strategy under drawdown constraint – p.5/22

Literature on Drawdown � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t Long term growth rate [GZ93] [CK95] �� � p � 1 X x,C,θ u ( x ) = sup lim sup t log E t t →∞ θ ∈A D � � � X x,C,θ � ∗ X x,C,θ ⇒ θ t = π − α Investment t t Optimal consumption-investment strategy under drawdown constraint – p.5/22

Literature on Drawdown � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t Long term growth rate [GZ93] [CK95] �� � p � 1 X x,C,θ u ( x ) = sup lim sup t log E t t →∞ θ ∈A D � � � X x,C,θ � ∗ X x,C,θ ⇒ θ t = π − α Investment t t Intertemporal power utility [R06] �� ∞ � e − βt C tp dt u ( x ) = sup E ( C,θ ) ∈A D 0 Optimal consumption-investment strategy under drawdown constraint – p.5/22

Our Problem � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t Intertemporal general utility function �� ∞ � e − βt U ( C t ) dt u ( x ) := sup E ( C,θ ) ∈A D 0 Optimal consumption-investment strategy under drawdown constraint – p.6/22

Our Problem � t � t X x,C,θ = x − 0 C r dr + 0 σθ r ( dW r + λdr ) Wealth: t � X x,C,θ � ∗ X x,C,θ Drawdown constraint: ≥ α t t Intertemporal general utility function �� ∞ � e − βt U ( C t ) dt u ( x ) := sup E ( C,θ ) ∈A D 0 Extra dependence on current maximum X x,C,θ ≥ α Z x,z,C,θ Drawdown constraint: t t � � ∗ with Z x,z,C,θ X x,C,θ := z ∨ t t �� ∞ � e − β t U ( C t ) dt u ( x, z ) := sup E ( C,θ ) ∈A D 0 Optimal consumption-investment strategy under drawdown constraint – p.6/22

Properties �� ∞ � e − β t U ( C t ) dt u ( x, z ) := sup E ( C,θ ) ∈A D 0 Optimal consumption-investment strategy under drawdown constraint – p.7/22

Properties �� ∞ � e − β t U ( C t ) dt u ( x, z ) := sup E ( C,θ ) ∈A D 0 Assumptions: U is C 1 , increasing, concave, satisf. Inada, U (0) = 0 , xU ′ ( x ) γ with γ := 2 β and p := lim sup U ( x ) < 1 ∧ (1 − α )(1 + γ ) , λ 2 x →∞ Optimal consumption-investment strategy under drawdown constraint – p.7/22

Properties �� ∞ � e − β t U ( C t ) dt u ( x, z ) := sup E ( C,θ ) ∈A D 0 Assumptions: U is C 1 , increasing, concave, satisf. Inada, U (0) = 0 , xU ′ ( x ) γ with γ := 2 β and p := lim sup U ( x ) < 1 ∧ (1 − α )(1 + γ ) , λ 2 x →∞ Properties of the value function: Optimal consumption-investment strategy under drawdown constraint – p.7/22

Properties �� ∞ � e − β t U ( C t ) dt u ( x, z ) := sup E ( C,θ ) ∈A D 0 Assumptions: U is C 1 , increasing, concave, satisf. Inada, U (0) = 0 , xU ′ ( x ) γ with γ := 2 β and p := lim sup U ( x ) < 1 ∧ (1 − α )(1 + γ ) , λ 2 x →∞ Properties of the value function: u is defined for { αz ≤ x ≤ z } Optimal consumption-investment strategy under drawdown constraint – p.7/22