Time-Consistent Mean-Variance Portfolio Selection in Discrete and - PowerPoint PPT Presentation

Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time Christoph Czichowsky Department of Mathematics ETH Zurich AnStAp 2010 Wien, 15th July 2010 Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 1 / 17

Mean-variance portfolio selection in one period Harry Markowitz (Portfolio selection, 1952): ◮ maximise return and minimise risk ◮ return=expectation ◮ risk=variance Mean-variance portfolio selection with risk aversion γ > 0 in one period: U ( ϑ ) = E [ x + ϑ ⊤ ∆ S ] − γ 2 Var[ x + ϑ ⊤ ∆ S ] = max ϑ ! Solution is the so-called mean-variance efficient strategy , i.e. ϑ := 1 � γ Cov[∆ S |F 0 ] − 1 E [∆ S |F 0 ] =: � ϑ. Question: How does this extend to multi-period or continuous time? Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 2 / 17

Basic problem Markowitz problem: � � � � � T � T − γ U ( ϑ ) = E x + 0 ϑ u dS u 2 Var x + 0 ϑ u dS u = max ! ( ϑ s ) 0 ≤ s ≤ T � T Static: criterion at time 0 determines optimal � 0 � ϑ implicitly via � g = ϑ dS . Question: more explicit dynamic description of � ϑ on [0 , T ] from � g ? Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 3 / 17

Basic problem Markowitz problem: � � � � � T � T − γ U ( ϑ ) = E x + 0 ϑ u dS u 2 Var x + 0 ϑ u dS u = max ! ( ϑ s ) 0 ≤ s ≤ T � T Static: criterion at time 0 determines optimal � 0 � ϑ implicitly via � g = ϑ dS . Question: more explicit dynamic description of � ϑ on [0 , T ] from � g ? Dynamic: Use � ϑ on (0 , t ] and determine optimal strategy on ( t , T ] via � � � � � � � T � T − γ � � U t ( ϑ ) = E x + 0 ϑ u dS u � F t 2 Var x + 0 ϑ u dS u � F t = max ! ( ϑ s ) t ≤ s ≤ T Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 3 / 17

Basic problem Markowitz problem: � � � � � T � T − γ U ( ϑ ) = E x + 0 ϑ u dS u 2 Var x + 0 ϑ u dS u = max ! ( ϑ s ) 0 ≤ s ≤ T � T Static: criterion at time 0 determines optimal � 0 � ϑ implicitly via � g = ϑ dS . Question: more explicit dynamic description of � ϑ on [0 , T ] from � g ? Dynamic: Use � ϑ on (0 , t ] and determine optimal strategy on ( t , T ] via � � � � � � � T � T − γ � � U t ( ϑ ) = E x + 0 ϑ u dS u � F t 2 Var x + 0 ϑ u dS u � F t = max ! ( ϑ s ) t ≤ s ≤ T Time inconsistent: this optimal strategy is different from � ϑ on ( t , T ]! Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 3 / 17

Basic problem Markowitz problem: � � � � � T � T − γ U ( ϑ ) = E x + 0 ϑ u dS u 2 Var x + 0 ϑ u dS u = max ! ( ϑ s ) 0 ≤ s ≤ T � T Static: criterion at time 0 determines optimal � 0 � ϑ implicitly via � g = ϑ dS . Question: more explicit dynamic description of � ϑ on [0 , T ] from � g ? Dynamic: Use � ϑ on (0 , t ] and determine optimal strategy on ( t , T ] via � � � � � � � T � T − γ � � U t ( ϑ ) = E x + 0 ϑ u dS u � F t 2 Var x + 0 ϑ u dS u � F t = max ! ( ϑ s ) t ≤ s ≤ T Time inconsistent: this optimal strategy is different from � ϑ on ( t , T ]! Time-consistent mean-variance portfolio selection: Find a strategy � ϑ , which is “optimal” for U t ( ϑ ) and time-consistent. Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 3 / 17

Previous literature Strotz (1955): “choose the best plan among those that [you] will actually follow.” → Recursive approach to time inconsistency for a different problem. In Markovian models: Deterministic functions, HJB PDEs and verification thm. Ekeland et al. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): General theory of Markovian time inconsistent stochastic optimal control problems (for various forms of time inconsistency.) Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 4 / 17

Previous literature Strotz (1955): “choose the best plan among those that [you] will actually follow.” → Recursive approach to time inconsistency for a different problem. In Markovian models: Deterministic functions, HJB PDEs and verification thm. Ekeland et al. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): General theory of Markovian time inconsistent stochastic optimal control problems (for various forms of time inconsistency.) 1) How to formulate this and to obtain the solution in a more general model? Financial market: R d -valued semimartingale S wlog. S = S 0 + M + A ∈ S 2 ( P ). � ϑ dS ∈ S 2 ( P ) } = L 2 ( M ) ∩ L 2 ( A ). Θ = Θ S := { ϑ ∈ L ( S ) | Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 4 / 17

Previous literature Strotz (1955): “choose the best plan among those that [you] will actually follow.” → Recursive approach to time inconsistency for a different problem. In Markovian models: Deterministic functions, HJB PDEs and verification thm. Ekeland et al. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): General theory of Markovian time inconsistent stochastic optimal control problems (for various forms of time inconsistency.) 1) How to formulate this and to obtain the solution in a more general model? Financial market: R d -valued semimartingale S wlog. S = S 0 + M + A ∈ S 2 ( P ). � ϑ dS ∈ S 2 ( P ) } = L 2 ( M ) ∩ L 2 ( A ). Θ = Θ S := { ϑ ∈ L ( S ) | 2) Rigorous justification of the continuous-time formulation? Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 4 / 17

Outline Discrete time 1 Continuous time 2 Convergence of solutions 3 Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 5 / 17

1 { k } ) ≥ 0 Local mean-variance efficiency in discrete time � T 0 ϑ u dS u = x + � T Use x + ϑ · S T := x + i =1 ϑ i ∆ S i and suppose d = 1. Definition A strategy � ϑ ∈ Θ is locally mean-variance efficient (LMVE) if U k − 1 ( � ϑ ) − U k − 1 ( � ϑ + δ P-a.s. for all k = 1 , . . . , T and any δ = ( ϑ − � ϑ ) ∈ Θ . allblad 2008): � Recursive optimisation (K¨ ϑ ∈ Θ is LMVE if and only if � � � ∆ S k , � T � i = k +1 � Cov ϑ i ∆ S i � F k − 1 ϑ k = 1 E [∆ S k |F k − 1 ] = 1 � γ λ k − ξ k ( � Var [∆ S k |F k − 1 ] − ϑ ) γ Var [∆ S k |F k − 1 ] for k = 1 , . . . , T . Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 6 / 17

1 { k } ) ≥ 0 Local mean-variance efficiency in discrete time � T 0 ϑ u dS u = x + � T Use x + ϑ · S T := x + i =1 ϑ i ∆ S i and suppose d = 1. Definition A strategy � ϑ ∈ Θ is locally mean-variance efficient (LMVE) if U k − 1 ( � ϑ ) − U k − 1 ( � ϑ + δ P-a.s. for all k = 1 , . . . , T and any δ = ( ϑ − � ϑ ) ∈ Θ . allblad 2008): � Recursive optimisation (K¨ ϑ ∈ Θ is LMVE if and only if � � � ∆ S k , � T � i = k +1 � Cov ϑ i ∆ S i � F k − 1 ϑ k = 1 E [∆ S k |F k − 1 ] = 1 � γ λ k − ξ k ( � Var [∆ S k |F k − 1 ] − ϑ ) γ Var [∆ S k |F k − 1 ] � �� � � � T � � � i = k +1 � E ∆ M k E ϑ i ∆ S i � F k � F k − 1 = 1 ∆ A k E [(∆ M k ) 2 |F k − 1 ] − E [(∆ M k ) 2 |F k − 1 ] γ for k = 1 , . . . , T . Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 6 / 17

Structure condition and mean-variance tradeoff process S satisfies the structure condition (SC) , i.e. there exists a predictable process λ such that � k � k � � (∆ M i ) 2 |F i − 1 A k = λ i E = λ i ∆ � M � i i =1 i =1 for k = 0 , . . . , T and the mean-variance tradeoff process (MVT) � � 2 k k k � � � E [∆ S i |F i − 1 ] λ 2 K k := Var [∆ S i |F i − 1 ] = i ∆ � M � i = λ i ∆ A i i =1 i =1 i =1 for k = 0 , . . . , T is finite valued, i.e. λ ∈ L 2 loc ( M ). If the LMVE strategy � ϑ exists, then λ ∈ L 2 ( M ), i.e. K T ∈ L 1 ( P ). Comments: 1) SC and MVT also appear naturally in other quadratic optimisation problems in mathematical finance; see Schweizer (2001). 2) No arbitrage condition: A ≪ � M � . Christoph Czichowsky (ETH Zurich) Mean-variance portfolio selection Wien, 15th July 2010 7 / 17