Better than Dynamic Mean-Variance Policy in Market with ALL Risky Assets

Transcription

1 Better than Dynamic Mean-Variance Policy in Market with ALL Risky Assets Xiangyu Cui and Duan Li Department of Systems Engineering & Engineering Management The Chinese University of Hong Kong June 15, 2010

2 Outline 1 Introduction 2 Discrete-time Dynamic Mean-Variance Portfolio Selection 3 Pseudo Efficiency and Revised Policies 4 Conclusions

3 Introduction Outline 1 Introduction 2 Discrete-time Dynamic Mean-Variance Portfolio Selection 3 Pseudo Efficiency and Revised Policies 4 Conclusions

4 Introduction Dynamic Mean-Variance Portfolio Selection In the early 1950s, Markowitz published his pioneering work on single-period mean-variance portfolio selection, which has paved a foundation of modern financial analysis. [Li and Ng 2000] solved the mean-variance formulation of the multi-period portfolio selection problem by adopting an embedding scheme. In the same year, [Zhou and Li 2000] also solved the mean-variance formulation in continuous-time by adopting the same embedding scheme. [Zhu, Li and Wang 2003] investigated the wealth reduction phenomena associated with the optimal multi-period mean-variance policy. [Basak and Chabakauri 2008] also recognized that investors may have incentives to deviate from the optimal dynamic mean-variance policy, which is termed pre-committed optimal policy, before reaching the terminal time.

5 Introduction Time consistent dynamic risk measure [Artzner, Delbaen, Eber and Heath 1997, 1999] introduced coherent risk measures. [Föllmer and Schied 2002], [Frittelli and Rosazza Gianin 2002] further introduced convex risk measure. Although Time Consistency requirements for dynamic risk measure introduced by [Rosazza Gianin 2002], [Boda and Filar 2006], [Artzner, Delbaen, Eber, Heath, Ku 2007], [Jobert and Rogers 2008] read differently, they all have their essence rooted in Bellman s dynamic programming. [Cui, Li, Wang and Zhu 2009] introduced the concept of Time Consistency in Efficiency for mean-risk model, which is rooted in multi-objective dynamic programming, and derived a better revised mean-variance policy in markets with a riskless asset.

6 Introduction Time consistency in efficiency Definition (Time Consistency in Efficiency) Assume that (π0,, π T 1 ) is the optimal policy of min {M 0 T (π 0,..., π T 1 x 0 )+λe(x T π 0,..., π T 1, x 0 )}, λ 0. π 0,,π T 1 Risk measure M (and its overall optimal policy) is said to satisfy time consistency in efficiency, if for all t = 1,..., T 1, (π t,..., π T 1) arg min πt,...,π T 1 {M t T (π t,..., π T 1 x t ) + λ t E(x T π t,..., π T 1, x t )}, holds for some nonpositive λ t and any possible wealth level x t.

7 Discrete-time Dynamic Mean-Variance Portfolio Selection Outline 1 Introduction 2 Discrete-time Dynamic Mean-Variance Portfolio Selection 3 Pseudo Efficiency and Revised Policies 4 Conclusions

8 Discrete-time Dynamic Mean-Variance Portfolio Selection Market Setting Consider a capital market consisted of only n + 1 risky assets within a finite time horizon T. e t = ( e 0 t,..., e n t ) : the vector of random total return rates of the n + 1 risky assets during period t with known first two moments, the mean and the covariance. Vectors e t, t = 0, 1,..., T - 1, are assumed to be statistically independent. x 0 : a given initial wealth level,. x t : the wealth level at the beginning of the t-th time period. u i t (i = 1, 2,..., n): the amount invested in the ith risky asset at the beginning of the t-th time period.

9 Discrete-time Dynamic Mean-Variance Portfolio Selection Problem Formulation The dynamic mean-variance portfolio problem is given by (MV) min Var(x T x 0 ) + λe(x T x 0 ) s.t. x t+1 = e 0 t x t + P tu t, t = 0, 1,..., T 1, (1) x 0 > 0 is given, where P t = (P 1 t, P 2 t,..., P n t ) = ((e 1 t e 0 t ), (e 2 t e 0 t ),..., (e n t e 0 t )) satisfies E(P t P t) 0, t = 0, 1,..., T 1, E((e 0 t ) 2 ) E(e 0 t P t)e 1 (P t P t)e(e 0 t P t ) > 0, t = 0, 1,..., T 1.

10 Discrete-time Dynamic Mean-Variance Portfolio Selection Pre-committed Optimal Policy The pre-committed optimal policy for (MV) [Li and Ng 2000]: ( ) u t (x t ) = E 1 (P t P t)e(e 0 µt+1 t P t )x t + Γ E 1 (P t P t)e(p t ). τ t+1 ( ) where Γ = 1 2 b 0 x 0 ν 0λ 2a 0 is termed risk attitude parameter. Furthermore, [Li and Ng 2000] give the minimum variance set of (MV) explicitly as follows, Var(x T x 0 ) = a 0 ν0 2 (E(x T x 0 ) (µ 0 + b 0 ν 0 )x 0 ) 2 + c 0 x0. 2 (2) It is easy to verify that, when E(x T x 0 ) (µ 0 + b 0 ν 0 )x 0, the mean-variance pair is efficient.

11 Discrete-time Dynamic Mean-Variance Portfolio Selection Parameters Define the following parameters: B t = E(P t)e 1 (P t P t)e(p t ) > 0, A 1 t = E(e 0 t ) E(P t)e 1 (P t P t)e(e 0 t P t ), A 2 t = E((e 0 t ) 2 ) E(e 0 t P t)e 1 (P t P t)e(e 0 t P t ) > 0, T 1 T 1 T 1 µ t = A 1 t, ν t = B 1 k, τ t = k=t k=t j=k+1 A 1 j T 1 a t = ν t 2 (v t) 2, b t = µ tν t a t = 2µ t 1 2ν t, c t = τ t (µ t ) 2 a t (b t ) 2. k=t A 2 k,

12 Pseudo Efficiency and Revised Policies Outline 1 Introduction 2 Discrete-time Dynamic Mean-Variance Portfolio Selection 3 Pseudo Efficiency and Revised Policies 4 Conclusions

13 Pseudo Efficiency and Revised Policies Three dimensional objective space The pre-committed mean-variance efficient pair, which satisfies equation (2) for (E(x T x 0 ), Var(x T x 0 )), is Pareto-optimal in the objective space of { max (expected terminal wealth), min (variance of the terminal wealth)}. In the real world, we d better consider the efficiency in an expanded three-dimensional objective space: { min (initial investment level), max (expected terminal wealth), min (variance of the terminal wealth)}.

14 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 1) Pseudo Efficiency (Type 1) Consider the revised mean-variance portfolio selection, (RMV 1 ) min Var(x T y 0 ) + λe(x T y 0 ) s.t. x t+1 = e 0 t x t + P tu t, t = 1, 2,..., T 1, x 1 = e 0 0y 0 + P 0 u 0, y 0 x 0. Definition For a wealth level x 0, if an efficient mean-variance pair for (MV) is dominated by a mean-variance pair of problem (RMV 1 ), i.e., ( x 0, E(x T x 0 ), Var(x T x 0 )) ( y 0, E(x T y 0 ), Var(x T y 0 )), (3) the given T-period mean-variance pair is termed pseudo efficient (type 1).

15 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 1) The Existence of Pseudo Efficiency (Type 1) Proposition Pseudo efficiency condition (3) x 0 > x 0 = Γµ 0/τ 0. For a given positive initial wealth x 0, condition x 0 > x 0 does not hold when 2(µ2 0 (1 2ν 0)) x 0 < 0, if µ 0 > 0, λ = µ 0 2(µ2 0 (1 2ν 0)) x 0 > 0, if µ 0 < 0. µ 0 Remark The concept of pseudo efficiency (type 1) can be extended to truncated (T s)-period problem.

16 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 1) The First Type of Revised Policies Proposed T-period revised portfolio policy for (MV): ( ) û k( x k) = E 1 (P k P k)e(e 0 µk+1 kp k ) x k + Γ k E 1 (P k P k)e(p k ); (4) τ k+1 x k, if x k x k, x k = 2µk(µkxk + 2νkΓk 1) x k +, if x 2ν kτ k + µ 2 k > x k, k x 0 = x 0 x k+1 = e 0 k x k + P kû k( x k), Γ k 1, if x k x k, Γ k = Γ k 1 + 2µkτk(xk x k ), if x 2ν kτ k + µ 2 k > x k, k Γ 1 = 1 ( ) b 0x 0 λ0ν0 2 2a 0 x k = Γk 1µk τ k.

17 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 1) Scheme Illustration of the First Revised Policy E(xT xs) ˆxs xs E(xT xs) ˆxs (E(xT xs), Var(xT xs)) 0 Var(xT xs) (E(xT xs), Var(xT xs)) xs 0 Var(xT xs) (a) µ k > 0 (b) µ k < 0 Figure: The scheme of the first revised policy

18 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 1) Performance The first type of revised policies keeps the conditional mean and variance unchanged, thus achieving the same mean-variance pair as does the pre-committed optimal mean-variance policy of the T-period problem (MV), while having a possibility to take positive free cash flow stream, {x k x k }, out of the market during the investment process, i.e., E( x T x 0 ) û = E(x T x 0 ) u, Var( x T x 0 ) û = Var(x T x 0 ) u, P{ N 1 k=1 [(x k x k ) + > 0] x 0 } > 0.

19 then the given T-period mean-variance pair is called pseudo efficient (type 2). Better than Dynamic Mean-Variance Policy in Market with ALL Risky Assets Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 2) Pseudo Efficiency (Type 2) Consider the revised mean-variance portfolio selection, (RMV 2 ) min Var(x T + x 0 y 0 x 0 ) + E(x T + x 0 y 0 x 0 ) s.t. x t+1 = e 0 t x t + P tu t, t = 1, 2,..., T 1, x 1 = e 0 0y 0 + P 0 u 0, y 0 x 0. Definition For a wealth level x 0, if an efficient mean-variance pair for (MV) is not pseudo efficient (type 1) and is, however, dominated by a total mean-variance pair of problem (RMV 2 ), i.e., (E(x T x 0 ), Var(x T x 0 )) (E(x T + x 0 y 0 x 0 ), Var(x T + x 0 y 0 x 0 )),

20 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 2) The Existence of Pseudo Efficiency (Type 2) Proposition Pseudo efficiency (type 2) condition (τ 0 µ 0 )x 0 > (µ ν 0 )Γ. For a given positive initial wealth x 0, condition (τ 0 µ 0 )x 0 > (µ ν 0 )Γ does not hold when 2(µ2 0 τ 0(1 2ν 0 )) x 0 < 0, if µ 0 > 1 2ν 0, λ = µ ν 0 2(µ2 0 τ 0(1 2ν 0 )) x 0 > 0, if µ 0 < 1 2ν 0. µ ν 0

21 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 2) The Second Type of Revised Policies Proposed T-period revised portfolio policy, ǔ k(ˇx k), k = 0,..., T 1: ( ) ǔ k(ˇx k) = E 1 (P k P k)e(e 0 µk+1 kp k )ˇx k + Γ k E 1 (P k P k)e(p k ); (5) τ k+1 x k, if (τ k µ k) x k (µ k 1 + 2ν k)γ k 1, ˇx k = (µ k 1 + 2ν k)[(µ k 1) x k + 2ν kγ k 1], 2ν k(τ k 1) + (µ k 1) 2 if (τ k µ k) x k > (µ k 1 + 2ν k)γ k 1, x 0 = x 0 x k+1 = e 0 kˇx k + P kǔ k(ˇx k), Γ k 1, if (τ k µ k) x k (µ k 1 + 2ν k)γ k 1, Γ k = (τ k µ k)[(µ k 1) x k + 2ν kγ k 1], if (τ 2ν k(τ k 1) + (µ k 1) 2 k µ k) x k > (µ k 1 + 2ν k)γ k 1, Γ 1 = 1 ( ) b 0x 0 λ0ν0. 2 2a 0

22 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 2) Scheme Illustration of the Second Revised Policy E(xT xk) E(xT xk) xk xk (E(xT xk), Var(xT xk)) xs xs (E(xT xk), Var(xT xk)) 0 Var(xT xk) 0 Var(xT xk) (a) µ k > 0 (b) µ k < 0 Figure: The scheme of the second revised policy

23 Pseudo Efficiency and Revised Policies Pseudo Efficiency (Type 2) Performance Denote x k = x k ˇx k. The second type of revised policies achieves the same total mean as the pre-committed optimal mean-variance policy of the T-period problem (MV) does, while having smaller total variance than the pre-committed optimal policy does, i.e., T 1 E( x T + x j x 0 ) ǔ = E(x T x 0 ) u, j=0 T 1 Var( x T + x j x 0 ) ǔ < Var(x T x 0 ) u. j=0

24 Conclusions Outline 1 Introduction 2 Discrete-time Dynamic Mean-Variance Portfolio Selection 3 Pseudo Efficiency and Revised Policies 4 Conclusions

25 Conclusions Conclusion The dynamic mean-variance portfolio selection in markets with all risky assets is not time consistent in efficiency, due to the inherent nonseparable nature of the involved variance term. By adding the initial investment level into the objective space, the concept of pseudo efficiency (type 1 or type 2) has been introduced. By relaxing the self-financing constraint, two revised policies have been proposed to tackle pseudo efficiency (type 1 or type 2), thus achieving better performance than the original dynamic mean-variance policy.

26 Conclusions Thank you for your attention!