Duality Methods in Portfolio Allocation with Transaction Constraints and Uncertainty

Transcription

1 Duality Methods in Portfolio Allocation with Transaction Constraints and Uncertainty Christopher W. Miller Department of Mathematics University of California, Berkeley January 9, 2014

2 Project Overview We examine optimal portfolio allocation with transaction constraints via duality methods Various models of asset returns and transaction costs Correlations and uncertainty in estimated parameters General algorithm to solve dual portfolio allocation problem.

3 Project Overview We examine optimal portfolio allocation with transaction constraints via duality methods Various models of asset returns and transaction costs Correlations and uncertainty in estimated parameters General algorithm to solve dual portfolio allocation problem. Rapidly approximate optimal allocations with a large number of assets and uncertain parameters.

4 Table of contents 1 Problem Description Lagrangian Relaxation 2 3

5 Problem Description Problem Description Lagrangian Relaxation Single-period investment model with n assets: Risk-return preference: f = f f n, Transaction costs: g = g g n, Investment constraint: w i [w i, w i ] = W i.

6 Problem Description Problem Description Lagrangian Relaxation Single-period investment model with n assets: Risk-return preference: f = f f n, Transaction costs: g = g g n, Investment constraint: w i [w i, w i ] = W i. Assume that we can rapidly optimize: min f i (w i ) + λg i (w i ). w i W i

7 Lagrangian Relaxation Problem Description Lagrangian Relaxation In general, we consider an investment problem of the form: p = min {f (w) : g(w) τ}. w W

8 Lagrangian Relaxation Problem Description Lagrangian Relaxation In general, we consider an investment problem of the form: p = min {f (w) : g(w) τ}. w W In this project, we consider the related problem: d = max min L(w, λ) λ 0 w W where L(w, λ) = f (w) + λ (g(w) τ).

9 Lagrangian Relaxation Problem Description Lagrangian Relaxation Proposition With the dual problem defined as above, p d.

10 Lagrangian Relaxation Problem Description Lagrangian Relaxation Proposition With the dual problem defined as above, p d. Proof. Let w W such that g(w ) τ and f (w ) = p.

11 Lagrangian Relaxation Problem Description Lagrangian Relaxation Proposition With the dual problem defined as above, p d. Proof. Let w W such that g(w ) τ and f (w ) = p. For all λ 0, p f (w ) + λ (g(w ) τ) min {f (w) + λ (g(w) τ)}. w W

12 Lagrangian Relaxation Problem Description Lagrangian Relaxation Proposition With the dual problem defined as above, p d. Proof. Let w W such that g(w ) τ and f (w ) = p. For all λ 0, p f (w ) + λ (g(w ) τ) min {f (w) + λ (g(w) τ)}. w W Then p max λ 0 min w W L(w, λ) = d.

13 In the binary model, we have fixed transaction costs for purchases and disallow short-sales: + if ξ < 0 g i (ξ) = 0 if ξ = 0 b i if ξ > 0.

14 In the binary model, we have fixed transaction costs for purchases and disallow short-sales: + if ξ < 0 g i (ξ) = 0 if ξ = 0 b i if ξ > 0. Theorem We can construct an optimal solution (λ, w ) to the dual problem in the binary model in time O(n).

15 In the binary model, we have fixed transaction costs for purchases and disallow short-sales: + if ξ < 0 g i (ξ) = 0 if ξ = 0 b i if ξ > 0. Theorem We can construct an optimal solution (λ, w ) to the dual problem in the binary model in time O(n). Corollary There is no duality gap in the binary model with unit transaction costs and integer-valued τ.

16 Theorem We can construct an optimal solution (λ, w ) to the dual problem in the binary model in time O(n).

17 Theorem We can construct an optimal solution (λ, w ) to the dual problem in the binary model in time O(n). Proof d = max min {f (w) + λ (g(w) τ)} λ 0 w W

18 Theorem We can construct an optimal solution (λ, w ) to the dual problem in the binary model in time O(n). Proof d = max min {f (w) + λ (g(w) τ)} λ 0 w W { } n = max λτ + min f i (w i ) + λg i (w i ) λ 0 w i W i i=1

19 Theorem We can construct an optimal solution (λ, w ) to the dual problem in the binary model in time O(n). Proof d = max min {f (w) + λ (g(w) τ)} λ 0 w W { } n = max λτ + min f i (w i ) + λg i (w i ) λ 0 w i W i i=1 { n { } } = max λτ + min f i (0), λb i + min f i (w i ). λ 0 0<w i w i i=1

20 d = max λ 0 { λτ + n min { fi 0, λb i + f + } }. i=1 i Maximization in O(n) via Quickselect algorithm.

21 In the ternary model, we have fixed transaction costs for both purchases and sales: s i if ξ < 0 g i (ξ) = 0 if ξ = 0 b i if ξ > 0. Furthermore, let us assume that we have the restriction on τ that 0 τ n max (s i, b i ). i=1

22 Proposition The solution of the dual problem under the ternary model may be written { d = 1 f 0 + min u ± (h + ) u + + (h ) u : u ± 0, for appropriate vectors f 0, h +, and h. u + + u 1, b u + + s u τ }

23 Proposition The solution of the dual problem under the ternary model may be written { d = 1 f 0 + min u ± (h + ) u + + (h ) u : u ± 0, for appropriate vectors f 0, h +, and h. Theorem u + + u 1, b u + + s u τ } We can construct an optimal solution (λ, w ) to the dual problem in the ternary model in polynomial time.

24 Proposition In the ternary model, it is not true that p = d in general.

25 Proposition In the ternary model, it is not true that p = d in general. Proposition Let (λ, w ) be an optimal solution to the dual problem obtained from the algorithm above. Then the duality gap is bounded by 0 p d λ (τ g(w )).

26 In the correlation model, we consider an objective function: f (w, ˆr) = 1 2 w Σw ˆr w, where ˆr R R n contains predicted returns and Σ is an estimate of the covariance matrix.

27 In the correlation model, we consider an objective function: f (w, ˆr) = 1 2 w Σw ˆr w, where ˆr R R n contains predicted returns and Σ is an estimate of the covariance matrix. The corresponding dual problem is: d = max λ 0 min max w W ˆr R {f (w, ˆr) + λ (g(w) τ)}

28 Branch-and-Bound Method

29 Branch-and-Bound Method

30 Branch-and-Bound Method

31 Branch-and-Bound Method

32 Tested under extra condition that n i=1 w i 1.

33 Conclusions Duality methods provide approximate allocations in difficult portfolio optimization problems: Usefulness of decomposability Opportunities for parallelization in branch-and-bound methods Approximate solution as input to solver for the primal problem.

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