Monte Carlo methods for mean-risk optimization and portfolio selection

Xu, Huifu; Zhang, Dali

doi:10.1007/s10287-010-0123-6

Cited by 15 publications

(12 citation statements)

References 23 publications

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“…Another field of research deals with sample average approximations, where ambiguity sets appear as confidence regions, see e.g. [5,37,59]. An overview of these and related topics can be found in the book by Consigli, Kuhn and Brandimarte [8].…”

Section: Ambiguity Setsmentioning

confidence: 99%

A Review on Ambiguity in Stochastic Portfolio Optimization

Pflug

Pohl

2017

Set-Valued Var. Anal

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In mean-risk portfolio optimization, it is typically assumed that the assets follow a known distribution P 0 , which is estimated from observed data. Aiming at an investment strategy which is robust against possible misspecification of P 0 , the portfolio selection problem is solved with respect to the worst-case distribution within a Wasserstein-neighborhood of P 0 . We review tractable formulations of the portfolio selection problem under model ambiguity, as it is called in the literature. For instance, it is known that high model ambiguity leads to equally-weighted portfolio diversification. However, it often happens that the marginal distributions of the assets can be estimated with high accuracy, whereas the dependence structure between the assets remains ambiguous. This leads to the problem of portfolio selection under dependence uncertainty. We show that in this case portfolio concentration becomes optimal as the uncertainty with respect to the estimated dependence structure increases. Hence, distributionally robust portfolio optimization can have two very distinct implications: Diversification on the one hand and concentration on the other hand.

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Section: Ambiguity Setsmentioning

confidence: 99%

A Review on Ambiguity in Stochastic Portfolio Optimization

Pflug

Pohl

2017

Set-Valued Var. Anal

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“…In this section we give new versions of the strong uniform law of large numbers with explicit rate of convergence and explicit tail behavior. Related results for F n (x, ξ n ) can be found in [2,6,21,27,36,43,46,47,49,50,51,52,55] and for H n (x, ξ n ) in [33,56].…”

Section: Calibration Of Complex Stochastic Simulation Modelsmentioning

confidence: 70%

“…max {0, y i } and some sufficiently large penalty coefficient L. For stochastic optimization problems such trick was used, e.g., in [35,56]; detailed discussion of the penalty function method can be found in [38].…”

mentioning

confidence: 99%

Sample Average Approximation Method for Compound Stochastic Optimization Problems

Ermoliev¹,

Norkin²

2013

SIAM J. Optim.

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The paper studies stochastic optimization (programming) problems with compound functions containing expectations and extreme values of other random functions as arguments. Compound functions arise in various applications. A typical example is a variance function of nonlinear outcomes. Other examples include stochastic minimax problems, econometric models with latent variables, multi-level and multicriteria stochastic optimization problems. As a solution technique a Sample Average Approximation (SAA) method (also known as statistical or empirical (sample) mean method) is used. The method consists in approximation of all expectation functions by their empirical means and solving the resulting approximate deterministic optimization problems. In stochastic optimization this method is widely used for optimization of standard expectation functions under constraints. In this paper SAA method is extended to general compound stochastic optimization problems. The conditions for convergence in mean, almost surely and rate of convergence are established. The study of the convergence rate is based on properties of Rademacher averages of functional sets, concentration inequalities for bounded random functions and on the concept of uniform normalized convergence of random variables. The convergence results are applicable both for discrete and continuous stochastic optimization problems.

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“…are closed and belong to the compact set {w + , w − ∈ R m : 1 T m(w + + w − ) ≤ η, w + , w − ≥ 0}. Moreover, we have uniform boundness of (y 1 −y 3 , y 2 −y 4 , y 5 ), and (y 1,N −y 3,N , y 2,N − y 4,N , y 5,N ) for N sufficiently large by Proposition 2.2 in[30] and the discussion below Proposition 3, we have S and S N are nonempty for N sufficiently large.Moreover, by Proposition 3, we have that ς N → ς and γ N → γ as N → ∞ w.p.1. Then by the uniform law of large numbers [4, Chapter 6, Proposition 7], the constraint functions of problem(14) uniformly converge to the constraint functions of problem(13).…”

mentioning

confidence: 71%

“…The result is directly implied by Proposition 2.3 in [30]. Note that in Proposition 2.3 of [30], they use the condition named "no nonzero abnormal multipliers constraint qualification (NNAMCQ)" to bound the Lagrangian multipliers. This constraint qualification is well known, see [5] and [32].…”

mentioning

confidence: 94%

Sparse markowitz portfolio selection by using stochastic linear complementarity approach

Wang¹,

Sun²

2018

Journal of Industrial &Amp; Management Optimization

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We consider the framework of the classical Markowitz mean-variance (MV) model when multiple solutions exist, among which the sparse solutions are stable and cost-efficient. We study a two-phase stochastic linear complementarity approach. This approach stabilizes the optimization problem, finds the sparse asset allocation that saves the transaction cost, and results in the solution set of the Markowitz problem. We apply the sample average approximation (SAA) method to the two-phase optimization approach and give detailed convergence analysis. We implement this methodology on the data sets of Standard and Poor 500 index (S&P 500), real data of Hong Kong and China market stocks (HKCHN) and Fama & French 48 industry sectors (FF48). With mock investment in training data, we construct portfolios, test them in the out-of-sample data and find their Sharpe ratios outperform the 2010 Mathematics Subject Classification. Primary: 90C15, 90C90; Secondary: 91G10.

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Monte Carlo methods for mean-risk optimization and portfolio selection

Cited by 15 publications

References 23 publications

A Review on Ambiguity in Stochastic Portfolio Optimization

A Review on Ambiguity in Stochastic Portfolio Optimization

Sample Average Approximation Method for Compound Stochastic Optimization Problems

Sparse markowitz portfolio selection by using stochastic linear complementarity approach

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