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Xun Yu Zhou - One of the best experts on this subject based on the ideXlab platform.
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Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances
arXiv: Methodology, 2018Co-Authors: Jose Blanchet, Lin Chen, Xun Yu ZhouAbstract:We revisit Markowitz's mean-Variance Portfolio selection model by considering a distributionally robust version, where the region of distributional uncertainty is around the empirical measure and the discrepancy between probability measures is dictated by the so-called Wasserstein distance. We reduce this problem into an empirical Variance minimization problem with an additional regularization term. Moreover, we extend recent inference methodology in order to select the size of the distributional uncertainty as well as the associated robust target return rate in a data-driven way.
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mean Variance Portfolio optimization with state dependent risk aversion
Mathematical Finance, 2014Co-Authors: Tomas Bjork, Agatha Murgoci, Xun Yu ZhouAbstract:The objective of this paper is to study the mean–Variance Portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time‐inconsistent control developed in Bjork and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.
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Mean-Variance Portfolio Selection under Partial Information
SIAM Journal on Control and Optimization, 2007Co-Authors: Jie Xiong, Xun Yu ZhouAbstract:This paper is concerned with a continuous-time mean-Variance Portfolio selection problem in a (possibly incomplete) market with multiple stocks and a bond. Only the past price movements of the stocks and the bond are the information available to the investors. A separation principle is shown to hold in this setting. Efficient strategies based on the aforementioned partial information are derived, which involve the optimal filter of the stock appreciation rate processes. The main methodological contribution of the paper is to employ the particle system representation to develop analytical and numerical approaches in obtaining the filter as well as solving the related backward stochastic differential equation.
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continuous time mean Variance Portfolio selection with bankruptcy prohibition
Mathematical Finance, 2005Co-Authors: Tomasz R Bielecki, Stanley R. Pliska, Hanqing Jin, Xun Yu ZhouAbstract:A continuous-time mean-Variance Portfolio selection problem is studied where all the market coefficients are random and the wealth process under any admissible trading strategy is not allowed to be below zero at any time. The trading strategy under consideration is defined in terms of the dollar amounts, rather than the proportions of wealth, allocated in individual stocks. The problem is completely solved using a decomposition approach. Specifically, a (constrained) Variance minimizing problem is formulated and its feasibility is characterized. Then, after a system of equations for two Lagrange multipliers is solved, Variance minimizing Portfolios are derived as the replicating Portfolios of some contingent claims, and the Variance minimizing frontier is obtained. Finally, the efficient frontier is identified as an appropriate portion of the Variance minimizing frontier after the monotonicity of the minimum Variance on the expected terminal wealth over this portion is proved and all the efficient Portfolios are found. In the special case where the market coefficients are deterministic, efficient Portfolios are explicitly expressed as feedback of the current wealth, and the efficient frontier is represented by parameterized equations. Our results indicate that the efficient policy for a mean-Variance investor is simply to purchase a European put option that is chosen, according to his or her risk preferences, from a particular class of options.
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markowitz s mean Variance Portfolio selection with regime switching a continuous time model
Siam Journal on Control and Optimization, 2003Co-Authors: Xun Yu ZhouAbstract:A continuous-time version of the Markowitz mean-Variance Portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks. The market parameters, including the bank interest rate and the appreciation and volatility rates of the stocks, depend on the market mode that switches among a finite number of states. The random regime switching is assumed to be independent of the underlying Brownian motion. This essentially renders the underlying market incomplete. A Markov chain modulated diffusion formulation is employed to model the problem. Using techniques of stochastic linear-quadratic control, mean-Variance efficient Portfolios and efficient frontiers are derived explicitly in closed forms, based on solutions of two systems of linear ordinary differential equations. Related issues such as a minimum-Variance Portfolio and a mutual fund theorem are also addressed. All the results are markedly different from those for the case when there is no regime switching. An interesting observation is, however, that if the interest rate is deterministic, then the results exhibit (rather unexpected) similarity to their no-regime-switching counterparts, even if the stock appreciation and volatility rates are Markov-modulated.
Imre Kondor - One of the best experts on this subject based on the ideXlab platform.
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replica approach to mean Variance Portfolio optimization
Journal of Statistical Mechanics: Theory and Experiment, 2016Co-Authors: Istvan Vargahaszonits, Fabio Caccioli, Imre KondorAbstract:We consider the problem of mean-Variance Portfolio optimization for a generic coVariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for r = N/T < 1, where N is the dimension of the Portfolio and T the length of the time series used to estimate the coVariance matrix. At the critical point r = 1 a phase transition is taking place. The out of sample estimation error blows up at this point as 1/(1 − r), independently of the coVariance matrix or the expected return, displaying the universality not only of the critical exponent, but also the critical point. As a conspicuous illustration of the dangers of in-sample estimates, the optimal in-sample Variance is found to vanish at the critical point inversely proportional to the divergent estimation error.
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replica approach to mean Variance Portfolio optimization
arXiv: Portfolio Management, 2016Co-Authors: Istvan Vargahaszonits, Fabio Caccioli, Imre KondorAbstract:We consider the problem of mean-Variance Portfolio optimization for a generic coVariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for $r=N/T<1$, where $N$ is the dimension of the Portfolio and $T$ the length of the time series used to estimate the coVariance matrix. At the critical point $r=1$ a phase transition is taking place. The out of sample estimation error blows up at this point as $1/(1-r)$, independently of the coVariance matrix or the expected return, displaying the universality not only of the critical index, but also the critical point. As a conspicuous illustration of the dangers of in-sample estimates, the optimal in-sample Variance is found to vanish at the critical point inversely proportional to the divergent estimation error.
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Replica approach to mean-Variance Portfolio optimization
LSE Research Online Documents on Economics, 2016Co-Authors: Istvan Varga-haszonits, Fabio Caccioli, Imre KondorAbstract:We consider the problem of mean-Variance Portfolio optimization for a generic coVariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for $r=N/T
Michael Viriato Araujo - One of the best experts on this subject based on the ideXlab platform.
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a generalized multi period mean Variance Portfolio optimization with markov switching parameters
Automatica, 2008Co-Authors: Oswaldo Luiz Do Valle Costa, Michael Viriato AraujoAbstract:In this paper, we deal with a generalized multi-period mean-Variance Portfolio selection problem with market parameters subject to Markov random regime switchings. Problems of this kind have been recently considered in the literature for control over bankruptcy, for cases in which there are no jumps in market parameters (see [Zhu, S. S., Li, D., & Wang, S. Y. (2004). Risk control over bankruptcy in dynamic Portfolio selection: A generalized mean Variance formulation. IEEE Transactions on Automatic Control, 49, 447-457]). We present necessary and sufficient conditions for obtaining an optimal control policy for this Markovian generalized multi-period mean-Variance problem, based on a set of interconnected Riccati difference equations, and on a set of other recursive equations. Some closed formulas are also derived for two special cases, extending some previous results in the literature. We apply the results to a numerical example with real data for risk control over bankruptcy in a dynamic Portfolio selection problem with Markov jumps selection problem.
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Generalized Mean-Variance Portfolio Selection Model with Regime Switching
IFAC Proceedings Volumes, 2008Co-Authors: Oswaldo Luiz Do Valle Costa, Michael Viriato AraujoAbstract:Abstract In this paper we deal with a generalized multi-period mean-Variance Portfolio selection problem with the market parameters subject to Markov random regime switchings. We present necessary and sufficient conditions for obtaining an optimal control policy for this Markovian generalized multi-period mean-Variance model, based on a recursive procedure. The analytical solution of our model provides the base for the solution of a great variety of mean-Variance formulations.
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Multi-period mean-Variance Portfolio optimization with markov switching parameters
Sba: Controle & Automação Sociedade Brasileira de Automatica, 2008Co-Authors: Oswaldo Luiz Do Valle Costa, Michael Viriato AraujoAbstract:In this paper we deal with a multi-period mean-Variance Portfolio selection problem with the market parameters subject to Markov random regime switching. We analytically derive an optimal control policy for this mean-Variance formulation in a closed form. Such a policy is obtained from a set of interconnected Riccati difference equations. Additionally, an explicit expression for the efficient frontier corresponding to this control law is identified and numerical examples are presented.
Fabio Caccioli - One of the best experts on this subject based on the ideXlab platform.
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replica approach to mean Variance Portfolio optimization
Journal of Statistical Mechanics: Theory and Experiment, 2016Co-Authors: Istvan Vargahaszonits, Fabio Caccioli, Imre KondorAbstract:We consider the problem of mean-Variance Portfolio optimization for a generic coVariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for r = N/T < 1, where N is the dimension of the Portfolio and T the length of the time series used to estimate the coVariance matrix. At the critical point r = 1 a phase transition is taking place. The out of sample estimation error blows up at this point as 1/(1 − r), independently of the coVariance matrix or the expected return, displaying the universality not only of the critical exponent, but also the critical point. As a conspicuous illustration of the dangers of in-sample estimates, the optimal in-sample Variance is found to vanish at the critical point inversely proportional to the divergent estimation error.
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replica approach to mean Variance Portfolio optimization
arXiv: Portfolio Management, 2016Co-Authors: Istvan Vargahaszonits, Fabio Caccioli, Imre KondorAbstract:We consider the problem of mean-Variance Portfolio optimization for a generic coVariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for $r=N/T<1$, where $N$ is the dimension of the Portfolio and $T$ the length of the time series used to estimate the coVariance matrix. At the critical point $r=1$ a phase transition is taking place. The out of sample estimation error blows up at this point as $1/(1-r)$, independently of the coVariance matrix or the expected return, displaying the universality not only of the critical index, but also the critical point. As a conspicuous illustration of the dangers of in-sample estimates, the optimal in-sample Variance is found to vanish at the critical point inversely proportional to the divergent estimation error.
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Replica approach to mean-Variance Portfolio optimization
LSE Research Online Documents on Economics, 2016Co-Authors: Istvan Varga-haszonits, Fabio Caccioli, Imre KondorAbstract:We consider the problem of mean-Variance Portfolio optimization for a generic coVariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for $r=N/T
R. G. Tompkins - One of the best experts on this subject based on the ideXlab platform.
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The general mean-Variance Portfolio selection problem. Discussion
Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences, 1994Co-Authors: Harry M. Markowitz, R. Lacey, J. Plymen, M. A. H. Dempster, R. G. TompkinsAbstract:This paper states the `general mean-Variance Portfolio analysis problem' and its solution, and briefly discusses its use in practice.
