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International Journal of Mathematics in Operational Research (17575850)30(3)pp. 392-414
In a financial market, the state of the underlying economy and investors’ mood affect market trends and consequently asset prices movements. Regime-switching models are used to describe changes in market states and trends. The main assumption in regime-switching models is that asset returns depend on the current state of the market. We generalise this assumption to the case where market states in the past, as well as the current state, affect asset returns. In fact, we assume that asset returns are market path-dependent. Under this assumption, we study a multi-period mean-variance portfolio selection problem in a Markovian regime-switching market when the time horizon is uncertain. Using the stochastic dynamic programming approach, we obtain the path-dependent optimal portfolio strategy and the mean-variance efficient frontier in a closed form. We show that the results obtained under conventional regime-switching model, can be obtained as special cases of the present model. Copyright © 2025 Inderscience Enterprises Ltd.
Journal of Mathematical Modeling (2345394X)13(3)pp. 485-496
Asset prices typically follow significant trends influenced by the economic environment or overall investor sentiment. Regime-switching is commonly employed to capture asset price dynamics, as it effectively describes significant trends and reflects the changing correlations of asset returns over various periods. This paper explores multi-period mean-variance portfolio optimization under regime-switching with path-dependent returns. Unlike conventional models, this paper assumes that asset returns depend on the entire path of market states rather than just the current one. Consequently, investors base their decisions on all observed states up to the current moment. Utilizing dynamic programming techniques, we derive the path-dependent optimal portfolio strategy and the mean-variance efficient frontier in closed form. Furthermore, we demonstrate that the results from the traditional regime-switching model, can be viewed as specific cases of our proposed model. © 2025 University of Guilan.
Journal of the Operations Research Society of China (21946698)
This paper considers the Markowitz’s mean–variance portfolio selection model in a multi-period setting with regime switching and uncertain time horizon. The returns of the assets depend on the state of the market modulated by a discrete-time Markov chain with a finite state space. The exit time from the market is a stopping time with respect to the market state filtration. So, the definitive decision to exit the market at any time depends only on the market states up to that time. The original problem with uncertain exit time is reformulated as a problem with certain exit time. The Lagrange duality method and the dynamic programming approach are used to derive explicit closed-form expressions for the efficient investment strategy and the mean–variance efficient frontier. Toward this objective, a market path-dependent value function method is introduced and it is shown that optimal portfolios are market path dependent. The two cases where the exit time is the first hitting time to a specified subset of the market state space and the market state contains a bankruptcy state are investigated separately. Moreover, it is shown that some results in the existing literature can be obtained as special cases of the results in this paper. Finally, some numerical examples are presented to illustrate the results. © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2024.
Market conditions profoundly influence investors’ decisions regarding market participation and exit strategies. This paper investigates the multi-period mean-variance portfolio selection problem within a regime-switching market framework, where the time horizon is uncertain and the exit time depends on observed market states. Although the exit time is market path-dependent, we do not regard it as a stopping time with respect to the market state filtration, and we also include exogenous stochastic factors in its determination. The Lagrangian duality method and dynamic programming are utilized to derive the analytical expressions for the optimal investment strategy and the mean-variance efficient frontier. A path-dependent version of the Bellman equation is derived, demonstrating that, at any given time, both the value function and the optimal portfolio are path-dependent. This is different from the standard regime-switching model, where they depend on the current state of the market at that time. Our framework encompasses models with current state-dependent exit times as a special case. This study provides a detailed case study analyzing the effectiveness of the exit mechanism and its implications on investment returns under different market scenarios. © The Author(s), under exclusive licence to Operational Research Society of India 2024.
International Journal of Mathematics in Operational Research (17575850)18(3)pp. 336-359
In this paper, we study optimal multi-period portfolio selection problem with uncertain exit-time under mean-variance criterion in a Markovian regime-switching market. The market state space contains an absorbing state which represents the bankruptcy state. It is assumed that all random key parameters, i.e., asset returns, the recovery rate, and the exit-time depend on the current market state. Three common mean-variance formulations are considered, i.e., minimum variance formulation, maximum expected return formulation and the trade-off formulation. First, the problem with an uncertain exit-time is reformulated as a problem with a certain exit-time. Then, by applying the Lagrange duality method and the dynamic programming approach, the optimal multi-period portfolio strategies and the efficient frontier are derived in a closed form. Moreover, the conditions under which the aforementioned three problems are (mutually) equivalent are given. A numerical example is provided to illustrate the results. Copyright © 2021 Inderscience Enterprises Ltd.
Operational Research (11092858)20(3)pp. 1231-1254
This paper studies three versions of the multi-period mean–variance portfolio selection problem, that are: minimum variance problem, maximum expected return problem and the trade-off problem, in a Markovian regime switching market where the exit-time is uncertain and exogenous. The underlying Markov chain contains an absorbing state, which denotes the bankruptcy state. When the Markov chain switches to this state, the investors only get a random fraction, known as the recovery rate, taking values in [0, 1] of their wealth. Asset returns, as well as recovery rate, depend on the market state. Dynamic programming and Lagrange duality method are used to derive analytical expressions for optimal investment strategies and the mean–variance efficient frontier. It is shown that portfolio selection models with no bankruptcy state and certain exit-time can be considered as special cases of our model. Some numerical examples are provided to demonstrate the effect of the recovery rate and exit-probabilities. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
RAIRO - Operations Research (28047303)53(4)pp. 1171-1186
In this paper, we deal with multi-period mean-variance portfolio selection problems with an exogenous uncertain exit-time in a regime-switching market. The market is modelled by a non-homogeneous Markov chain in which the random returns of assets depend on the states of the market and investment time periods. Applying the Lagrange duality method, we derive explicit closed-form expressions for the optimal investment strategies and the efficient frontier. Also, we show that some known results in the literature can be obtained as special cases of our results. A numerical example is provided to illustrate the results. © EDP Sciences, ROADEF, SMAI 2019.
Pakistan Journal Of Statistics And Operation Research (22205810)14(2)pp. 399-414
The only source of uncertainty in the standard Markowitz's static Mean-Variance portfolio selection model is the future price of assets. This paper studies the static Mean-Variance portfolio selection model under general sources of uncertainty which generalizes the Markowitz's model. It is shown that how the generalized problem can be reformulated as a quadratic program. Sufficient conditions are provided under which the standard and the generalized models produce the same set of optimal portfolios. Some sources of uncertainty and relevant examples are investigated. An illustrative example is provided to demonstrate the model. © 2018, University of the Punjab.
RAIRO - Operations Research (28047303)51(4)pp. 921-930
Various reward-risk performance measures and ratios have been considered in reward-risk portfolio selection problems. This paper investigates the optimal portfolio corresponding to the CVaR (STARR) ratio. Considering the LP solvability of CVaR, a method is proposed for detecting the optimal portfolio by using the corresponding Mean-CVaR optimization problem. By applying LP tools, a method is suggested for producing the optimal portfolio as a by-product during the procedure of computing the efficient frontier of the Mean-CVaR problem. © EDP Sciences, ROADEF, SMAI 2017.
Pakistan Journal Of Statistics And Operation Research (22205810)12(4)pp. 765-773
The standard Markowitz Mean-Variance optimization model is a single-period portfolio selection approach where the exit-time (or the time-horizon) is deterministic. In this paper the Mean-Variance portfolio selection problem has been studied with uncertain exit-time when each asset has individual uncertain exit-time, which generalizes the Markowitz's model. Some conditions are provided under which the optimal portfolio of the generalized problem is independent of the exit-times distributions. Also, it is shown that under some general circumstances, the sets of optimal portfolios in the generalized model and the standard model are the same. © 2016, Pak.j.stat.oper.res. All rights reserved.
RAIRO - Operations Research (12903868)47(3)pp. 311-320
One-fund theorem states that an efficient portfolio in a Mean-Variance (M-V) portfolio selection problem for a set of some risky assets and a riskless asset can be represented by a combination of a unique risky fund (tangency portfolio) and the riskless asset. In this paper, we introduce a method for which the tangency portfolio can be produced as a corner portfolio. So, the tangency portfolio can be computed easily and fast by any algorithm designed for tracing out the M-V efficient frontier via computing the corner portfolios. Moreover, we show that how this method can be used for tracing out the M-V efficient frontier when problem contains a riskless asset in which the borrowing is not allowed. © EDP Sciences, ROADEF, SMAI 2013.
RAIRO - Operations Research (12903868)46(2)pp. 149-158
A risk measure in a portfolio selection problem is linear programming (LP) solvable, if it has a linear formulation when the asset returns are represented by discrete random variables, i.e., they are defined by their realizations under specified scenarios. The efficient frontier corresponding to an LP solvable model is a piecewise linear curve. In this paper we describe a method which realizes and produces a tangency portfolio as a by-product during the procedure of tracing out of the efficient frontier of risky assets in an LP solvable model, when our portfolio contains some risky assets and a riskless asset, using nonsmooth optimization methods. We show that the test of finding the tangency portfolio can be limited only for two portfolios. Also, we describe that how this method can be employed to trace out the efficient frontier corresponding to a portfolio selection problem in the presence of a riskless asset. © EDP Sciences, ROADEF, SMAI 2012.
