Multi-period Portfolio Research Articles

Multi-Period Mean-Variance Portfolio Selection with State-Dependent Exit Probability and Bankruptcy State

Based on the mean-variance portfolio selection under multi-period criterion, this paper focuses on the study of the uncertain time horizon and the regime-switching market including the bankruptcy state, where the conditional distribution of exit time is followed by the market state. When the market enters the bankruptcy state, investors are assumed to get back δ part of the wealth from the bankrupt company, where δ refers to the retrieval rate. By introducing the Lagrange multiplier λ, we create an innovative expression for the wealth process and the iterative representation of the value function to obtain the analytical expression of the optimal strategy and the corresponding efficient frontier. Besides, some special cases and numerical examples are presented to demonstrate the effects of state-dependent exit probability and bankruptcy state on the investment strategy.

Skewed target range strategy for multiperiod portfolio optimization using a two-stage least squares Monte Carlo method

In this paper, we propose a novel investment strategy for portfolio optimization problems. The proposed strategy maximizes the expected portfolio value bounded within a targeted range, composed of a conservative lower target representing a need for capital protection and a desired upper target representing an investment goal. This strategy favorably shapes the entire probability distribution of returns, as it simultaneously seeks a desired expected return, cuts off downside risk and implicitly caps volatility and higher moments. To illustrate the effectiveness of this investment strategy, we study a multiperiod portfolio optimization problem with transaction costs and develop a two-stage regression approach that improves the classical least squares Monte Carlo (LSMC) algorithm when dealing with difficult payoffs, such as highly concave, abruptly changing or discontinuous functions. Our numerical results show substantial improvements over the classical LSMC algorithm for both the constant relative risk-aversion (CRRA) utility approach and the proposed skewed target range strategy (STRS). Our numerical results illustrate the ability of the STRS to contain the portfolio value within the targeted range. When compared with the CRRA utility approach, the STRS achieves a similar mean-variance efficient frontier while delivering a better downside risk-return trade-off.

An Incremental-Hybrid-Yager’s Entropy Model for Dynamic Portfolio Selection with Fuzzy Variable

To settle down the resolutional uncertainty in optimum portfolio strategy, this paper addresses an incremental-hybrid-Yager’s entropy model to newly describe the relationship between return and risk. Different from the traditional multiperiod portfolio, we design the ratio threshold to divide asset price into different time interval and use state instead of time point to model the dynamic portfolio process. In addition, fuzzy variables are utilized to represent prices of assets, while historical data based on Markov chain is exploited to estimate membership functions of fuzzy prices. At last, a compromised genetic algorithm is designed, and the numerical example shows that the proposed model achieves solid returns compared against the mean-variance model and Markov chain Monte Carlo method.

An Effective League Championship Algorithm for the Stochastic Multi-Period Portfolio Optimization Problem

The multi-period portfolio optimization models were introduced to overcome the weaknesses of the single-period models via considering a dynamic optimization system. However, due to the nonlinear nature of the problem and rapid growth of the size complexity with increasing the number of periods and scenarios, this study is devoted to developing a novel league championship algorithm (LCA) to maximize the portfolio’s mean-variance function subject to different constraints. A Vector Auto Regression model is also developed to estimate the return on risky assets in different time periods and to simulate different scenarios of the rate of return accordingly. Besides, we proved a valid upper bound of the objective function based on the idea of using surrogate relaxation of constraints. Our computational results based on sample data collected from S&P 500 and 10-year T. Bond indices indicate that the quality of portfolios, in terms of the mean-variance measure, obtained by LCA is 10 to 20 percent better than those of the commercial software. This sounds promising that our method can be a suitable tool for solving a variety of portfolio optimization problems.

Sharp Target Range Strategies with Application to Dynamic Portfolio Selection

In this paper, we propose a novel investment strategy for portfolio optimization problems. The proposed strategy maximizes the expected portfolio value bounded within a targeted range, composed of a conservative lower target representing a need for capital protection and a desired upper target representing an investment goal. This strategy favorably shapes the entire probability distribution of returns, as it simultaneously seeks a desired expected return, cuts off downside risk and implicitly caps volatility and higher moments. To illustrate the effectiveness of this investment strategy, we study a multiperiod portfolio optimization problem with transaction costs and develop a two-stage regression approach that improves the classical least squares Monte Carlo (LSMC) algorithm when dealing with difficult payoffs, such as highly concave, abruptly changing or discontinuous functions. Our numerical results show substantial improvements over the classical LSMC algorithm for both the constant relative risk-aversion (CRRA) utility approach and the proposed skewed target range strategy (STRS). Our numerical results illustrate the ability of the STRS to contain the portfolio value within the targeted range. When compared with the CRRA utility approach, the STRS achieves a similar mean–variance efficient frontier while delivering a better downside risk–return trade-off.

A linear programming model for selection of sparse high-dimensional multiperiod portfolios

This paper studies the mean-variance (MV) portfolio problems under static and dynamic settings, particularly for the case in which the number of assets (p) is larger than the number of observations (n). We prove that the classical plug-in estimation seriously distorts the optimal MV portfolio in the sense that the probability of the plug-in portfolio outperforming the bank deposit tends to 50% for p ≫ n and a large n. We investigate a constrained ℓ1 minimization approach to directly estimate effective parameters that appear in the optimal portfolio solution. The proposed estimator is implemented efficiently with linear programming, and the resulting portfolio is called the linear programming optimal (LPO) portfolio. We derive the consistency and the rate of convergence for LPO portfolios. The LPO procedure essentially filters out unfavorable assets based on the MV criterion, resulting in a sparse portfolio. The advantages of the LPO portfolio include its computational superiority and its applicability for dynamic settings and non-Gaussian distributions of asset returns. Simulation studies validate the theory and illustrate its finite-sample properties. Empirical studies show that the LPO portfolios outperform the equally weighted portfolio and the estimated optimal portfolios using shrinkage and other competitive estimators.

Possibilistic Moment Models for Multi-period Portfolio Selection with Fuzzy Returns

The aim of this paper is to investigate the effects of higher moments on multi-period portfolio selection with fuzzy returns. This paper gives the definitions of possibilistic mean and variance about the product of multiple fuzzy numbers. Based on these definitions, three multi-period fuzzy portfolio optimization models are proposed. The proposed models aim to maximize terminal wealth and minimize terminal risk by taking into account some realistic constraints including higher moments, budget constraint, round-lot constraint, cardinality constraint and bound constraint. To ensure the selection of the best solutions, a novel fuzzy programming approach-based self-adaptive differential evolution algorithm is designed to solve the proposed models. A numerical example is given to demonstrate the application of the proposed models. Computational results show that the designed algorithm is effective for solving complex portfolio selection model with realistic constraints.

Multiperiod portfolio optimization for asset-liability management with quadratic transaction costs

This paper investigates the multiperiod asset-liability management problem with quadratic transaction costs. Under the mean-variance criteria, we construct tractability models with/without the riskless asset and obtain the pre-commitment and time-consistent investment strategies through the application of embedding scheme and backward induction approach, respectively. In addition, some conclusions in the existing literatures can be regarded as the degenerated cases under our setting. Finally, the numerical simulations are given to show the difference of frontiers derived by different strategies. Also, some interesting findings on the impact of quadratic transaction cost parameters on efficient frontiers are discussed.

Multi-period portfolio selection with drawdown control

In this article, model predictive control is used to dynamically optimize an investment portfolio and control drawdowns. The control is based on multi-period forecasts of the mean and covariance of financial returns from a multivariate hidden Markov model with time-varying parameters. There are computational advantages to using model predictive control when estimates of future returns are updated every time new observations become available, because the optimal control actions are reconsidered anyway. Transaction and holding costs are discussed as a means to address estimation error and regularize the optimization problem. The proposed approach to multi-period portfolio selection is tested out of sample over two decades based on available market indices chosen to mimic the major liquid asset classes typically considered by institutional investors. By adjusting the risk aversion based on realized drawdown, it successfully controls drawdowns with little or no sacrifice of mean–variance efficiency. Using leverage it is possible to further increase the return without increasing the maximum drawdown.

Multi-period mean–semivariance portfolio optimization based on uncertain measure

In this paper, we discuss a multi-period portfolio selection problem when security returns are given by experts’ estimations. By considering the security returns as uncertain variables, we propose a multi-period mean–semivariance portfolio optimization model with real-world constraints, in which transaction costs, cardinality and bounding constraints are considered. Furthermore, we provide an equivalent deterministic form of mean–semivariance model under the assumption that the security returns are zigzag uncertain variables. After that, a modified imperialist competitive algorithm is developed to solve the corresponding optimization problem. Finally, a numerical example is given to illustrate the effectiveness of the proposed model and the corresponding algorithm.

Lifetime Active Portfolio Selection for Investments and Consumption A Bull Bear Market Cycle Based Probabilistic Approach

Beyond the single period Modern Portfolio Theory (Markowitz, 1955), the seminal work by Robert C. Merton (1969) solved elegantly a multi-period (finite time horizon) continuous portfolio optimization problem under the random walk market assumption. Due to the mathematical challenges, there has been little further theoretical breakthrough in nearly half century. Instead, popular financial planning and investment practice such as time diversification and target-date/glide-path based funds, deviate from the Merton/Samuelson’s teachings of long term static allocation. From a more realistic view of capital market and financial planning economics, what is the right asset allocation policy for lifetime investments and consumption? In this study, I replace the simplified stock market assumption in the classical Merton model with a Markov chain Bull/Bear market regime switching formulation (Wonham, 1965). The bull or bear market probability is calculated based on the price history of a broad market index such as the S&P 500 Composite. To maximize the total utility of discounted consumptions and bequest wealth value at the end of lifetime planning horizon, I derive a quantitative portfolio selection rule. It turns out that dynamic weight to risky asset is approximated solely as a quadratic function of the conditional bull market probability. From a market timing perspective, the active probabilistic rule has both trend following and mean reversion mechanisms at different stages of the anticipated market cycle. For practical purposes, I demonstrate the analytical approach in two sets of back-tests under different investment/consumption preferences. First, for a pure investor with zero consumption requirements, the objective reduces to maximizing terminal wealth. I found the optimal dynamic investment leverage or exposure to a risky stock market index depends not only on investor risk preference and the index’s average attributes of market return premium and volatility, but also the “strength” of the bull/bear market cycles, and the point-in-time market regime probability itself. Second, random walk or bull/bear market characteristics assume no impact on how an investor values the utility of his/her bequest wealth relative to lifetime consumption. In this case, the probabilistic portfolio selection is shown to out-perform easily in total utilities either the buy-and-hold static allocation or a typical linearly de-risking target-date glide path. By integrating the cyclical capital market view into a multi-period portfolio optimization framework, the current probabilistic formulation is ground breaking, despite the need for further refinement and analysis. With a fast changing landscape in global markets, the current approach of market cycle based dynamic allocation and active management is particularly important. It has the potential to seriously impact the theory and practice of investment management and financial planning, for the long term.

Multi-period portfolio optimization in a defined contribution pension plan during the decumulation phase

This paper studies a multi-period portfolio selection problem for retirees during the decumulation phase. We set a series of investment targets over time and aim to minimize the expected losses from the time of retirement to the time of compulsory annuitization by using a quadratic loss function. A target greater than the expected wealth is given and the corresponding explicit expressions for the optimal investment strategy are obtained. In addition, the withdrawal amount for daily life is assumed to be a linear function of the wealth level. Then according to the parameter value settings in the linear function, the withdrawal mechanism is classified as deterministic withdrawal, proportional withdrawal or combined withdrawal. The properties of the investment strategies, targets, bankruptcy probabilities and accumulated withdrawal amounts are compared under the three withdrawal mechanisms. Finally, numerical illustrations are presented to analyze the effects of the final target and the interest rate on some obtained results.

RETRACTION: Peng Zhang, Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection

In this paper, we propose a new multiperiod mean absolute deviation uncertain chance-constrained portfolio selection model with transaction costs, borrowing constraints, threshold constraints and cardinality constraints. In proposed model, the return rate of asset is quantified by uncertain expected value and the risk is characterized by uncertain absolute deviation. The chance constraints are that the uncertain expected return of the portfolio selection is bigger than the preset return of investors under the given confidence level. Cardinality constraints limit the number of assets in the optimal portfolio and threshold constraints limit the amount of capital to be invested in each asset and prevent very small investments in any asset. Based on uncertain theories, the model is converted to a dynamic optimization problem. Because of the transaction costs and cardinality constraints, the multiperiod portfolio selection is a mix integer dynamic optimization problem with path dependence, which is NP hard problem. The proposed model is approximated to a mix integer dynamic programming model. A novel discrete iteration method is designed to obtain the optimal portfolio strategy, and is proved linearly convergent. Finally, an example is given to illustrate the behavior of the proposed model and the designed algorithm using real data from the Shanghai Stock Exchange.

Multiperiod mean absolute deviation uncertain portfolio selection with real constraints

Absolute deviation is a commonly used risk measure, which has attracted more attentions in portfolio optimization. Most of existing mean---absolute deviation models are devoted to stochastic single-period portfolio optimization. However, practical investment decision problems often involve the uncertain dynamic information. Considering transaction costs, borrowing constraints, threshold constraints, cardinality constraints and risk control, we present a novel multiperiod mean absolute deviation uncertain portfolio selection model, which an optimal investment policy can be generated to help investors not only achieve an optimal return, but also have a good risk control. In proposed model, the return rate of asset and the risk are quantified by uncertain expected value and uncertain absolute deviation, respectively. Cardinality constraints limit the number of risky assets in the optimal portfolio. Threshold constraints limit the amount of capital to be invested in each asset and prevent very small investments in any asset. Based on uncertainty theories, the model is transformed into a crisp dynamic optimization problem. Because of the transaction costs and cardinality constraints, the multiperiod portfolio selection is a mix integer dynamic optimization problem with path dependence, which is "NP hard" problem that is very difficult to solve. The proposed model is approximated to a mix integer dynamic programming model. A novel discrete iteration method is designed to obtain the optimal portfolio strategy and is proved linearly convergent. Finally, an example is given to illustrate the behavior of the proposed model and the designed algorithm using real data from the Shanghai Stock Exchange.

Time-Consistent Portfolio Policy for Asset-Liability Mean-Variance Model with State-Dependent Risk Aversion

In reality, when facing a multi-period asset-liability portfolio selection problem, the risk aversion attitude of a mean-variance investor may depend on the wealth level and liability level. Thus, in this paper, we propose a state-dependent risk aversion model for the investor, in which risk aversion is a linear function of current wealth level and current liability level. Due to the time inconsistency of the resulting multi-period asset-liability mean-variance model, we investigate its time-consistent portfolio policy by solving a nested mean-variance game formulation. We derive the analytical time-consistent portfolio policy, which takes a linear form of current wealth level and current liability level. We also analyze the influence of the risk aversion coefficients on the time-consistent portfolio policy and the investment performance via a numerical example.

Stable Value Funds Performance

Little in the scholarly economics literature is directed specifically to the performance of stable value funds, although they occupy a leading place among retirement investment vehicles. They are currently offered in more than one-third of all defined contribution plans in the USA, with more than $800 billion of assets under management. This paper rigorously examines their performance throughout the entire period since their inception in 1973. We produce a composite index of stable value returns. We next conduct mean-variance analysis, Sharpe and Sortino ratio analysis, stochastic dominance analysis, and optimal multi-period portfolio composition analysis. Our evidence suggests that stable value funds dominate (on average) two major asset classes based on a historical analysis, and that they often occupy a significant position in optimized portfolios across a broad range of risk aversion levels. We discuss factors that contributed to stable value funds’ past performance and whether they can continue to perform well into the future. We also discuss considerations regarding whether or not to include stable value as an element in target date funds within defined contribution pension plans.

A Class of Multi-Period Portfolio Selection with a Special Investment Policy Structure

A Class of Multi-Period Portfolio Selection with a Special Investment Policy Structure

Strong and Weak Multivariate First-Order Stochastic Dominance

In this article we deal with three types of multivariate first-order stochastic dominance which serve for comparing random vectors. The first one is the strongest and it is generated by all non-decreasing multivariate utility functions. The second one, called weak multivariate stochastic dominance is defined by comparison of cumulative distribution functions of considered random vectors. The last one applies the univariate stochastic dominance notion to linear combinations of marginals. We compare these multivariate stochastic dominance relations among each other. Then we present important properties of strong and weak multivariate first-order stochastic dominance, in particular we describe their generators in the sense of von Neumann-Morgenstern utility functions and we explain their relation to joint and marginal distribution functions. Moreover, we discuss formulations of strong multivariate first-order stochastic dominance between random vectors with discrete distributions. Finally, we apply this stochastic dominance relation as a constraint to multivariate and multiperiod portfolio optimization problems.

MULTIPERIOD CREDIBILITIC MEAN SEMI-ABSOLUTE DEVIATION PORTFOLIO SELECTION

In this paper, we discuss a multiperiod portfolio selection problem with fuzzy returns. We present a new credibilitic multiperiod mean semi- absolute deviation portfolio selection with some real factors including transaction costs, borrowing constraints, entropy constraints, threshold constraints and risk control. In the proposed model, we quantify the investment return and risk associated with the return rate on a risky asset by its credibilitic expected value and semi- absolute deviation. Since the proposed model is a nonlinear dynamic optimization problem with path dependence, we design a novel forward dynamic programming method to solve it. Finally, we provide a numerical example to demonstrate the performance of the designed algorithm and the application of the proposed model.

Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint

• A multi-period uncertain portfolio selection problem is discussed. • A new method is presented for modeling the multi-period uncertain portfolio selection problem. • A deterministic transformation method is proposed for solving problem. The complexity of financial markets leads to different types of indeterminate asset returns. For example, asset returns are considered as random variables, when the available data is enough. When the available data is too small or even no available data to estimate a probability distribution, we have to invite some domain experts to evaluate the belief degrees of asset returns. Then, asset returns can be described as uncertain variables. In this paper, we discuss a multi-period portfolio selection problem under uncertain environment, which maximizes the final wealth and minimizes the risk of investment. Unlike the common method to describe the multi-period portfolio selection problem as a bi-objective optimization model, we formulate this uncertain multi-period portfolio selection problem by a new method in three steps with two single objective optimization models. And, we consider the influence of transaction cost and bankruptcy of investor. Then, the proposed uncertain optimization models are transformed into the corresponding crisp optimization models and we use the genetic algorithm combined with penalty function method to solve them. Finally, a numerical example is given to show the effectiveness and practicability of proposed models and method.