Multi-period Portfolio Optimization Problem Research Articles

Multi-period portfolio optimization using a deep reinforcement learning hyper-heuristic approach

Portfolio optimization concerns with periodically allocating the limited funds to invest in a variety of potential assets in order to satisfy investors’ appetites for risk and return goals. Recently, Deep Reinforcement Learning (DRL) has shown its promising capabilities in sequential decision making problems. However, traditional DRL algorithms directly operate in the space of low-level actions, which exhibits poor scalability and becomes intractable in real-world problem instances when the dimensionality of the environment increases. To deal with this, in this work, a novel DRL hyper-heuristic framework is proposed for multi-period portfolio optimization problem. Instead of exploiting the entire action domain, our proposed approach is more effective by searching for low-level well-developed trading strategies. In addition, our proposed approach is data-driven and respects the nature of the problem by taking advantage of expert domain knowledge and posing it multidimensional states to further leverage additional diverse information from alternative views of the environment. The proposed approach is evaluated on five real-world capital market problem instances and numerous experimental results demonstrate our proposed method can achieve notable performance gains compared to state-of-art trading strategies as well as traditional DRL baseline method. The data we used are from five stock indices, covering the period from the 2012 to 2022. Our study can have salient policy implications for investment strategy formulation and effective regulatory frameworks establishment.

Multi-Period Portfolio Optimization under Cash-In Strategies

Portfolio optimization has been crucial in its applications to finance. Investors allocate capital in terms of various assets and wisely adjust each proportion in quest of higher returns and lower risks. However, common optimization strategies are usually short- sighted, lacking the ability of gaining higher returns over a long period. This paper addresses Multi-Period Portfolio Optimization problem in more applicable settings to provide efficient strategies. Firstly, a summary of conclusions from former works and scholars is given. Since most previous works assume self-financing conditions, this work extends this constraint and considers more complex cash-in and cash-out situations. It also derives the expression of the optimal policy for trading at each period as well as the objective functions. Finally, numerical experiments are done under both self-financing and cash-in situations to prove the validity of conclusions. This work has profound implications in finance because it adopts conditions more consistent with real-world scenarios and provides more farsighted strategies. Moreover, the solution offers a prototype for similar optimization in other fields like electrical engineering and stochastic processes.

Multi-period portfolio selection based on uncertainty theory with bankruptcy control and liquidity

Due to the impact of uncertain events, such as the 2008 financial crisis and the outburst of COVID-19 pandemic, the experts’ evaluations information is becoming increasingly important. This paper considers a multi-period portfolio optimization problem under uncertain circumstance, and the return rates of risky securities are regarded as uncertain variables, where the uncertainty theory is used to deal with experts’ evaluations. In light of the complexity of financial markets, we formulate an uncertain multi-period mean-entropy-variance model, where bankruptcy, liquidity, diversification and self-financing are considered as realistic constraints. Furthermore, the maximum return and the minimum risk are both achieved in a single-objective model through the normalization method. Then the equivalent deterministic forms of two secondary models for main model are provided. In addition, we develop a modified root system growth algorithm, which is more suitable for the proposed model. Finally, the effectiveness of the proposed model and designed algorithm is confirmed by numerical simulations.

Multi-period uncertain portfolio selection model with prospect utility function.

In this paper, we discuss a multi-period portfolio optimization problem based on uncertainty theory and prospect theory. We propose an uncertain multi-period portfolio selection model, in which the return utility and risk of investment are measured by prospect theory utility function and uncertain semivariance. More realistically, the influence of transaction costs and bankruptcy of investor are also considered. Moreover, to solve the portfolio model, this paper designs a new artificial bee colony algorithm by combining sine cosine search method. Finally, a numerical experiment is presented to demonstrate the proposed model and the effectiveness of the designed algorithm.

Multi-period portfolio optimization using model predictive control with mean-variance and risk parity frameworks

We employ model predictive control for a multi-period portfolio optimization problem. In addition to the mean-variance objective, we construct a portfolio whose allocation is given by model predictive control with a risk-parity objective, and provide a successive convex program algorithm that provides 30 times faster and robust solutions in the experiments. We provide a comprehensive comparison of the models in regard of planning horizon, parameter estimation, as well as objective function choice. Computational results on a multi-asset universe show that multi-period models perform better than their single period counterparts in out-of-sample period, 2006–2020, in the presence of market impact costs. The out-of-sample risk-adjusted performance of both mean-variance and risk-parity formulations beat the fix-mix benchmarks, and achieve Sharpe ratio of 0.64 and 0.97, respectively. We also include tests on different asset universes (Fama French industry portfolios) and alternative parameter estimation methods (Bayes-Stein and Black-Litterman) with consistent findings.

Multi-period uncertain portfolio optimization model with minimum transaction lots and dynamic risk preference

An accurate variable description of security returns is necessary to establish a valid portfolio optimization model. It is usually assumed to be a random variable when the historical data of security returns are sufficient. However, when historical data are too limited to estimate its probability distribution, uncertain variables are employed to characterize security returns to be effective. This paper focuses on a multi-period portfolio optimization problem in uncertain environment with the consideration of minimum transaction lots. Different from the previous multi-period work assuming the total available wealth in the end of the investment horizon is consistently in an exponential format, our study provides a simplified additive format of the total wealth, which may make the process of experimental calculation concise, since it is a linear function of decision variables. Besides, we consider the investor’s dynamic risk preference along the whole investment horizon. With these realistic constraints derived from the complex financial markets, we build a multi-period mean-VaR (value-at-risk) model with the objective of maximizing the terminal wealth under the risk control over the whole investment. Genetic algorithm is used to solve the proposed model, and two numerical examples are given to illustrate the effectiveness of the proposed approach.

Econometric Model Used in the Portfolio Optimization over Several Periods

"The classic problem of optimizing a portfolio can be extended to a multi-stage programming problem. The purpose of the multi-period portfolio optimization problem is to determine the optimal portfolio for a certain finite time horizon. In a multi-period model in which investors are allowed to change the composition of the portfolio, it is essential to take into account trading costs, a solution in this regard being the use of tree-type scenarios. The study undertaken by the authors considered the construction of a portfolio optimization model in case there is a certain constraint on returns. ARMA type processes were used to model the conditional mean equation."

Multi-period Portfolio Optimization using Model Predictive Control with Mean-Variance and Risk Parity Frameworks

We employ model predictive control for a multi-period portfolio optimization problem. In addition to the mean-variance objective, we construct a portfolio whose allocation is given by model predictive control with a risk-parity objective, and provide a successive convex program algorithm that provides 30 times faster and robust solutions in the experiments. Computational results on the multi-asset universe show that multi-period models perform better than their single period counterparts in out-of-sample period, 2006-2020. The out-of-sample risk-adjusted performance of both mean-variance and risk-parity formulations beat the fix-mix benchmark, and achieve Sharpe ratio of 0.64 and 0.97, respectively.

Multiperiod portfolio selection models under uncertain measure and with multiple criteria

This paper explores a multiperiod portfolio optimization problem under uncertain measure involving background risk, liquidity constraints and V-shaped transaction costs. Unlike traditional studies, we establish multiperiod mean-variance portfolio optimization models with multiple criteria in which security returns, background asset returns and turnover rates are assumed to be uncertain variables that can be estimated by experienced experts. When the returns of the securities and background assets follow normal uncertainty distributions, we use the deterministic forms of the multiperiod portfolio optimization model. The uncertain multiperiod portfolio selection models are practical but complicated. Therefore, the models are solved by employing a genetic algorithm. The uncertain multiperiod model with multiple criteria is compared with an uncertain multiperiod model without background risk and an uncertain multiperiod model without liquidity constraint respectively, we discuss how background risk and liquidity affect optimal terminal wealth. Finally, we give two numerical examples to demonstrate the effectiveness of the proposed approach and models.

Deep-Learning Solution to Portfolio Selection with Serially-Dependent Returns

This paper investigates a deep-learning solution to high-dimensional multi-period portfolio optimization problems with bounding constraints on the control. We propose a deep neural network (DNN) architecture to describe the underlying control process. The DNN consists of $K$ subnetworks, where $K$ is the total number of decision steps. The feedback control function is determined solely by the network parameters. In this way, the multi-period portfolio optimization problem is linked to a training problem of the DNN, that can be efficiently computed by the standard optimization techniques for network training. We offer a sufficient condition for the algorithm to converge for a general utility function and general asset return dynamics including serially-dependent returns. Specifically, under the condition that the global minimum of the DNN training problem is attained, we prove that the algorithm converges with the quadratic utility function when the risky asset returns jointly follow multivariate AR(1) models and/or multivariate GARCH(1,1) models. Numerical examples demonstrate the superior performance of the DNN algorithm in various return dynamics for a high-dimensional portfolio (up to 100 dimensions).

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 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.

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 mean–variance fuzzy portfolio optimization model with transaction costs

This paper examines the multi-period portfolio optimization problem with transaction costs and fuzzy variables to count for the uncertainty of future returns and liquidities on assets. The portfolio risk is quantified by using the variance of fuzzy returns. Two conflicting optimization objectives, namely, maximizing the terminal wealth and minimizing the cumulative risk of portfolios over the entire investment horizon, are taken into consideration. For solving the proposed model we introduce a new multi-objective evolutionary algorithm. Finally the performance of the proposed algorithm is compared with the NSGAII and MOEA/D with the assistance of real data from FTSE-100 in London.

Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer

We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements, and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.

Quantum Computing: Implications for Portfolio Managers

Computers that leverage the power (and mysteries) of quantum mechanics are available today, and for some specialized problems, they have been shown to provide significant speed advantages over digital computers. What are the implications of this technology for portfolio managers and others working in the field of computational finance? This article introduces some of the fundamental concepts of quantum computing and presents an example of using a quantum annealer for solving a multiperiod portfolio optimization problem. <b>TOPICS:</b>Quantitative methods, portfolio construction

Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer

We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements, and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.

Volatility vs. Downside Risk: Optimally Protecting Against Drawdowns and Maintaining Portfolio Performance

As a consequence of recent market conditions an increasing number of investors are realizing the importance of controlling tail risk to reduce drawdowns thus increasing possibilities of achieving long-term objectives. Recently, so called volatility control strategies and volatility target approaches to investment have gained a lot of interest as strategies able to mitigate tail risk and produce better risk-adjusted returns. Essentially these are rule-based backward looking strategies in which no optimization is considered. In this contribution we focus on the role of volatility in downside risk reduction and, in particular, in tail risk reduction. The first contribution of our paper is to provide a viable way to integrate a target volatility approach, into a multiperiod portfolio optimization model, through the introduction of a local volatility control approach. Our optimized volatility control is contrasted with existing rule-based target volatility strategies, in an out-of sample simulation on real data, to assess the improvement that can be obtained from the optimization process. A second contribution of this work is to study the interaction between volatility control and downside risk control. We show that combining the two tools we can enhance the possibility of achieving the desired performance objectives and, simultaneously, we reduce the cost of hedging. The multiperiod portfolio optimization problem is formulated in a stochastic programming framework that provides the necessary flexibility for dealing with different constraints and multiple sources of risk.

A multi-period portfolio selection optimization model by using interval analysis

Due to few historical data that can be obtained in an emerging securities market, the future returns, risk and liquidity of securities cannot be forecasted precisely. The investment environment is usually fuzzy and uncertain. To handle these imprecise data, this paper discusses a fuzzy multi-period portfolio optimization problem where the returns, risk, and liquidity of securities are represented by interval variables. By taking the return, risk, liquidity and diversification degree of portfolio into consideration, an interval multi-period portfolio selection optimization model is proposed with the objective of maximizing the terminal wealth under the constraints of the return, risk and diversification degree of portfolio at each period. In the proposed model, a proportion entropy is employed to measure the diversification degree of portfolio. Using the fuzzy decision-making theory and multi-objective programming approach, the proposed model is transformed into a crisp nonlinear programming. Then, we design an improved particle swarm optimization algorithm for solution. Finally, a numerical example is given to illustrate the application of our model and demonstrate the effectiveness of the designed algorithm.