Dynamic Mean-variance Portfolio Research Articles

Dynamic mean-variance portfolio selection under factor models

Utilizing insights from financial literature and empirical financial data, we introduce a comprehensive system of factor models designed to capture both return and risk dynamics. Our focus extends to addressing the multi-period mean-variance portfolio selection challenge within the framework of these proposed factor models. Through rigorous analysis, we formulate a semi-analytical optimal portfolio policy, characterized by a linear relationship with the current wealth level. The coefficients of this optimal policy are intricately linked to a specific stochastic process known as the future investment opportunity (FIO), reflecting the investor's anticipation of future investment prospects. Furthermore, empirical examination within the U.S. market context underscores the efficacy of our approach. By incorporating the factor models for return and risk, our optimal portfolio policy exhibits superior out-of-sample Sharpe ratio compared to benchmark policies.

The self-coordination mean-variance strategy in continuous time

The dynamic mean-variance portfolio selection problem is time inconsistent. In the literature, scholars try to derive the pre-committed strategy, the time consistent strategy and the self-coordination strategy. The pre-committed strategy only concern the global investment interest of the investor. The time consistent strategy only concerns the local investment interests of the investor, while the self-coordination strategy balances between the global investment interest and local investment interests of the investor. However, the self-coordination strategy is only studied for the discrete time mean-variance setting. We study the self-coordination strategy for the continuous time mean-variance setting in this paper. With the help of mean-field reformulation, we derive the analytical self-coordination mean-variance strategy and show that the pre-committed strategy and time consistent strategy are special cases of the self-coordination strategy.

Research on dynamic mean-variance portfolio selection

The covariance matrix of portfolio returns constructed by static mean-variance model will change with time, so the optimal portfolio with fixed weights may not be optimal. Based on the static mean-variance model, this paper introduces two new parameters of the length of dynamic historical period and holding period to construct a dynamic mean-variance model. Numerical analysis shows that the proposed dynamic portfolio strategy can achieve good returns.

Constrained Dynamic Mean-Variance Portfolio Selection in Continuous-Time

This paper revisits the dynamic MV portfolio selection problem with cone constraints in continuous-time. We first reformulate our constrained MV portfolio selection model into a special constrained LQ optimal control model and develop the optimal portfolio policy of our model. In addition, we provide an alternative method to resolve this dynamic MV portfolio selection problem with cone constraints. More specifically, instead of solving the correspondent HJB equation directly, we develop the optimal solution for this problem by using the special properties of value function induced from its model structure, such as the monotonicity and convexity of value function. Finally, we provide an example to illustrate how to use our solution in real application. The illustrative example demonstrates that our dynamic MV portfolio policy dominates the static MV portfolio policy.

Mean-Variance Portfolio Analysis for Continuous Compound Return

We consider the dynamic mean-variance portfolio optimization problem in a continuous-time frame- work. Our objective is to maximize the expected continuous compound return and minimize its variance. We derive the optimal investment strategy and present a closed form mean-variance efficient frontier. We also show that the mean-variance optimization is equivalent to the CRRA utility maximization. The connection between the mean-variance analysis and the Kelly criterion is discussed, and we show that the mean-variance optimal investment strategy is indeed the fractional Kelly criterion.

Learning Equilibrium Mean-Variance Strategy

We study a dynamic mean-variance portfolio optimization problem under the reinforcement learning framework, where an entropy regularizer is introduced to induce exploration. Due to the time-inconsistency involved in a mean-variance criterion, we aim to learn an equilibrium strategy. Under an incomplete market setting, we obtain a semi-analytical, exploratory, equilibrium mean-variance strategy that turns out to follow a Gaussian distribution. We then focus on a Gaussian mean return model and propose an algorithm to find the equilibrium strategy using the reinforcement learning technique. Thanks to a thoroughly designed policy iteration procedure in our algorithm, we can prove our algorithm's convergence under mild conditions, despite that dynamic programming principle and the usual policy improvement theorem fail to hold for an equilibrium solution. Numerical experiments are given to demonstrate our algorithm.

The surprising robustness of dynamic Mean-Variance portfolio optimization to model misspecification errors

In single-period portfolio optimization settings, Mean-Variance (MV) optimization can result in notoriously unstable asset allocations due to small changes in the underlying asset parameters. This has resulted in the widespread questioning of whether and how MV optimization should be implemented in practice, and has also resulted in a number of alternatives being proposed to the MV objective for asset allocation purposes. In contrast, in dynamic or multi-period MV portfolio optimization settings, preliminary numerical results show that MV investment outcomes can be remarkably robust to model misspecification errors, which arise when the investor derives an optimal investment strategy based on some chosen model for the underlying asset dynamics (the investor model), but implements this strategy in a market driven by potentially completely different dynamics (the true model). In this paper, we systematically investigate the causes of this surprising robustness of dynamic MV portfolio optimization to model misspecification errors under both the pre-commitment MV (PCMV) and time-consistent MV (TCMV) approaches. We identify particular combinations of parameters that play a key role in explaining the observed model misspecification errors. We investigate the impact of the chosen dynamic MV approach, underlying model formulation, portfolio rebalancing frequency and the application of multiple realistic investment constraints on the robustness of investment outcomes, as well as the implications for model calibration.

Uniqueness of equilibrium strategies in dynamic mean-variance problems with random coefficients

This paper is concerned with the uniqueness issue of open-loop equilibrium investment strategies of dynamic mean-variance portfolio selection problems with random coefficients. A unified method is developed to treat both the problems with deterministic risk-free return rate, state-dependent risk aversion, and those with full random coefficients, constant risk aversion. To do so, some new necessity conditions for the existence of equilibrium investment strategies are established, which considerably extends the analogue in [9] with distinctive ideas. Some new interesting facts are revealed. For example, if risk-free return rate is random, it is shown that there exists a unique open-loop equilibrium investment strategy relying on initial wealth, even when risk aversion is merely a constant but not state-dependent.

DYNAMIC MEAN-VARIANCE PORTFOLIOS WITH RISK BUDGET

We study a dynamic mean-variance portfolio selection problem subject to possible limit of market risk. Three measures of market risk are considered: value-at-risk, expected shortfall, and median shortfall. They are all calculated in a dynamic consistent sense. After applying the technique of delta-normal approximation, we can explicitly solve for the optimal solution and calculate the economic loss brought by the risk budget constraint. With the analytical results obtained, influential factors of economic loss are then explored by which some guidelines on trading practice are proposed. The guidelines are independent of risk measures, and are valuable to both institutions and regulators, for they suggest that an institutional investor would spontaneously obey good investment discipline to avoid potential impact of risk constraint. This result meets the purpose of external regulation from the perspective of market discipline.

Robust Time-Inconsistent Stochastic Control Problems

This paper establishes a general analytical framework for continuous-time stochastic control problems for an ambiguity-averse agent (AAA) with time-inconsistent preference, where the control problems do not satisfy Bellman's principle of optimality. The AAA is concerned about model uncertainty in the sense that she is not completely confident in the reference model of the controlled Markov state process and rather considers some similar alternative models. The problems of interest are studied within a set of dominated models and the AAA seeks for an optimal decision that is robust with respect to model risks. We adopt a game-theoretic framework and the concept of subgame perfect Nash equilibrium to derive an extended dynamic programming equation and extended Hamilton -- Jacobi -- Bellman -- Isaacs equations for characterizing the robust dynamically optimal control of the problem. We also prove a verification theorem to theoretically support our construction of robust control. To illustrate the tractability of the proposed framework, we study an example of robust dynamic mean-variance portfolio selection under two cases: 1. constant risk aversion; and 2. state-dependent risk aversion.

Portfolio Selection with Random Liability and Affine Interest Rate in the Mean-Variance Framework

Abstract This paper studies a dynamic mean-variance portfolio selection problem with random liability in the affine interest rate environment, where the financial market consists of three assets: one risk-free asset, one risky asset and one zero-coupon bond. Assume that short rate is driven by affine interest rate model and liability process is described by the drifted Brownian motion, in addition, stock price dynamics is affected by interest rate dynamics. The investors expect to look for an optimal strategy to minimize the variance of the terminal surplus for a given expected terminal surplus. The efficient strategy and the efficient frontier are explicitly obtained by applying dynamic programming principle and Lagrange duality theorem. A numerical example is given to illustrate our results and some economic implications are analyzed.

Self-coordination in time inconsistent stochastic decision problems: A planner–doer game framework

Time inconsistency has been a thorny issue in many economic and financial decision making problems, especially when risk measures are involved in performance criteria. We develop in this paper a two-tier planner–doer game framework with self-coordination. By aligning the global interest of the planner and the local interests of the doers by applying suitable penalty functions, a degree of internal harmony (measured quantitatively by the expected cost of self-coordination) can be achieved by the self-coordination policy. We establish an axiom system for modified preferences, under which the self-coordination policy can be obtained by solving a corresponding optimization problem. We then apply our game framework successfully in dynamic mean-variance portfolio selection.

Unified Framework of Mean-Field Formulations for Optimal Multi-Period Mean-Variance Portfolio Selection

When a dynamic optimization problem is not decomposable by a stage-wise backward recursion, it is nonseparable in the sense of dynamic programming. The classical dynamic programming-based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies. Among these notorious nonseparable problems, the dynamic mean-variance portfolio selection formulation had posed a great challenge to our research community until recently. Different from the existing literature that invokes embedding schemes and auxiliary parametric formulations to solve the dynamic mean-variance portfolio selection formulation, we propose in this paper a novel mean-field framework that offers a more efficient modeling tool and a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the multi-period mean-variance-type portfolio selection problems.

Unified Framework of Mean-Field Formulations for Optimal Multi-Period Mean-Variance Portfolio Selection

When a dynamic optimization problem is not decomposable by a stage-wise backward recursion, it is nonseparable in the sense of dynamic programming. The classical dynamic programming-based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies. Among these notorious nonseparable problems, the dynamic mean-variance portfolio selection formulation had posed a great challenge to our research community until recently. Different from the existing literature that invokes embedding schemes and auxiliary parametric formulations to solve the dynamic mean-variance portfolio selection formulation, we propose in this paper a novel mean-field framework that offers a more efficient modeling tool and a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the multi-period mean-variance-type portfolio selection problems.

Mean-variance portfolio selection for defined-contribution pension funds with stochastic salary.

This paper focuses on a continuous-time dynamic mean-variance portfolio selection problem of defined-contribution pension funds with stochastic salary, whose risk comes from both financial market and nonfinancial market. By constructing a special Riccati equation as a continuous (actually a viscosity) solution to the HJB equation, we obtain an explicit closed form solution for the optimal investment portfolio as well as the efficient frontier.

Dynamic Mean-Variance Model with Borrowing Constraint under the Constant Elasticity of Variance Process

This paper studies a continuous-time dynamic mean-variance portfolio selection problem with the constraint of a higher borrowing rate, in which stock price is governed by a constant elasticity of variance (CEV) process. Firstly, we apply Lagrange duality theorem to change an original mean-variance problem into an equivalent optimization one. Secondly, we use dynamic programming principle to get the Hamilton-Jacobi-Bellman (HJB) equation for the value function, which is a more sophisticated nonlinear second-order partial differential equation. Furthermore, we use Legendre transform and dual theory to transform the HJB equation into its dual one. Finally, the closed-form solutions to the optimal investment strategy and efficient frontier are derived by applying variable change technique.

Dynamic Mean-Variance Portfolio Selection with Transaction Costs

This paper investigates a mean-variance portfolio selection problem in continuous time with fixed and proportional transaction costs. Utilizing the dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation is derived, and the explicit closed form solution is obtained. Furthermore, the optimal strategies and efficient frontiers are also proposed for the original mean-variance problem. Numerical experiments present the variation of the transaction region and efficient frontier with the transaction costs change, which demonstrate the proposed method performs effective.

Dynamic Mean-Variance Asset Allocation

We solve the dynamic mean-variance portfolio problem and derive its time-consistent solution using dynamic programming. Previous literature, in contrast, only determines either myopic or precommitment (committing to follow the initially optimal policy) solutions. We provide a fully analytical simple characterization of the dynamically optimal mean-variance portfolios within a general incomplete-market economy. We also identify a probability measure that incorporates intertemporal hedging demands and facilitates tractability. We illustrate this by easily computing portfolios explicitly under various stochastic investment opportunities. A calibration exercise shows that the mean variance hedging demands are economically significant.

Dynamic Mean-Variance Portfolio Selection with No-Shorting Constraints

This paper is concerned with mean-variance portfolio selection problems in continuous-time under the constraint that short-selling of stocks is prohibited. The problem is formulated as a stochastic optimal linear-quadratic (LQ) control problem. However, this LQ problem is not a conventional one in that the control (portfolio) is constrained to take nonnegative values due to the no-shorting restriction, and thereby the usual Riccati equation approach (involving a completion of squares) does not apply directly. In addition, the corresponding Hamilton--Jacobi--Bellman (HJB) equation inherently has no smooth solution. To tackle these difficulties, a continuous function is constructed via two Riccati equations, and then it is shown that this function is a viscosity solution to the HJB equation. Solving these Riccati equations enables one to explicitly obtain the efficient frontier and efficient investment strategies for the original mean-variance problem. An example illustrating these results is also presented.