Dynamic Mean Variance Research Articles

Dynamic Mean–Variance Portfolio Optimization with Value-at-Risk Constraint in Continuous Time

Recognizing the importance of incorporating different risk measures in the portfolio management model, this paper examines the dynamic mean-risk portfolio optimization problem using both variance and value at risk (VaR) as risk measures. By employing the martingale approach and integrating the quantile optimization technique, we provide a solution framework for this problem. We demonstrate that, under a general market setting, the optimal terminal wealth may exhibit different patterns. When the market parameters are deterministic, we derive the closed-form solution for this problem. Examples are provided to illustrate the solution procedure of our method and demonstrate the benefits of our dynamic portfolio model compared to its static counterpart.

A non-zero-sum investment and reinsurance game between two mean–variance insurers with dynamic CVaR constraints

This paper is devoted to investigating a non-zero-sum game between two competing insurers. The insurers can diversify their insurance risks by purchasing proportional reinsurance and investing their collected premiums into a financial market composed of one risk-free asset and one stock. The reinsurance premiums charged by the reinsurer follow the generalized mean–variance premium principle. Moreover, the dynamic CVaR constraints are incorporated in the game problem to control risks. With the dynamic mean–variance objective, we introduce two forward deterministic auxiliary processes to represent the expectations of the insurers’ wealth processes and transform the original time inconsistent game problem into a standard time consistent game problem with two state variables for each insurer. By adopting the dynamic programming principle and the Lagrange duality method, we derive the Nash equilibrium investment–reinsurance strategies for the two insurers. Finally, the effects of several important model parameters on the optimal policies are analyzed by numerical examples.

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.

Optimal Strategy of the Dynamic Mean-Variance Problem for Pairs Trading under a Fast Mean-Reverting Stochastic Volatility Model

We discuss the dynamic mean-variance (MV) problem for pairs trading with the assumptions that one of the security prices satisfies a stochastic volatility model (SVM) and the corresponding price spread follows an Ornstein–Uhlenbeck (OU) process. We provide a semi-closed-form of the optimal strategy based on the solution of a PDE, which is difficult to solve explicitly. Thus, we assume that one of the security prices satisfies the Scott model, a fast-mean-reverting volatility model, and give a closed-form approximation for the optimal strategy. Empirical studies, by using historical data from Chinese security markets, show that the Scott model produces a more stable strategy by better capturing mean-reverting volatility.

A non-zero-sum stochastic differential game between two mean-variance insurers with inside information

This paper is devoted to the study of a non-zero-sum investment-reinsurance game between two insurers with different opinions about some inside information. Each insurer is concerned about her terminal wealth level and the relative performance of wealth. We assume that the surplus processes of the insurers are correlated jump-diffusion processes. The insurers can invest in the financial market consisting of a risk-free asset and a risky asset. From the beginning of the transaction, the insurers have some inside information about the risky asset price in the future, which is disturbed by noise. However, one insurer trusts it while the other does not. We use the filtration expansion technique to transform the wealth processes for these two insurers with inside information. Then, under the dynamic mean-variance criterion, the explicit solutions for the equilibrium investment-reinsurance strategy and the equilibrium value function are derived by solving the extended HJB equations. Finally, the numerical examples are provided to analyze the effects of the inside information and the model parameters on the equilibrium strategy and the effective frontier.

Multi-Period Mean Expected-Shortfall Strategies: ‘Cut Your Losses and Ride Your Gains’

Dynamic mean-variance (MV) optimal strategies are inherently contrarian. Following periods of strong equity returns, there is a tendency to de-risk the portfolio by shifting into risk-free investments. On the other hand, if the portfolio still has some equity exposure, the weight on equities will increase following stretches of poor equity returns. This is essentially due to using variance as a risk measure, which penalizes both upside and downside deviations relative to a satiation point. As an alternative, we propose a dynamic trading strategy based on an expected wealth (EW), expected shortfall (ES) objective function. ES is defined as the mean of the worst β fraction of the outcomes, hence the EW-ES objective directly targets left tail risk. We use stochastic control methods to determine the optimal trading strategy. Our numerical method allows us to impose realistic constraints: no leverage, no shorting, infrequent rebalancing. For 5 year investment horizons, this strategy generates an annualized alpha of 180 bps compared to a 60:40 stock-bond constant weight policy. Bootstrap resampling with historical data shows that these results are robust to parametric model misspecification. The optimal EW-ES strategy is generally a momentum-type policy, in contrast to the contrarian MV optimal strategy.

Dynamic mean–variance problem with frictions

Dynamic mean–variance problem with frictions

Time consistent in efficiency dynamic mean–variance policy

The dynamic mean–variance formulation for a market with all risky assets is not time consistent in efficiency (TCIE), in the sense that the pre-committed policy, which is optimal for the entire investment horizon, could become mean–variance inefficient when facing a truncated time-horizon problem at an intermediate time instant. By removing the self-financing restriction and fully avoiding the irrational investment behaviours in intermediate time periods, we develop the TCIE dynamic mean–variance policy, which is TCIE. Comparing to the polices in the literature, such as the strictly dominating policy, the pre-committed policy and the time consistent policy, the TCIE policy can achieve better global investment performance.

Optimal dynamic mean–variance portfolio subject to proportional transaction costs and no-shorting constraint

This paper studies mean–variance portfolio selection problem subject to proportional transaction costs and no-shorting constraint. We do not impose any distributional assumptions on the asset returns. By adopting dynamic programming, duality theory, and a comparison approach, we manage to derive a semi-closed form solution of the optimal dynamic investment policy with the boundaries of buying, no-transaction, selling, and liquidation regions. Numerically, we illustrate the properties of the optimal policy by depicting the corresponding efficient frontiers under different rates of transaction costs and initial wealth allocations. We find that the efficient frontier is distorted due to the transaction cost incurred. We also examine how the width of the no-transaction region varies with different transaction cost rates. Empirically, we show that our transaction-cost-aware policy outperforms the transaction-cost-unaware policy in a realistic trading environment that incurs transaction costs

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.

PRACTICAL INVESTMENT CONSEQUENCES OF THE SCALARIZATION PARAMETER FORMULATION IN DYNAMIC MEAN–VARIANCE PORTFOLIO OPTIMIZATION

We consider the practical investment consequences of implementing the two most popular formulations of the scalarization (or risk-aversion) parameter in the time-consistent dynamic mean–variance (MV) portfolio optimization problem. Specifically, we compare results using a scalarization parameter assumed to be (i) constant and (ii) inversely proportional to the investor’s wealth. Since the link between the scalarization parameter formulation and risk preferences is known to be nontrivial (even in the case where a constant scalarization parameter is used), the comparison is viewed from the perspective of an investor who is otherwise agnostic regarding the philosophical motivations underlying the different formulations and their relation to theoretical risk-aversion considerations, and instead simply wishes to compare investment outcomes of the different strategies. In order to consider the investment problem in a realistic setting, we extend some known results to allow for the case where the risky asset follows a jump-diffusion process, and examine multiple sets of plausible investment constraints that are applied simultaneously. We show that the investment strategies obtained using a scalarization parameter that is inversely proportional to wealth, which enjoys widespread popularity in the literature applying MV optimization in institutional settings, can exhibit some undesirable and impractical characteristics.

Robo-advising: a dynamic mean-variance approach

Robo-advising: a dynamic mean-variance approach

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.

Bayesian Filtering for Multi-period Mean–Variance Portfolio Selection

For a long investment time horizon, it is preferable to rebalance the portfolio weights at intermediate times. This necessitates a multi-period market model. Usually, dynamic programming techniques are applied to optimize the portfolio for the multi-period model. However, this assumes a known distribution for the parameters of the financial time series. We consider the situation where the distribution of parameters is unknown and is estimated directly from the dynamically arriving data. We implement the Bayesian filtering method through dynamic linear models to sequentially update the parameters. We also acknowledge the uncertain investment lifetime to make the model more adaptive to the market conditions. These updated parameters are put into the dynamic mean–variance problem to arrive at optimal efficient portfolios. Implementing this model to the S&P500 illustrates that the data strongly favor the Bayesian updating and is practically implementable.

Equilibrium excess-of-loss reinsurance and investment strategies for an insurer and a reinsurer

In this paper, we consider the equilibrium excess-of-loss reinsurance and investment problem for both an insurer and a reinsurer. The risk process of the insurer is described by a classical Cramér-Lundberg (C-L) risk model and the insurer can purchase excess-of-loss reinsurance from the reinsurer. Both the insurer and the reinsurer are allowed to invest in a financial market consisting of a risk-free asset and two risky assets. The market price of risk depends on a Markovian, affine-form and square-root stochastic factor process. Dynamic mean-variance criterion is considered in this paper. We aim to maximize the weighted sum of the insurer’s and the reinsurer’s objectives with different risk averse coefficients. By solving the corresponding extended Hamilton-Jacobi-Bellman (HJB) equations, we derive the equilibrium reinsurance and investment strategies and the corresponding equilibrium value function. Finally, the economic implications of our findings are illustrated.

On the Distribution of Terminal Wealth under Dynamic Mean-Variance Optimal Investment Strategies

We compare the distributions of terminal wealth obtained from implementing the optimal investment strategies associated with the different approaches to dynamic mean-variance (MV) optimization avai...

Robo-Advising: A Dynamic Mean-Variance Approach

In contrast to traditional financial advising, robo-advising needs to elicit investors’ risk profile via several simple online questions and provide advice consistent with conventional investment wisdom, e.g., rich and young people should invest more in risky assets. To meet the two challenges, we propose to do the asset allocation part of robo- advising using a dynamic mean-variance criterion over the the portfolio’s log-returns. The model yields analytical and time-consistent optimal portfolio policies under jump-diffusion models and regime-switching models.

Indefinite mean-field type linear–quadratic stochastic optimal control problems

This paper focuses on indefinite stochastic mean-field linear–quadratic (MF-LQ, for short) optimal control problems, which allow the weighting matrices for state and control in the cost functional to be indefinite. The solvability of stochastic Hamiltonian system and Riccati equations is presented under indefinite case. The optimal controls in open-loop form and closed-loop form are derived, respectively. In particular, dynamic mean–variance portfolio selection problem can be formulated as an indefinite MF-LQ problem to tackle directly. Another example also sheds light on the theoretical results established.

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.