Xiangyu Cui

Better than Dynamic Mean‐Variance: Time Inconsistency and Free Cash Flow Stream

As the dynamic mean‐variance portfolio selection formulation does not satisfy the principle of optimality of dynamic programming, phenomena of time inconsistency occur, i.e., investors may have incentives to deviate from the precommitted optimal mean‐variance portfolio policy during the investment process under certain circumstances. By introducing the concept of time inconsistency in efficiency and defining the induced trade‐off, we further demonstrate in this paper that investors behave irrationally under the precommitted optimal mean‐variance portfolio policy when their wealth is above certain threshold during the investment process. By relaxing the self‐financing restriction to allow withdrawal of money out of the market, we develop a revised mean‐variance policy which dominates the precommitted optimal mean‐variance portfolio policy in the sense that, while the two achieve the same mean‐variance pair of the terminal wealth, the revised policy enables the investor to receive a free cash flow stream (FCFS) during the investment process. The analytical expressions of the probability of receiving FCFS and the expected value of FCFS are derived.

Optimal multi-period mean–variance policy under no-shorting constraint

We consider in this paper the mean–variance formulation in multi-period portfolio selection under no-shorting constraint. Recognizing the structure of a piecewise quadratic value function, we prove that the optimal portfolio policy is piecewise linear with respect to the current wealth level, and derive the semi-analytical expression of the piecewise quadratic value function. One prominent feature of our findings is the identification of a deterministic time-varying threshold for the wealth process and its implications for market settings. We also generalize our results in the mean–variance formulation to utility maximization with no-shorting constraint.

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.

Continuous time mean-variance portfolio optimization with piecewise state-dependent risk aversion

In this paper, a continuous time mean-variance portfolio optimization problem is considered within a game theoretic framework, where the risk aversion function is assumed to depend on the current wealth level and the discounted (preset) investment target. We derive the explicit time consistent investment policy, and find that if the current wealth level is less (larger) than the discounted investment target, the future wealth level along the time consistent investment policy is always less (larger) than the discounted investment target.

Dynamic Trading with Reference Point Adaptation and Loss Aversion

We formalize the reference point adaptation process by relating it to a way people perceive prior gains and losses. We then develop a dynamic trading model with reference point adaptation and loss aversion, and derive its semi-analytical solution. The derived optimal stock holding has an asymmetric V-shaped form with respect to prior outcomes, and the related sensitivities are directly determined by the sensitivities of reference point shifts with respect to the outcomes. We also find that the effects of reference point adaptation can be used to shed light on some well documented trading patterns, e.g., house money, break even, and disposition effects.

A mean-field formulation for optimal multi-period mean–variance portfolio selection with an uncertain exit time

Abstract A multi-period mean–variance portfolio selection problem with an uncertain exit time is one of the nonseparable dynamic optimization problems as the principle of optimality of dynamic programming no longer applies. In this paper, we introduce a mean-field formulation to tackle this multi-period nonseparable problem directly without introducing an embedding scheme. Moreover, we shed light on the efficient feature of the mean-field formulation when dealing with the issue of dynamic nonseparability.

Time consistent behavioral portfolio policy for dynamic mean–variance formulation

Abstract When one considers an optimal portfolio policy under a mean-risk formulation, it is essential to correctly model investors’ risk aversion which may be time variant or even state dependent. In this paper, we propose a behavioral risk aversion model, in which risk aversion is a piecewise linear function of the current excess wealth level with a reference point at the discounted investment target (either surplus or shortage), to reflect a behavioral pattern with both house money and break-even effects. Due to the time inconsistency of the resulting multi-period mean–variance model with adaptive risk aversion, we investigate the time consistent behavioral portfolio policy by solving a nested mean–variance game formulation. We derive a semi-analytical time consistent behavioral portfolio policy which takes a piecewise linear feedback form of the current excess wealth level with respect to the discounted investment target. Finally, we extend the above results to time consistent behavioral portfolio selection for dynamic mean–variance formulation with a cone constraint.

Classical Mean Variance Model Revisited: Pseudo Efficiency

Investigating the inverse problem of the classical Markowitz mean-variance formulation: Given a mean-variance pair, find initial investment levels and their corresponding portfolio policies such that the given mean-variance pair can be realized, we reveal that any mean-variance pair inside the reachable region can be achieved by multiple portfolio policies associated with different initial investment levels. Therefore, in the mean-variance world for a market of all risky assets, the common belief of monotonicity: ‘The larger you invest, the larger expected future wealth you can expect for a given risk (variance) level’ does not hold, which stimulates us to extend the classical two-objective mean-variance framework to an expanded three-objective framework: to maximize the mean and minimize the variance of the final wealth as well as to minimize the initial investment level. As a result, we eliminate from the policy candidate list the set of pseudo efficient policies that are efficient in the original mean-variance space, but inefficient in this newly introduced three-dimensional objective space.

Dynamic mean–VaR portfolio selection in continuous time

The value-at-risk (VaR) is one of the most well-known downside risk measures due to its intuitive meaning and wide spectra of applications in practice. In this paper, we investigate the dynamic mean–VaR portfolio selection formulation in continuous time, while the majority of the current literature on mean–VaR portfolio selection mainly focuses on its static versions. Our contributions are twofold, in both building up a tractable formulation and deriving the corresponding optimal portfolio policy. By imposing a limit funding level on the terminal wealth, we conquer the ill-posedness exhibited in the original dynamic mean–VaR portfolio formulation. To overcome the difficulties arising from the VaR constraint and no bankruptcy constraint, we have combined the martingale approach with the quantile optimization technique in our solution framework to derive the optimal portfolio policy. In particular, we have characterized the condition for the existence of the Lagrange multiplier. When the opportunity set of the market setting is deterministic, the portfolio policy becomes analytical. Furthermore, the limit funding level not only enables us to solve the dynamic mean–VaR portfolio selection problem, but also offers a flexibility to tame the aggressiveness of the portfolio policy.

Multiperiod Mean-CVaR Portfolio Selection

Due to the time inconsistency issue of multiperiod mean-CVaR model, two important policies of the model with finite states, the pre-committed policy and the time consistent policy, are derived and discussed. The pre-committed policy, which is global optimal for the model, is solved through linear programming. A detailed analysis shows that the pre-committed policy doesn’t satisfy time consistency in efficiency either, i.e., the truncated pre-committed policy is not efficient for the remaining short term mean-CVaR problem. The time consistent policy, which is the subgame Nash equilibrium policy of the multiperson game reformulation of the model, takes a piecewise linear form of the current wealth level and the coefficients can be derived by a series of integer programming problems and two linear programming problems. The difference between two polices indicates the degree of time inconsistency.