Utility-Deviation-Risk Portfolio Selection

Abstract

Abstract. We here provide a comprehensive study of the utility-deviation-risk portfolio selection problem. By considering the first-order condition for the corresponding objective function, we first derive the necessary condition that the optimal terminal wealth satisfying two mild regularity conditions solves for a primitive static problem, called Nonlinear Moment Problem. We then illustrate the application of this general necessity result by revisiting the non-existence of the optimal solution for the mean-semivariance problem. Secondly, we establish an alternative version of the verification theorem serving as the sufficient condition that the solution, which satisfies another mild condition different from that for necessity, of the Nonlinear Moment Problem is the optimal terminal wealth of the original utility-deviation-risk portfolio selection problem. We then apply this general sufficiency result to revisit the various well-posed mean-risk problems already known in the literature, and to also establish the existence of the optimal solutions for both utility-downside-risk and utility-strictlyconvex-risk problems under the assumption that the underlying utility satisfies the Inada Condition. To the best of our knowledge, the positive answers to the latter two problems have long been absent in the literature. In particular, the existence result in the utility-downside-risk problem is in contrast to the well-known non-existence of an optimal solution for the mean-downside-risk problem. As a corollary, the existence result in utility-semivariance problem allows us to utilize the semivariance as a proper risk measure in the classical portfolio management paradigm.

Constrained Utility Deviation-Risk Optimization and Time-Consistent HJB Equation

“…Specifically, we want to maximize the expected utility of the terminal wealth less the deviationrisk of the terminal wealth, adjusted by the stochastic risk aversion which depends on the initial time and state, and solve the problem time-consistently. This utility deviation-risk setup is similar to the model discussed in Wong et al [20] with the following fundamental differences: [20] focuses on the existence of a pre-commitment optimal solution in a complete Black-Scholes market and characterizes its solution with a system of algebraic equations, whereas in this paper we discuss a dynamic portfolio optimization problem in a possibly incomplete market with control constraints and solve it with a time-consistent HJB equation, so called to reflect the time-consistent nature of our approach for the equilibrium value function.…”

Closed-Loop Equilibrium Strategies for General Time-Inconsistent Optimal Control Problems

“…[2,11,12,23,28]). For example, the objective function in these papers is additively separable in the terminal state variable and its conditional expectation, which excludes the nonseparable utility-deviation-risk portfolio selection problem (Wong et al [27]) that is a special case of our model. To the best knowledge of the authors, this is the first result in the literature on equilibrium strategies for general time-inconsistent and coupled utility deviation risk portfolio selection problems in continuous time framework.…”

A Level-Set Approach for Stochastic Optimal Control Problems Under Controlled-Loss Constraints

“…In the finance literature, optimization under risk-measure constraints has been at the cornerstone of modern portfolio selection theory since the pioneering work [3]. We refer the interested reader to [4,5] for an exposition of the different models that have emerged in portfolio selection and their solution methods, and to [6][7][8] for additional examples of risk-measure constrained portfolio selection problems. In particular, [4] presents a comprehensive analysis of utilitydeviation-risk portfolio selection problems.…”

“…We refer the interested reader to [4,5] for an exposition of the different models that have emerged in portfolio selection and their solution methods, and to [6][7][8] for additional examples of risk-measure constrained portfolio selection problems. In particular, [4] presents a comprehensive analysis of utilitydeviation-risk portfolio selection problems. In that study, a deviation-risk-measure term, designed as the expected value of a function of the spread between the underlying portfolio and its mean at the terminal date, appears in the objective function as a penalization to the expected utility.…”