Multi-period mean-variance asset-liability portfolio selection

Abstract

The thesis is concerned with multi-period mean-variance asset-liability portfolio selection. It is a nonseparable problem in the sense of dynamic programming as it cannot be decomposed by a stage-wise backward recursion. In this thesis, we resort to tackling the nonseparability of the problem and seeking analytical optimal solutions and efficient frontiers. On the one hand, we formulate the mean-variance model by fixing the terminal mean and deal with it using the parameterized method. By a variable substitution and Lagrange multiplier method, we can turn the nonseparable problem to a solvable stochastic linear quadratic optimal control problem. One prominent feature of the dynamic mean-variance formulations is that the optimal portfolio policy is always linear with respect to the current wealth and liability. According to this feature, we derive the analytical optimal policies and efficient frontiers. The analytical form of the Lagrange multiplier is also given in expression of the expectation of the final surplus. The results are much more explicit and accurate compared with the similar model solved by the embedding technique. It is worth mentioning that the relationship of returns between the assets and liability plays an important role in the whole derivation. We consider different cases such as the returns of assets of liability are stochastically correlated at the same period and in different periods as well as uncorrelated, compare their differences and illustrate their effects on optimal strategy and efficient frontier theoretically and numerically. On the other hand, by putting weights on the two criteria, we transform the mean-variance problem into a single-objective optimization problem. Instead of the parameterized method, we employ the mean-field formulation to solve different asset-liability mean-variance model with various constraints such as uncertain exit time, and bankruptcy control, respectively. In fact, when uncertain exit time or bankruptcy are considered in the model, the parameterized method and the embedding technique will not work smoothly. We shed light on the efficiency and accuracy of mean-field formulation when dealing with the issue of dynamic nonseparability in those models. By taking {212040}mean of the constraints and some simple calculation, the state space and the control space are enlarged in the language of optimal control. The objective function then becomes separable in the expanded space which enables us to solve the problem by dynamic programming. The analytical form of optimal policy and efficient frontier are derived. It is showed that when the uncertain exit time reduces to terminal exit time or the control over bankruptcy is left out and deterministic expected return is taken, the results of the parameterized method and mean-field formulation are proved to be the same. This further suggests that the two approaches to solve multi-period mean-variance model are accurate.