Mean-field formulation for multi-period asset-liability mean-variance portfolio selection with cash flow

Abstract

This thesis introduces a mean-field formulation to investigate the multi-period mean-variance model with cash flow, liability and uncertain exit time. As this model cannot be decomposed by a stage-wise backward recursion stage by stage on the basis of dynamic programming, it is a nonseparable problem. This thesis devotes to resolving this nonseparability as well as searching analytical optimal solutions and numerical example. On the one hand, the original bi-objective mean-variance problem can be trans­formed into a single-objective one by putting weights on the mean and variance. In substitution of the parameterized method, a mean-field formulation is employed to tackle various optimal multi-period mean-variance policy problems with cash flow and uncertain exit time, respectively. As a matter of fact, parameterized method and embedding technique cannot work smoothly when these constraints are considered. We will illuminate the efficiency and accuracy of mean-field formulation when models are not separable in dynamic programming. By taking expectation of the constraints with some calculations, in the language of optimal control, the state space and the control space will be enlarged, then the objective function becomes separable enabling us to use dynamic programming to solve this problem in expanded spaces. An analytical form of optimal policy and efficient frontier are also derived in this thesis. On the other hand, we take into account the liability on mean-variance model. Since in dynamic mean-variance problems, the optimal portfolio policy is always linear with current wealth and liability. Therefore, we employ the mean-field method and derive analytical optimal policies whose results are more explicit and accurate compared with the solution from embedding technique. During the whole derivation, the relationship among investment, cash flow and liability plays an important role. We investigate several cases such correlated or uncorrelated return rates at the same period, and we also illustrate the differences as well as the effects on optimal strategies theoretically and numerically.