مصرف خانگی، سرمایه گذاری و بیمه عمر

This paper develops a continuous-time Markov model for utility optimization of households. The household optimizes expected future utility from consumption by controlling consumption, investments and purchase of life insurance for each person in the household. The optimal controls are investigated in the special case of a two-person household, and we present graphics illustrating how differences between the two persons affect the controls.

Original consumption-investment problems are formulated in terms of optimizing utility of consumption and a terminal utility over a fixed time horizon for a single person; see Merton, 1969 and Merton, 1971. Richard (1975) included the problem of finding an optimal life insurance strategy, and formulated the problem of optimizing expected utility over an uncertain life time, where utility now arose from consumption and from leaving a positive amount of money upon death. Apart from introducing life insurance, Richard (1975) also modeled a continuous life time income, and found that the expected life time income had a positive effect on the demand for life insurance. Actually, the inclusion of an insurance decision in the personal finance optimization problem was first formulated in a discrete-time setting by Yaari (1965). Since the path-breaking article of Hoem (1969), the continuous-time finite state Markov chain has played a prominent role in the theory of life insurance, and (Kraft and Steffensen, 2008) applied the continuous-time finite state Markov chain to the ideas established by Richard (1975). Kraft and Steffensen (2008) motivated the set-up by a personal finance model which allowed the customer to insure himself against disability, unemployment and similar personal risks. Inspired by Kraft and Steffensen (2008) we use the Markov chain set-up for modeling household finance in the sense of optimizing expected future utility for a household consisting of economically and probabilistically dependent persons. The modeling is flexible enough to capture several interesting differences between the members of the household, and leads to closed form solutions for the optimal controls of investments, consumption of the household and purchase of life insurance for each of its members. The paper is organized as follows. In Section 2, we present the general Markov model including the dynamics of the wealth of the household. Furthermore, we describe the assumptions concerning utility, and the general optimal value function for the problem. Section 3 presents the problem and the solution in the case of a one-person household, thereby setting the foundation for the multiple-person models. In Section 4, we solve the problem for a two-person household. We comment on the optimal control processes regarding consumption, investment and life insurance purchase, and in Section 5, we show numerical examples of these based on expectations to the investment market. In Section 6, we explain the mathematical induction techniques used for solving the multiple-person problem and write up the optimal controls in this case. Finally, in Section 7, we present ideas for further development of the model.