| This dissertation examines the demand for investment life insurance in the portfolio of finance assets, using the continuous-time dynamic models. Based on Yaari (1965), Richard (1975), Ye (2003), Merton (1969,1971) and other scholars, this dissertation constructs a framework of analyzing investment life insurance demand, by linking the life uncertainty and financial assets selection. On one hand, in the previous literature on financial assets selection, probably due to different study emphasis, they tend to neglect two facts as follow:[1] life is uncertain; [2] investment life insurance is one financial asset in the portfolio. Thus, these literatures are not fit to analyze the problem of life insurance demand. On the other hand, the literatures on life insurance demand generally take insurance as term life insurance, with few literature on investment life insurance. This dissertation models this problem effectively by defining the conception of investment life insurance.Like Yarri (1965), Merton (1969,1971), this dissertation constructs models in the continuous-time environment. The decision-maker is assumed to optimize his utility in the period [0,∞),with gradually increasing mortality in his lifetime. He has a Marshall Utility, e.t, consumption utility and bequest utility included. And the bequest is often assumed to include two parts:the wealth value and the payment by insurance company, at the time of death. The traditional literature on life insurance demand is only on the bequest utility, while we assume the investment life insurance can produce influence on both wealth accumulation and bequest. Specifically, the premiums of investment life insurance include two parts:one is to purchase the term life insurance; the other is to invest like securities invest. Thus, in order to study the demand for investment life insurance, we need to take both term life insurance and securities invest into consideration. As to the way of accumulating wealth, there are three options:one is risk-free saving, one is risky asset-investment life insurance, the third is risky asset-stock. In the environment of multi-financial assets being together, rational decision-maker should allocate his assets according to his risk attitude.Undoubtedly, the introduction of investment life insurance into the previous models complicates the decisions. In order to maximize his utility, the decision-maker needs to decide the level of consumption and term life insurance premium in the first step. Obviously, if he consumes more today, he will certainly increase his consumption utility. However, this choice will probably lead to less expense on term life insurance. Once he dies, his family will obtain less payment from insurance company. Besides, even if he survives, more consumption will also lead to less wealth left, which will affect the wealth accumulation in the future. Further, it will affect the consumption and term life insurance premium in the future. So is the term life insurance premium. After making the decisions on consumption and term life insurance premium, the wealth left will be used to invest in the three financial assets: savings, investment life insurance and stock. Savings can produce risky-free interest in the slow speed. Risky assets-investment life insurance and stock can yield higher return with certain risks. Rational decision-maker need to select suitable assets to optimize his allocation. In sum, the decision-maker need to tradeoff between all these selections. This dissertation provides the solution to this complicated problem.In the order of from the simple to the complicated, this dissertation provides three models to describe the problems. In the chapter 3 of basic model, we assume that the decision-maker only purchase term life insurance in the simple financial environment where the sole financial asset is saving. The conclusions are as follow: interest rate, initial wealth, and hazard rate exert positive effect on insurance premium. Risk-aversion parameter, utility discount rate affects the preference on insurance products. The objective of chapter 3 is to provide the basic framework and assumptions for the future analysis.In chapter 4, investment life insurance is introduced into the basic model in chapter 3. In the first step, we study the simple case in which the return of investment life insurance is certain. We find out that the equilibrium of consumption and term life insurance is the same as the chapter 3, while the investment ratio of investment life insurance is random between 0 and 1. More importantly, the total premium of investment life insurance begins to change. It is now not only affected by term life insurance, but also by investment part. In the second step of chapter 4, we analyze the case in which the return of investment life insurance is a stochastic process. In this case, the premium on term life insurance and investment in the equilibrium is liner with wealth, affected by many parameters such as risk aversion, market interest,e.t.. We calculate the effects of each parameter on insurance demand and then employ the numerical method to simulate the results obtained by models. It is shown that utility discount rate determines the preference of life insurance, risk attitude affects total insurance premium negatively, while the yield and variance of investment insurance produce positive and negative effects on total insurance demand, individually. Initial wealth has positive effects on insurance demand, while the effect of hazard rate is not clear in the model.In the chapter 5, we add another risky asset-stock-into the single-stochastic model. Under the constrains of two stochastic factors, the HJB equation changes its form. So, we try to derive the equilibrium conditions from the optimal principles by Bellman (1957). Further, we provide the explicit solution in the CRRA case and make a comparative dynamic analysis of these factors. Numerical method is again employed to study the effects of each parameter. We find out that most conclusions are similar to that in chapter 4. The difference is the introduction of another 3 paratermers into the model. It is shown that the effects of the yield and variance of stock is related to coefficient between stochastic items. When the coefficient is negative, the yield and variance of stock produces positive and negative effect on insurance demand, individually. When the coefficient is positive, the yield and variance of stock has the reverse effects, compared with the case of negative coefficient. The coefficient produces dramatic effect on insurance demand. Generally, the higher the positive coefficient, the less demand for investment insurance.The innovations of the dissertation include the following three points.1. This dissertation provides a framework of analyzing investment life insurance demand for the first time. The previous literature on insurance demand, whether static or dynamic models, whether discrete-time models or continuous-time models, did not consider investment life insurance demand. In this dissertation, we effectively make an analysis of investment life insurance demand by decomposing the investment life insurance into two parts:term life insurance and security investing.2. This dissertation, based on the work of Yaari (1965), Ye (2006), and Merton (1969,1971), Bellman (1957), derives the HJB equation that differs from the common case. Further, we obtain the explicit solutions for CRRA utility functions. Comparative dynamic analysis and numerical method are employed in each chapter to examine the effect of the parameters on equilibrium solutions.3. This dissertation solves the optimal problems under constrains of two stochastic process being together. With this model, we can analyze how equilibrium consumption, allocation of two risky assets are determined. Further, we can examine how the total premium of investment life insurance changes in the portfolio of financial assets. Most of the existing literatures on life insurance demand conduct the research in the single constraints of stochastic process, like Ye (2006). These models generally take the stock price as stochastic process, with others non-stochastic process. Some literatures on life insurance demand also consider two stochastic process constraints. However, they put their emphasis on stochastic income or stochastic interest, which does not come into the wealth motion equation (Ding Chuanming, 2003). In this dissertation, two stochastic processes are intensely connected with wealth motion equation, which leads to the changes of HJB equations as well as equilibrium solutions.
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