خرید بیمه زندگی بهینه و مصرف / سرمایه گذاری در طول عمر نامطلوب

In this paper, we consider optimal insurance and consumption rules for a wage earner whose lifetime is random. The wage earner is endowed with an initial wealth, and he also receives an income continuously, but this may be terminated by the wage earner’s premature death. We use dynamic programming to analyze this problem and derive the optimal insurance and consumption rules. Explicit solutions are found for the family of CRRA utilities, and the demand for life insurance is studied by examining our solutions and doing numerical experiments

In this paper, we consider the optimal life insurance purchase and consumption strategies for a wage earner subject to mortality risk in a continuous-time economy. Decisions are made continuously about these two strategies for all time t ∈ [0, T], where the fixed planning horizon T can be interpreted as the retirement time of the wage earner. The wage earner also receives his/her income at rate y(t) ⩾ 0 continuously, but this is terminated by the wage earner’s death or retirement, whichever happens first. We use a random variable to model the wage earner’s lifetime. The life insurance offered has an instantaneous term: the bigger the insurance premium rate paid by the wage earner, the bigger the claim paid to his/her family upon premature death. Income not consumed or used to buy insurance is invested at a riskless interest rate. The problem is to find the strategies that are best in terms of both the family’s consumption for all t ⩽ T as well as the terminal time – T wealth. Beginning in the 1960s, many researchers constructed quantitative models to analyze the demand for life insurance and the rate of investment for an individual under uncertainty. We are going to review some papers which contributed to this effort. Yaari (1965) is a starting point for modern research on the demand for life insurance. Yaari considered the problem of life insurance under an uncertain lifetime for an individual; this is the sole source of the uncertainty. The individual’s objective was to maximize View the MathML sourceE∫0TU(c(t))dt, Turn MathJax on where T , the individual’s lifetime, is the random variable which takes values on View the MathML source[0,T¯], View the MathML sourceT¯ is some given positive number which represents for the maximum possible lifetime for the consumer, and U is a utility function. Note that the horizon is random in the above functional, but the above functional can be rewritten into the following equivalent form: View the MathML source∫0T¯F¯(t)U(c(t))dt, Turn MathJax on where View the MathML sourceF¯(t) is the probability that the individual will be alive at time t. Note that now the horizon is a fixed time. This simple idea provides a useful method to analyze the optimization problem with a random life time. Since then, numerous literature has been built on Yaari’s pioneering work. However, Leung (1994) pointed out that Yaari’s model cannot have an interior solution which lasts until the maximum lifetime for the optimal consumption. What’s more, we cannot employ dynamic programming to analyze this kind of model since we can not appropriately define the terminal condition for the HJB equation within the frame of Yaari’s model (e.g., Ye, 2006). Campbell, 1980, Fischer, 1973, Lewis, 1989 and Iwaki and Komoribayashi, 2004 examined the demand for life insurance from different perspectives. Campbell (1980) considered the insurance problem in a very short time, [t, t + Δt], used a local analysis (Taylor expansion) to greatly simplify the problem, and then derived the insurance policy in terms of the present value of the future income, the current wealth, and other parameters. Since the local analysis has no information about the present value of the future income even if one knows the income stream, Campbell had to assume that the present value of the future income is given exogenously. Lewis (1989) examined the demand for life insurance from the perspective of the beneficiaries. Iwaki and Komoribayashi (2004) considered the optimal insurance from the perspective of households using a martingale method. Households are only allowed to buy life insurance at time 0 in their model, and so the households cannot change the amount of life insurance regardless of what happens after time 0, although the market for life insurance exists. Meanwhile, Merton, 1969 and Merton, 1971 developed a well known continuous-time model for optimal consumption and investment, a model that ignored insurance. Then Richard (1975) combined Merton’s model with the insurance literature, but he made the questionable assumption that the wage earner’s lifetime is bounded by a fixed planning horizon. The main contribution of our paper is to develop a version of Richard’s model where the wage earner’s lifetime can exceed the planning horizon, thereby allowing the planning horizon to be interpreted as the wage earner’s retirement date. Then how do we incorporate the uncertain life into the financial model such that it is theoretically consistent and has practical use? In our paper we use the model of uncertain life found in reliability theory; this kind of model is commonly used for industrial life-testing and actuarial science. We setup the wage earner’s objective (see (13) below) and convert the random horizon to fixed horizon (see Lemma 1 below). Then we employ the technique of dynamic programming to analyze our model. This paper is organized as follows. In the next section we setup our model. In Section 3, we state our HJB equation and then derive the optimal feedback control. In Section 4, we obtain the explicit solutions for the family of CRRA utilities, and examine the economic implications of our solutions. We conclude with some final remarks in Section 5.

We model the optimal insurance purchase and consumption under an uncertain lifetime for a wage earner in a simple economic environment, obtaining successfully the explicit solutions in the case of CRRA utilities, and explaining how factors affect the demand of life insurance purchase via numerical experiments. Our numerical experiments show in particular that insurance companies should increase the loading factor moderately from one in order to acquire maximal possible profit from the life insurance business. It should be noted from Fig. 5 and Fig. 6 that the premium payment rate p can be negative, especially when the wage earner is nearing retirement. This means the wage earner is collecting money at the rate −p, but his or her estate pays out a lump sum upon his or her death. But this negative payment rate is limited since the bequest is nonnegative from (32). This requirement rules out the possibility that the wage earner might be indebted to insurance companies if premature death occurs. For example, if the wage earner’s current wealth is $100,000, and the premium-insurance ratio is 0.01, then his or her payment rate must satisfy p ⩾ −$1000. This amount relative to the current wealth is typically very small, but the penalty is very large if premature death occurs. So allowing the wage earner to sell life insurance changes his or her behavior little. It is reasonable that the wage earner would find it optimal to sell life insurance close to retirement time. Future research will investigate this problem with the addition of a nonnegative constraint on the premium payment rate. Nowadays, the merger of the capital and insurance markets is accelerating due to financial services deregulation and globalization of financial services. In a forthcoming paper, we will examine optimal life insurance purchase, consumption and portfolio investment rules for a wage earner with an uncertain lifetime in a more complex economic environment where investment opportunities are stochastic.