سرمایه گذاری بهینه و استراتژی های بیمه عمر تحت محدودیت های حداقل و حداکثر

We derive optimal strategies for an individual life insurance policyholder who can control the asset allocation as well as the sum insured (the amount to be paid out upon death) throughout the policy term. We first consider the problem in a pure form without constraints (except nonnegativity on the sum insured) and then in a more general form with minimum and/or maximum constraints on the sum insured. In both cases we also provide the optimal life insurance strategies in the case where risky-asset investments are not allowed (or not taken into consideration), as in basic life insurance mathematics. The optimal constrained strategies are somewhat more complex than the unconstrained ones, but the latter can serve to ease the understanding and implementation of the former.

This paper is concerned with optimal strategies regarding life insurance (i.e., coverage against death) and investment for an individual policyholder in a life insurance company or a pension fund (referred to as the company henceforth). More specifically, we consider a life insurance policy comprising life insurance as well as retirement saving during [0,T][0,T], where T>0T>0 is a fixed finite time horizon. The policyholder is allowed to choose, in a continuous manner, the sum insured, which is the sum to be paid out from the company upon death of the policyholder, as well as the investment strategy. Our main focus is on the issue of optimal life insurance rather than optimal investment (although, as we shall see, these issues should not be viewed separately), and we therefore work throughout with a simple and well-known model for the financial market offering the assets available for investment. We impose in general the very realistic constraint that the sum insured must be nonnegative, but as the title of the paper suggests we shall also consider the optimization problem under more restrictive minimum and maximum constraints in the form of lower and upper boundaries for the sum insured. The motivation for the company to impose an upper boundary is rather obvious, since it puts a limit on the company’s immediate risk at any time during the policy term, but the motivation for a lower boundary may not be obvious. However, a lower boundary is sometimes imposed e.g. in pension schemes that are mandatory for employees within a certain line of business in order to ensure that a minimum coverage against death is provided automatically, i.e., without a specific request from each employee. The unconstrained case, i.e., without minimum or maximum constraints (except nonnegativity), is a special case of the constrained case since the lower and upper boundaries can be set to 0 and ∞∞, respectively. However, for educational purposes, and to ease the overall presentation of the results, we have chosen to include the solution of the unconstrained case separately (in Section 2), also because the results from the unconstrained case play an important role as convenient references in the constrained case. The solution of the unconstrained case in itself is not a main contribution of this paper to the literature, though, since it, from a mathematical perspective, actually is equivalent to a certain purely financial consumption/investment problem with a well-known solution (as will be noted). From a mathematical point of view it is interesting to note that the unconstrained case can be solved by dynamic programming (as is done in this paper), whereas the more general constrained case is quite troublesome (at least) to solve by this approach. The latter case is thus substantially facilitated by the martingale methodology, emphasizing one of the major strengths of this technique, namely that it can lead to (more or less explicit) solutions in problems with binding constraints, where the dynamic programming approach typically is not easy to apply. An interesting aspect of the optimal insurance strategy in the unconstrained case is that although the optimal sum insured depends heavily on the development of the financial market, its range is the entire interval [0,∞)[0,∞) at any time during the policy term. Thus, the boundaries imposed in the constrained case are strictly binding whenever they are non-trivial (i.e., strictly positive and finite, respectively). In general the investment strategy is taken to be unconstrained (except for technical conditions), i.e., we allow all positions in the risky assets. However, we also provide the optimal life insurance strategy in the special case of a market without risky assets, or equivalently, under the constraint that no risky-asset investments (long or short) are allowed. There are two main motivations for this “sub-problem”: Firstly, it is more in line with basic life insurance mathematics, where the interest rate is typically assumed constant (or deterministic), see e.g. Møller and Steffensen (2007), so it constitutes an interesting problem in its own right, at least from an actuarial perspective. Various alternative interpretations of the interest rate (which in the general setup below is the risk-free money market rate) are then possible; in particular it may play the role of the so-called second order rate or bonus rate, see Møller and Steffensen (2007). Secondly, the optimal life insurance strategy as such stands out more clearly and is thus perhaps easier to interpret and analyze. However, we do not provide detailed proofs of our results pertaining to this case (the proofs are similar to the proofs provided in the general setup; the details are left to the interested reader). Studying the optimal demand for life insurance for an economic agent dates back to Yaari (1965) and has been followed up by Richard (1975), who was the first to study the combined problem of optimal life insurance and investment (and consumption as well), Campbell (1980), and others. The problem variations studied in the literature concern whether some or all of the processes regarding investment, life insurance, and consumption are considered as decision processes, whether the agent has non-capital (wage) income, and whether the problems are solved in discrete or continuous time. More recent contributions to this body of literature are provided by Chen et al. (2006), who allow for stochastic income, Hong and Ríos-Rull (2007), who take a family point of view and also take social security into account, and Ye and Pliska (2007), who study a problem close to the one studied by Richard (1975) and also provide a nice survey of the literature. The main contribution of the present paper is the solution of the continuous-time problem where all processes are decision processes (as in Richard (1975)), and with constraints on the life insurance decision. To the knowledge of the authors, the problem with such constraints has not been considered previously in the literature. Dynamic utility optimization is studied recently in the context of non-life insurance by Moore and Young (2006). Another related body of research concerns optimal investment with the objective of minimizing the lifetime ruin probability (and generalizations); this constitutes a relevant personal finance problem in the absence of life insurance, see e.g. Bayraktar and Young (2007). The remainder of the paper is organized as follows: Section 2 introduces the general setting and the basic optimization problem, and the unconstrained case is treated. In Section 3 we solve the problem in the general constrained case; this section thus contains the main results of the paper. Section 4 concludes. Some basic notations: All vectors are column vectors. The transposed of a vector or matrix aa is denoted by a′a′. The dd-dimensional vector of 1’s is denoted by 1d1d.

We briefly summarize our results, focusing on the structural forms of the optimal strategies. In the unconstrained case the optimal investment strategy corresponds to that obtained by Merton (1971) in the case with non-capital income, with the reserve interpreted as the current financial wealth, and the actuarial present value of the future premiums interpreted as the present value of future income (human capital). The optimal life insurance strategy is to let the sum insured at any time t∈[0,T]t∈[0,T] equal A(t)/h(t)A(t)/h(t), where A(t)A(t) is the sum of the current reserve and the actuarial present value of the future premiums and thus can be interpreted as the total assets, and where h(⋅)h(⋅) is a certain function depending on the individual preferences of the policyholder (see Section 2 for a closer analysis). In particular, the range of the sum insured is the entire interval (0,∞)(0,∞). Moreover, the investment and life insurance strategies can be viewed as being independent in the sense that at any given time, the optimal risky-asset allocation and the optimal sum insured are independent. In the constrained case with lower and upper boundaries View the MathML sources¯(⋅) and View the MathML sources¯(⋅) on the sum insured, the total assets should be decomposed into three parts as A(⋅)=A1(⋅)−A2(⋅)+A3(⋅).A(⋅)=A1(⋅)−A2(⋅)+A3(⋅). Turn MathJax on The first component, A1(⋅)A1(⋅), corresponds to the total asset process of the unconstrained case: It should be invested according to the same (constant) risky-asset proportions as in the unconstrained case, and it forms the basis for the optimal life insurance strategy in the sense that the sum insured at any time t∈[0,T]t∈[0,T] should be set to A1(t)/h(t)A1(t)/h(t) unless View the MathML sourceA1(t)/h(t)<s¯(t) or View the MathML sourceA1(t)/h(t)>s¯(t), in case of which it should be set to the relevant boundary. The other two asset components, A2(⋅)A2(⋅) and A3(⋅)A3(⋅), act as hedging instruments ensuring that the optimal insurance strategy can be financed (in the sense of the actuarial equivalence principle; see Remark 2.1 and the discussion following Definition 2.4). Thus, A3(⋅)A3(⋅) is (the value of) a continuum of European put options written on A1(⋅)/h(⋅)A1(⋅)/h(⋅) with strike View the MathML sources¯(⋅) paying off the deficit View the MathML source[s¯(t)−A1(t)/h(t)]+ in the event of death at time tt, and A2(⋅)A2(⋅), in which the policyholder holds a short position, is (the value of) a continuum of European call options written on A1(⋅)/h(⋅)A1(⋅)/h(⋅) with strike View the MathML sources¯(⋅) paying off the surplus View the MathML source[A1(t)/h(t)−s¯(t)]+ in the event of death at time tt. The amount allocated to the first component is determined by the budget constraint π0+g(0)=A1(0)−A2(0)+A3(0).π0+g(0)=A1(0)−A2(0)+A3(0). Turn MathJax on In contrast to the unconstrained case, the optimal investment and life insurance strategies should not be viewed separately since the optimal investment strategy depends on the upper and lower boundaries and is specifically designed so as to hedge against the “otherwise optimal” sum insured A1(⋅)/h(⋅)A1(⋅)/h(⋅) being outside the boundaries.