444 NILS H. HAKANSSON The components de in Section 2 are assembled into formal model in Sections 3, 4.6. 7.8 The fundamental approach taken is that the portfolio composition the financing decision, the consumption decision and, where applicable the insurance decision, are all analyzed in one model The vehicle of analysis is discrete-time dynamic programming Sections 4, 6, 7, and 8 consider the four possible combinations of no bequest motive/ bequest motive and no insurance/insurance Explicit solutions are derived, where possible, for that class of one-period utility functions whose proportional risk aversion indices are positive constants and are found have the same form as when the horizon is known. A review of the prop erties and implications of these solutions is given in Section 5: it is noted that due to the solvency constraint, the solution does not always exist in this form for all functions in the class In Section 9, the amount of insurance to be purchased in each period is included among the decision variables. when this is done the solution found to be of the indicated form only under highly specialized conditions the optimal insurance strategy is found to be linear increasing in the future installments of the non-capital income stream In Sectio is shown that the models developed in this paper give rise o an induced theory of the firm under risk, which may be viewed as ar extension of the theory developed for the case in which the horizon is certai is shown in Section 11 that when the premium charged is fair", an individual can in most instances increase his expected utility by selling insurance to others. Thus, any given individual may be able to make himself better off both by the purchase of insurance on his own life and the sale of insurance on the lives of others. Furthermore, both a supply of and a demand for insurance will exist in an economy of individuals whose utility functions belong to the class examined ASSUMPTIONS AND NOTATION In this section, the postulates concerning the individuals preferences,re- sources, and opportunities will be specified. As the various building blocks are introduced, we also give the notation to be used in the following sections 2. 1. Resources and opportunities. We assume that the individual has the opportunity to make decisions at diserete points, called decision points, which are equally spaced in time. The space of time intervening between the two adjacent decision points 3 and j+1 will be denoted period Let pi>0 be the individuals probability of dying in the j-th period, j= 1,.,n, where 2j=1pi=l; thus n is the last period in which death may occur, We now observe that (1) pm≡p∑p m,y=1,…,n(m≤j expresses the probability that the individual will pass away in period j given that he is alive at the beginning of period m. e denote the amount of the individuals monetary(capital) resources at This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM444 NILS H. HAKANSSON The components developed in Section 2 are assembled into formal models in Sections 3, 4, 6, 7, 8, and 9. The fundamental approach taken is that the portfolio composition decision, the financing decision, the consumption decision, and, where applicable, the insurance decision, are all analyzed in one model. The vehicle of analysis is discrete-time dynamic programming. Sections 4, 6, 7, and 8 consider the four possible combinations of no bequest motive/bequest motive and no insurance/insurance. Explicit solutions are derived, where possible, for that class of one-period utility functions whose proportional risk aversion indices are positive constants, and are found to have the same form as when the horizon is known. A review of the prop￾erties and implications of these solutions is given in Section 5; it is noted that due to the solvency constraint, the solution does not always exist in this form for all functions in the class. In Section 9, the amount of insurance to be purchased in each period is included among the decision variables. When this is done, the solution is found to be of the indicated form only under highly specialized conditions; the optimal insurance strategy is found to be linear increasing in the future installments of the non-capital income stream. In Section 10, it is shown that the models developed in this paper give rise to an induced theory of the firm under risk, which may be viewed as an extension of the theory developed for the case in which the horizon is certain [6]. Finally, it is shown in Section 11 that when the premium charged is "fair", an individual can in most instances increase his expected utility by selling insurance to others. Thus, any given individual may be able to make himself better off both by the purchase of insurance on his own life and the sale of insurance on the lives of others. Furthermore, both a supply of and a demand for insurance will exist in an economy of individuals whose utility functions belong to the class examined. 2. ASSUMPTIONS AND NOTATION In this section, the postulates concerning the individual's preferences, re￾sources, and opportunities will be specified. As the various building blocks are introduced, we also give the notation to be used in the following sections. 2.1. Resources and opportunities. We assume that the individual has the opportunity to make decisions at discrete points, called decision points, which are equally spaced in time. The space of time intervening between the two adjacent decision points j and j + 1 will be denoted period j. Let ]5j > 0 be the individual's probability of dying in the j-th period, j = 1, *.., n, where E =fpj = 1; thus n is the last period in which death may occur. We now observe that (1) Pmj-Pi l2Pac m,=, i n(,m< j) k-m expresses the probability that the individual will pass away in period j given that he is alive at the beginning of period m. We denote the amount of the individual's monetary (capital) resources at This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions