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《投资学原理 Investments》课程教学资源（阅读资料）optimal investment and consumption strategies under risk, an uncertain lifetime and insurance

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INTERNATIONAL ECONOMIC REVIEW Vol. 10, No 3, October, 1969 OPTIMAL INVESTMENT AND CONSUMPTION STRATEGIES UNDER RISK, AN UNCERTAIN LIFETIME, AND INSURANCE* BY NILS H. HAKANSSON 1. INTRODUCTION AND SUMMARY IN A PREVIOUS ARTICLE [8, a normative model of the individuals economic decision problem under risk was presented. In addition, certain implications of the model with respect to individual behavior were deduced for the class of utility functions, 2i=rai-lu(ej),0<a<l, where c; is the amount of con sumption in period 3, such that either the risk aversion index -u(e)/u'(a), or the risk aversion index -ou'(e)lu'(e), is a positive constant for all a20. In a second paper [6], it was further shown that this model, developed with the individual in mind, also gives rise to an induced theory of the firm under risk for the same class of utility functions. In the foregoing model, it was assumed that the individual' s horizon was infinite(or known with certainty ) In this paper, we consider the same basic model with three modifications. First, we postulate that the individuals lifetime is a random variable with a known probability distribution. Second we introduce a utility function intended to represent the individuals bequest motive. Third, we offer the individual the opportunity to purchase insurance on his life. It is found that when some or all of these modifications are made all of the more important properties possessed by the optimal consumption and investment strategies under a certain horizon are preserved, albeit only d In Section 2, the various components of the decision process are constructe In the earlier model, the individuals objective was assumed to be the maxi- mization of expected utility from consumption over time. Here, we postulate more generally, that his objective is to maximize expected utility from con sumption as long as he lives and from the bequest left upon his death. As before, the individual's resources are assumed to consist of an initial capital position(which may be negative)and a non-capital income stream. The latter, which may possess any time-shape, is assumed to be known with certainty and to terminate upon his death. In addition to insurance available at a"fair rate, the individual faces both financial opportunities(borrowing and lending) and an arbitrary number of productive investment opportunities. The interest rate is presumed to be known but may have any time shape. The returns from the productive opportunities are assumed to be random variables, whose probability distributions may differ from period to period but are assumed to satisfy the"no-easy-money"condition. While no limit is placed on borrow- ing the individual is required to be solvent at the time of his death with probability 1, that is, all debt must be fully secured at all times Manuscript received November 22, 1967, revised June 3, 1968. This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM

INTERNATIONAL ECONOMIC REVIEW Vol. 10, No. 3, October, 1969 OPTIMAL INVESTMENT AND CONSUMPTION STRATEGIES UNDER RISK, AN UNCERTAIN LIFETIME, AND INSURANCE* BY NILs H. HAKANSSON 1. INTRODUCTION AND SUMMARY IN A PREVIOUS ARTICLE [8], a normative model of the individual's economic decision problem under risk was presented. In addition, certain implications of the model with respect to individual behavior were deduced for the class of utility functions, Z' , ai-lu(ci), 0 < a < 1, where cj is the amount of consumption in period j, such that either the risk aversion index -u"(x)/u'(x), or the risk aversion index -xu"'(x)/u'(x), is a positive constant for all x > 0. In a second paper [6], it was further shown that this model, developed with the individual in mind, also gives rise to an induced theory of the firm under risk for the same class of utility functions. In the foregoing model, it was assumed that the individual's horizon was infinite (or known with certainty). In this paper, we consider the same basic model with three modifications. First, we postulate that the individual's, lifetime is a random variable with a known probability distribution. Second, we introduce a utility function intended to represent the individual's bequest motive. Third, we offer the individual the opportunity to purchase insurance on his life. It is found that when some or all of these modifications are made, all of the more important properties possessed by the optimal consumption and investment strategies under a certain horizon are preserved, albeit only under special conditions. In Section 2, the various components of the decision process are constructed. In the earlier model, the individual's objective was assumed to be the maximization of expected utility from consumption over time. Here, we postulate, more generally, that his objective is to maximize expected utility from con-- sumption as long as he lives and from the bequest left upon his death. As before, the individual's resources are assumed to consist of an initial capital position (which may be negative) and a non-capital income stream. The latter, which may possess any time-shape, is assumed to be known with certainty and to terminate upon his death. In addition to insurance available at a "fair'" rate, the individual faces both financial opportunities (borrowing and lending) and an arbitrary number of productive investment opportunities. The interest rate is presumed to be known but may have any time-shape. The returns from the productive opportunities are assumed to be random variables, whose probability distributions may differ from period to period but are assumed to satisfy the "no-easy-money" condition. While no limit is placed on borrowing, the individual is required to be solvent at the time of his death with probability 1, that is, all debt must be fully secured at all times. * Manuscript received November 22, 1967, revised June 3, 1968. 443 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

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 AM

444 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 properties 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, resources, 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

INVESTMENT AND CONSUMPTION 445 the j-th decision point, given that he is alive at that point, by j. In the event the individual passes away in period j-1, the amount of his resources at the end of that period will be termed his estate and will be designated wj We assume that the individual may also be the recipient of a non-capital income stream during all or part of his life-time. If the individual is alive at decision point j, he will be paid the(finite)installment pertaining to period 3, 1320, at the end of that period; if he is not alive, he will receive nothing. In this paper, we make the fairly strong assumption that the individuals tential non-capital income stream is exogenously determined and is known in advance. It may be thought of as consisting of the income from labor, pensions, unemployment compensation, ete We postulate that the individual faces both financial and productive oppor- tunities in each period. The first of these is the opportunity to borrow or lend arbitrary amounts of money in each period at the riskless(finite)rate r-1>0 on the condition that any borrowings (including interest)must be fully secured. The amount saved at decision point j will be denoted z1i; negative aii will then indicate borrowing For cont ce, we shall define (2) Y rirj+i where Y, may be interpreted as the present value of the individual's potential non-capital income stream at the i-th decision point. The productive opportunities faced by the individual consist of the possi bility of making risky investments. Let the total number of different risky (productive) opportunities available to the individual at decision point j be Mi-l, of which the first S,-1 s M-l may be sold short. A short sale will be defined as the opposite of a long investment, that is, if the individual ells opportunity i short in the amount e, he will receive a immediately (te do with as he pleases)in return for the obligation to pay the transformed value of e at the end of the period. The net proceeds realized at the end of of that period will be denoted Bij. Thus, returns to scale are assumed to period j from each unit of capital invested in opportunity i at the beginni stochastically constant, all investments are assumed to be realized in cash at the end of each period, and taxes and conversion costs, if any, are assumed to be proportional to the amount invested The amount invested in opportunity ,讠=2,……,M, at the j-th decision point will be denoted zij, and is,as indicated earlier, a decision variable along with z1i It will be assumed that the joint distribution functions F, given by (3)Fa2,m,…,mM)≡Pr{a≤m,月3s3,…,M;≤},j=1,…… are known and independent In addition, we shall postulate that the [Bish 1 In real world situations, the individual would, of course, be forced to derive h own subjective probability distributions. Numerous descriptions of how this may be ccomplished, on the basis of postulates presupposing certain consistencies in behavior, are available in the literature; see, for example the accounts of Savage [ 14]an Marschak [1 This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM

INVESTMENT AND CONSUMPTION 445 the j-th decision point, given that he is alive at that point, by x;. In the event the individual passes away in period j - 1, the amount of his resources at the end of that period will be termed his estate and will be designated x. We assume that the individual may also be the recipient of a non-capital income stream during all or part of his life-time. If the individual is alive at decision point j, he will be paid the (finite) installment pertaining to period j, yj ? 0, at the end of that period; if he is not alive, he will receive nothing. In this paper, we make the fairly strong assumption that the individual's potential non-capital income stream is exogenously determined and is known in advance. It may be thought of as consisting of the income from labor, pensions, unemployment compensation, etc. We postulate that the individual faces both financial and productive opportunities in each period. The first of these is the opportunity to borrow or lend arbitrary amounts of money in each period at the riskless (finite) rate rj- 1 > 0 on the condition that any borrowings (including interest) must be fully secured. The amount saved at decision point j will be denoted z,j; negative zlj will then indicate borrowing. For convenience, we shall define (2) yj - yj Yi+1 ... + Yn j1* ,n r3 rjrj+i ri ... rn where Yj may be interpreted as the present value of the individual's potential non-capital income stream at the j-th decision point. The productive opportunities faced by the individual consist of the possibility of making risky investments. Let the total number of different risky (productive) opportunities available to the individual at decision point j be Mj- 1, of which the first Sj - 1 < Mj -- 1 may be sold short. A short sale will be defined as the opposite of a long investment, that is, if the individual sells opportunity i short in the amount 0, he will receive a immediately (to do with as he pleases) in return for the obligation to pay the transformed value of 0 at the end of the period. The net proceeds realized at the end of period j from each unit of capital invested in opportunity i at the beginning of that period will be denoted ,Bj. Thus, returns to scale are assumed to be stochastically constant, all investments are assumed to be realized in cash at the end of each period, and taxes and conversion costs, if any, are assumed to be proportional to the amount invested. The amount invested in opportunity i, i = 2, ..., Mi, at the j-th decision point will be denoted zij, and is, as indicated earlier, a decision variable along with zlj. It will be assumed that the joint distribution functions Fj given by (3) Fi(x2, X3, * * *, XMj) - Pr{j92j < X2, l3j < X3* X8, ', imjj x XMj} 'j = 1, **, n are known and independent'. In addition, we shall postulate that the {J9ij} 1 In real world situations, the individual would, of course, be forced to derive his own subjective probability distributions. Numerous descriptions of how this may be accomplished, on the basis of postulates presupposing certain consistencies in behavior, are available in the literature; see, for example, the accounts of Savage [14] and Marschak [11]. 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

446 NILS HAKANSSON satisfy the following conditions for all j and all finite Bi; such that 0 i20 for all i>s, and 0,j+0 for at least one i.( 5) is known as the"no-easy-money condition for the case when the lending rate equals the borrowing rate [8]. This condition states that no combination of productive investment opportunities exists in any period which provides, with probability 1, a return at least as high as the(borrowing)rate of interest; no combination of short sales is available in which the probability is zero that a loss will exceed the (lending)rate of interest; and no combination of productive investments made from the proceeds of any short sale can guarantee against loss In some variants of the basic model the individual has the opportunity to purchase term insurance on his own life and to sell (purchase)term insurance on the lives of others in each period. Let t20 denote the premium paid by the individual at the j-th decision point for life insurance on his own life during period 3. If the individual dies during this period, which by(1)has probability pii of happening, we assume that his estate will receive ti/pi; at the end of period 3; if he is alive at decision point 3+1, he will receive nothing Since in this contract the mathematical expectation of the"return equals the cost we shall say that the insurance is available at a"fair" rate We assume that insurance is issued only when pjj<l, i.e. at decision points We shall allow the possibility of contracting in advance for purchases of insurance on the individuals own life. Such an arrangement will be called an insurance contract. The unexpired portion of such a contract at decision point 3 will be denoted (ti, ti+i,., tw-i, where tu/pek is the amount of insurance the individual will keep in force in period k given that he is alive at the ke-th decision point (when the premium tk is paid) For convenience we define 7=2+ We also assume that t≤x+B, where B, denotes the maximum an individual may borrow at the j-th decision point on the security of his non-capital income stream and his insurance contract. Since no insurance can be issued at the nth decision point, it is clear that (8) Bw=yn/r. and that t计+1+B This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM

446 NILS H. HAKANSSON satisfy the following conditions: (4) O? jij <oo, 2, M; j = 1, n Mi (5) Pr E (jj - rj)Oij < 0 > 0 for all j and all finite Oij such that Oij > 0 for all i > Sj and Oiji 0 for at least one i. (5) is known as the "no-easy-money" condition for the case when the lending rate equals the borrowing rate [8]. This condition states that no combination of productive investment opportunities exists in any period which provides, with probability 1, a return at least as high as the (borrowing) rate of interest; no combination of short sales is available in which the probabilityis zero that a loss will exceed the (lending) rate of interest; and no combination of productive investments made from the proceeds of any short sale can guarantee against loss. In some variants of the basic model the individual has the opportunity to purchase term insurance on his own life and to sell (purchase) term insurance on the lives of others in each period. Let tj ? 0 denote the premium paid by the individual at the j-th decision point for life insurance on his own life during period j. If the individual dies during this period, which by (1) has probability pjj of happening, we assume that his estate will receive tj/pjj at the end of period j; if he is alive at decision point j + 1, he will receive nothing. Since in this contract the mathematical expectation of the "return"' equals the cost, we shall say that the insurance is available at a "fair" rate. We assume that insurance is issued only when pjj < 1, i.e., at decision points. I, * **, n-1. We shall allow the possibility of contracting in advance for purchases of insurance on the individual's own life. Such an arrangement will be called an insurance contract. The unexpired portion of such a contract at decision point j will be denoted (tj, tj+?, * *, tn-1), where tklPkk is the amount of insurance the individual will keep in force in period k given that he is alive. at the k-th decision point (when the premium tk is paid). For convenience we define (6) Tj-tj+ ti+ ... + tn-i j-1,...,n-1. rj ri ... rn-2 We also assume that (7) tj < xj +Bj, j-1, * l ,nwhere Bj denotes the maximum an individual may borrow at the j-th decision point on the security of his non-capital income stream and his insurance contract. Since no insurance can be issued at the n-th decision point, it is, clear that (8) Bn = yn/rn, and that (9) Bj = min { r__+ ' _ jtj B, } = X-1 ri ri~~~I 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

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