OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 221 be offered except for the simple(but ad- Rather, any sequence of portfolio deci vantageous)strategy of taking one thing sions must be contingent upon the out at a time. Such a partial analysis has comes of previous periods and at the been justified by picturing the investor same time take into account information as providing for a series of future on future probability distributions, It is sumption dates by dividing total wealth only when the last period is reached, and into separate portfolios for each con- the final decision is to be taken, that the sumption date, with each such portfolio simple models of the preceding section to be managed independently. This pro- are applicable cedure must be rejected as clearly sub- At the beginning of the last period (n) optimal and hardly represents a satisfac- the investor's problem is simply to make tory solution. Neither the decision on a decision(call it dn), dividing his wealth how much to consume in any given pe- as of that time, An-1, among the different riod nor the management of any given assets such that Em[U(A,)] is maximized subportfolio could generally be independ -(the notation En indicates expectation ent of actual performance of other port- with respect to probability distributions folios. The first attempts to consider of yields during the nth period). But once squarely the interrelations between con- he has thus chosen his optimal decision sumption and portfolio decisions appear (depending, in general, upon An-1),the be represented by the still unpublished maximum o of expected utility of final papers by Dreze and Modigliani and by wealth is determined solely in terms of In our version of the theory, an in vestor, starting out with a given initial max En[U(An)=φn(An) wealth Ao, will make a first-period deci- The function n-1 is referred to as the sion on the allocation of this wealth to indirect""or "derived"utility functio different assets, then wait until the end of the period when a wealth level A1 ma- and is the appropriate representation of terializes. He then makes a second-period preferences over probability distributions decision on the allocation of Al, and so for An-1. Therefore, the optimal decision d,-1 to choose at the beginning of period 1 is the one which maximizes It is clear that for such a multiperiod planning problem it is rarely optimal, if En-1lom-1(An-1)]=En-lmax En[U(An)Il at all possible, to specify a sequence of single-period decisions once and for all Is way Nor could it generally be optimal to the next-to-last decision as a simple one- simply make a first-period decision that period problem, granted that the objec- would maximize expected utility of tive is appropriately defined in terms of wealth at the end of that period while the "derived" utility function. But to do disregarding the investment opportuni- so obviously requires the investor to ties in the second and later periods. specify the optimal last-period decision for every possible outcome of yield dur- eriod n is by means of such J. Dreze and F. Modigliani, "Consumption a backward-recursive procedure that it is Decisions under uncertai possible to determine an optimal first and Portfolio Choice"(manuscript in preparation ). period decision his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and ConditionsOPTIMAL MULTIPERIOD PORTFOLIO POLICIES 221 be offered except for the simple (but ad￾vantageous) strategy of taking one thing at a time. Such a partial analysis has been justified by picturing the investor as providing for a series of future con￾sumption dates by dividing total wealth into separate portfolios for each con￾sumption date, with each such portfolio to be managed independently.7 This pro￾cedure must be rejected as clearly sub￾optimal and hardly represents a satisfac￾tory solution. Neither the decision on how much to consume in any given pe￾riod nor the management of any given subportfolio could generally be independ￾ent of actual performance of other port￾folios. The first attempts to consider squarely the interrelations between con￾sumption and portfolio decisions appear to be represented by the still unpublished papers by Dreze and Modigliani and by Sandmo.8 In our version of the theory, an in￾vestor, starting out with a given initial wealth Ao, will make a first-period deci￾sion on the allocation of this wealth to different assets, then wait until the end of the period when a wealth level A 1 ma￾terializes. He then makes a second-period decision on the allocation of A1, and so on. It is clear that for such a multiperiod planning problem it is rarely optimal, if at all possible, to specify a sequence of single-period decisions once and for all. Nor could it generally be optimal to simply make a first-period decision that would maximize expected utility of wealth at the end of that period while disregarding the investment opportuni￾ties in the second and later periods. Rather, any sequence of portfolio deci￾sions must be contingent upon the out￾comes of previous periods and at the same time take into account information on future probability distributions. It is only when the last period is reached, and the final decision is to be taken, that the simple models of the preceding section are applicable. At the beginning of the last period (n) the investor's problem is simply to make a decision (call it d.), dividing his wealth as of that time, An-1 among the different assets such that E4[U(An)] is maximized (the notation En indicates expectation with respect to probability distributions of yields during the nth period). But once he has thus chosen his optimal decision (depending, in general, upon An-1), the maximum of expected utility of final wealth is determined solely in terms of An-1, that is, max En[U(An)] =na (An-l) . dn The function 4n-1 is referred to as the "indirect" or "derived" utility function and is the appropriate representation of preferences over probability distributions for An-1. Therefore, the optimal decision d._1 to choose at the beginning of period n - 1 is the one which maximizes En-J[?n-i(An-] = En-,{max En[U(An)] } dn In this way it is possible to consider the next-to-last decision as a simple one￾period problem, granted that the objec￾tive is appropriately defined in terms of the "derived" utility function. But to do so obviously requires the investor to specify the optimal last-period decision for every possible outcome of yield dur￾ing period n - 1. It is by means of such a backward-recursive procedure that it is possible to determine an optimal first￾period decision. 7lbid. 8 J. Dreze and F. Modigliani, "Consumption Decisions under Uncertainty" (manuscript in prepa￾ration); A. Sandmo, "Capital Risk, Consumption, and Portfolio Choice" (manuscript in preparation). This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions