18 THE REVIEW OF ECONOMICS AND STATISTICS shows this point can be obtained directly without has been determined,the investor completes the calculating the remainder of the efficient set.)optimization of his total investment position Given the same assumptions,(iii)the para- by selecting the point on the ray through M meters of the investor's particular utility within which is tangent to a utility contour in the the relevant set determine only the ratio of his standard manner.If his utility contours are as total gross investment in stocks to his total net in the Ui set in chart 1,he uses savings accounts investment (including riskless assets and borrow-and does not borrow.If his utility contours are ing);and(i)the investor's wealth is also,conse-as in U;set,he borrows in order to have a gross quently,relevant to determining the absolute size investment in his best stock mix greater than his of his investment in individual stocks,but not to net investment balance. the relalive distribution of his gross investment in stocks among individual issues. Risk Aversion,Normality and the Separation Theorem The Geometry of the Separation Theorem and Its The above analysis has been based on the Corrolaries assumptions regarding markets and investors The algebraic derivations given above can stated at the beginning of this section.One be represented graphically as in chart 1.Any crucial premise was investor risk-aversion in the given available stock portfolio is characterized by a pair of values (ar,F)which can be repre- form of preference for expected return and prefer- sented as a point in a plane with axes ay and ence against relurn-variance,ceteris paribus.We noted that Tobin has shown that either concave- Our assumptions insure that the pointsrepresent- guadratic utility functions or multivariate nor- ing all available stock mixes lie in a finite region, all parts of which lie to the right of the vertical mality (of probability assessments)and any con- cave utility were sufficient conditions to validate axis,and that this region is bounded by a closed this premise,but they were not shown (or alleged) curve.18 The contours of the investor's utility to be necessary conditions.This is probably for- function are concave upward,and any movement tunate because the quadratic utility of income in a north and or west direction denotes con- (or wealth!)function,in spite of its popularity in tours of greater utility.Equation(3)shows that theoretical work,has several undesirably restric- all the (y)pairs attainable by combining, tive and implausible properties,20 and,despite borrowing,or lending with any particular stock portfolio lie on a ray from the point (0,*) rowing and lending rates is clear.The optimal set of produc- though the point corresponding to the stock mix tion opportunities available is found by moving along the en- in question.Each possible stock portfolio thus velope function of efficient combinations of projects onto ever higher present value lines to the highest attainable.This best determines a unique"market opportunity line". set of production opportunities is independent of the investor's Given the properties of the utility function,it is particular utility function which determines only whether he obvious that shifts from one possible mix to then lends or borrows in the market (and by how much in either case)to reach hi best over-all position.The only diff. another which rotale the associated market op- erences between this case and ours lie in the concurrent nature portunity line counter colckwise will move the inves- of the comparisons (instead of inter-period),and the rotation tor to preferred positions regardless of the point on of the market opportunity lines around the common pivot of the riskless return (instead of parallel shifts in present value the line he had tentatively chosen.The slope of lines).See Fisher [4]and also Hirschlaifer [7],figure 1 and this market-opportunity line given by (3)is 0, section Ia. and the limit of the favorable rotation is given 2In brief,not only does the quadratic function imply negative marginal utilities of income or wealth much"too soon" by the maximum attainable 0,which identifies in empirical work unless the risk-aversion parameter is very the optimal mix M.19 Once this best mix,M, small-in which case it cannot account for the degree of risk- aversion empirically found,-it also implies that,over a major graph of this section),the modest narrowing of the relevant part of the range of empiricaldata,common stocks,like potatoes range of Markowitz'Efficient Set suggested by Baumol [2]is in Ireland,are "inferior"goods.Offering more return at the still larger than needed by a factor strictly proportionate to the same risk would so sate investors that they would reduce their number of portfolios he retains in his truncated set!This is risk-investments because they were more attractive.(Thereby, true since the relevant set is a single portfolio under these con- as Tobin [2]noted,denying the negatively sloped demand ditions. curves for riskless assets which are standard doctrine in"liqui- isSee Markowitz [14]as cited in the appendix,note I. dity preference theory"-a conclusion which cannot,inciden- 1The analogy with the standard Fisher two-period pro-tally,be avoided by "limit arguments"on quadratic utilities duction-opportunity case in perfect markets with equal bor- such as he used,once borrowing and leverage are admitted.) This content downloaded from 202.120.21.61 on Mon,06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms18 THE REVIEW OF ECONOMICS AND STATISTICS shows this point can be obtained directly without calculating the remainder of the efficient set.) Given the same assumptions, (iii) the para- meters of the investor's particular utility within the relevant set determine only the ratio of his total gross investment in stocks to his total net investment (including riskless assets and borrow- ing); and (iv) the investor's wealth is also, conse- quently, relevant to determining the absolute size of his investment in individual stocks, but not to the relative distribution of his gross investment in stocks among individual issues. The Geometry of the Separation Theorem and Its Corrolaries The algebraic derivations given above can be represented graphically as in chart 1. Any given available stock portfolio is characterized by a pair of values (0r, r) which can be repre- sented as a point in a plane with axes a- and y. Our assumptions insure that the points represent- ing all available stock mixes lie in a finite region, all parts of which lie to the right of the vertical axis, and that this region is bounded by a closed curve.'8 The contours of the investor's utility function are concave upward, and any movement in a north and or west direction denotes con- tours of greater utility. Equation (3) shows that all the (o-, y) pairs attainable by combining, borrowing, or lending with any particular stock portfolio lie on a ray from the point (0, r*) though the point corresponding to the stock mix in question. Each possible stock portfolio thus determines a unique "market opportunity line". Given the properties of the utility function, it is obvious that shifts from one possible mix to another which rotate the associated market op- portunity line counter colckwise will move the inves- tor to preferred positions regardless of the point on the line he had tentatively chosen. The slope of this market-opportunity line given by (3) is 0, and the limit of the favorable rotation is given by the maximum attainable 0, which identifies the optimal mix M.-9 Once this best mix, M, has been determined, the investor completes the optimization of his total investment position by selecting the point on the ray through M which is tangent to a utility contour in the standard manner. If his utility contours are as in the Ui set in chart 1, he uses savings accounts and does not borrow. If his utility contours are as in Uj set, he borrows in order to have a gross investment in his best stock mix greater than his net investment balance. Risk Aversion, Normality and the Separation Theorem The above analysis has been based on the assumptions regarding markets and investors stated at the beginning of this section. One crucial premise was investor risk-aversion in the form of preference for expected return and prefer- ence against return-variance, ceteris paribus. We noted that Tobin has shown that either concave- quadratic utility functions or multivariate nor- mality (of probability assessments) and any con- cave utility were sufficient conditions to validate this premise, but they were not shown (or alleged) to be necessary conditions. Ihis is probably for- tunate because the quadratic utility of income (or wealth!) function, in spite of its popularity in theoretical work, has several undesirably restric- tive and implausible properties,20 and, despite graph of this section), the modest narrowing of the relevant range of Markowitz' Efficient Set suggested by Baumol [2] iS still larger than needed by a factor strictly proportionate to the number of portfolios he retains in his truncated set! This is true since the relevant set is a single portfolio under these con- ditions. See Markowitz II4] as cited in the appendix, note I. The analogy with the standard Fisher two-period pro- duction-opportunity case in perfect markets with equal bor- rowing and lending rates is clear. The optimal set of produc- tion opportunities available is found by moving along the en - velope function of efficient combinations of projects onto ever higher present value lines to the highest attainable. This best set of production opportunities is independent of the investor's particular utility function which determines only whether he then lends or borrows in the market (and by how much in either case) to reach hi best over-all position. The only diff- erences between this case and ours lie in the concurrent nature of the comparisons (instead of inter-period), and the rotation of the market opportunity lines around the common pivot of the riskless return (instead of parallel shifts in present value lines). See Fisher [4] and also Hirschlaifer [7], figure 1 and section Ia. 20In brief, not only does the quadratic function imply negative marginal utilities of income or wealth much"too soon" in empirical work unless the risk-aversion parameter is very small - in which case it cannot account for the degree of risk- aversion empirically found,- it also implies that, over a major part of the range of empiricaldata,commonstocks,like potatoes in Ireland, are "inferior" goods. Offering more return at the same risk would so sate investors that they would reduce their risk-investments because they were more attractive. (Thereby, as Tobin [2I] noted, denying the negatively sloped demand curves for riskless assets which are standard doctrine in "liqui- dity preference theory" - a conclusion which cannot, inciden- tally, be avoided by "limit arguments" on quadratic utilities such as he used, once borrowing and leverage are admitted.) This content downloaded from 202.120.21.61 on Mon, 06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms