16 THE REVIEW OF ECONOMICS AND STATISTICS With respect to an investor's criterion for optimal mix of risk assets conditional on a given choices among different attainable combinations gross investment in this portfolio,and then for- of assets,we assume that(3)if any two mixtures mally proving the critical invariance property of assets have the same expected return,the inves-stated in the theorem.Tobin used more restric- tor will prefer the one having the smaller variance tive assumptions that we do regarding the avail- of return,and if any two mixtures of assets have able investment opportunities and he permitted the same variance of returns,he will prefer the no borrowing.u Under our somewhat broadened one having the greater expected value.Tobin [21,assumptions in these respects,the problem fits pp.75-76 has shown that such preferences are neatly into a traditional Fisher framework,with implied by maximization of the expected value different available combinations of expected of a von Neumann-Morgenstern utility function values and standard deviations of return on al- if either (a)the investor's utility function is con-ternative slock portfolios taking the place of cave and quadratic or (b)the investor's utility the original"production opportunity"set and function is concave,and he has assigned probabil- with the alternative investment choices being ity distributions such that the returns on all pos- concurrent rather than between time periods. sible portfolios differ at most by a location and scale Within this framework,alternative and more parameler,(which will be the case if the joint dis-transparent proofs of the separation theorem tribution of all individual stocks is multivariate are available which do not involve the actual normal). calculation of the best allocation in stocks over individual stock issues.As did Fisher,we shall Alternative Proofs of the Separation Theorem present a simple algebraic proofi2,set out the Since the interest rates on riskless savings logic of the argument leading to the theorem,and bank deposits ("loans to the bank")and on bor- depict the essential geometry of the problem.13 rowed funds are being assumed to be the same, As a preliminary step,we need to establish the we can treat borrowing as negative lending. relation between the investor's total investment Any portfolio can then be described in terms of in any arbitrary mixture or portfolio of individual (i)the gross amount invested in stocks,(ii)the stocks,his total net return from all his invest- fraction of this amount invested in each indivi- ments (including riskless assets and any borrow- dual stock,and (iii)the net amount invested in ing),and the risk parameters of his investment loans(a negative value showing that the investor position.Let the interest rate on riskless assets has borrowed rather than lent).But since the or borrowing be r*,and the uncertain relurn(divi- total net investment (the algebraic sum of stocks dends plus price appreciation)per dollar invested plus loans)is a given amount,the problem sim- in the given portfolio of stocks be r.Let w rep- ply requires finding the jointly optimal values resent the ralio of gross investment in stocks to for(1)the ratio of the gross investment in stocks uTobin considered the special case where cash with no return was the only riskless asset available.While he formally to the total net investment,and(2)the ratio of required that all assets be held in non-negative quantities the gross investment in each individual stock to (thereby ruling out short sales),and that the total value of risk the total gross investment in stocks.It turns out assets held not be greater than the investment balance available that although the solution of (1)depends upon without borrowing,these non-negativity and maximum value constraints were not introduced into his formal solution of the that of (2),in our context the latter is indepen- optimal investment mix,which in turn was used in proving the dent of the former.Specifically,the separalion invariance property stated in the theorem.Our proof of the theorem is independent of the programming constraints neglec- theorem asserts that: ted in Tobin's proof.Later in this section we show that when Given the assumptions about borrowing, short sales are properly and explicitly introduced into the set lending,and investor preferences stated earlier in of possible portfolios,the resulting equations for the optimum portfolio mix are identical to those derived by Tobin,but that this section,the optimal proportionate composition insistence on no short sales results in a somewhat more complex of the stock (risk-asset)portfolio (i.e.the solution programming problem (when covariances are non-zero),which to sub-problem 2 above)is independent of the may however,be readily handled with computer programs now available. ralio of the gross investment in stocks to the total net 12An alternative algebraic proof using utility functions inves!ment. explicitly is presented in the appendix,note I. Tobin proved this important separation theo- 1Lockwood Rainhard,Jr.hasalsoindependently developed and presented a similar proof of the theorem in an unpublished rem by deriving the detailed solution for the seminar paper. 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/terms16 THE REVIEW OF ECONOMICS AND STATISTICS With respect to an investor's criterion for choices among different attainable combinations of assets, we assume that (3) if any two mixtures of assets have the same expected return, the inves- tor will prefer the one having the smaller variance of return, and if any two mixtures of assets have the same variance of returns, he will prefer the one having the greater expected value. Tobin [21, pp. 75-761 has shown that such preferences are implied by maximization of the expected value of a von Neumann-Morgenstern utility function if either (a) the investor's utility function is con- cave and quadratic or (b) the investor's utility function is concave, and he has assigned probabil- ity distributions such that the returns on all pos- sible portfolios differ at most by a location and scale parameter, (which will be the case if the joint dis- tribution of all individual stocks is multivariate normal). Alternative Proofs of the Separation Theorem Since the interest rates on riskless savings bank deposits ("loans to the bank") and on bor- rowed funds are being assumed to be the same, we can treat borrowing as negative lending. Any portfolio can then be described in terms of (i) the gross amount invested in stocks, (ii) the fraction of this amount invested in each indivi- dual stock, and (iii) the net amount invested in loans (a negative value showing that the investor has borrowed rather than lent). But since the total net investment (the algebraic sum of stocks plus loans) is a given arnount, the problem sim- ply requires finding the jointly optimal values for (1) the ratio of the gross investment in stocks to the total net investment, and (2) the ratio of the gross investment in each individual stock to the total gross investment in stocks. It turns out that although the solution of (1) depends upon that of (2), in our context the latter is indepen- dent of the former. Specifically, the separation theorem asserts that: Given the assumptions about borrowing, lending, and investor preferences stated earlier in this section, the optimal proportionate composition of the stock (risk-asset) portfolio (i.e. the solution to sub-problem 2 above) is independent of the ratio of the gross investment in stocks to the total net investment. Tobin proved this important separation theo- ren by deriving the detailed solution for the optimal mix of risk assets conditional on a given gross investment in this portfolio, and then for- mally proving the critical invariance property stated in the theorem. Tobin used more restric- tive assumnptions that we do regarding the avail- able investment opportunities and he pernmitted no borrowing." Under our somewhat broadened assumptions in these respects, the problem fits neatly into a traditional Fisher framework, with different available combinations of expected values and standard deviations of return on al- ternative stock portfolios taking the place of the original "production opportunity" set and with. the alternative investment choices being concurrent rather than between time periods. Within this frarmework, alternative and more transparent proofs of the separation theorem are available which do not involve the actual calculation of the best allocation in stocks over individual stock issues. As did Fisher, we shall present a simple algebraic proof 12, set out the logic of the argument lea-ding to the theorem, and depict the essential geomretry of the problemr.13 As a preliminary step, we need to establish the relation between the investor's total investment in any arbitrary mixture or portfolio of individual stocks, his total net return from all his invest- nments (including risliless assets and any borrow- ing), and the risk parameters of his investment position. Let the interest rate on riskless assets or borrowing be r*, and the uncertain return (divi- dends plus price appreciation) per dollar invested in the given portfolio of stocks be r. Let w rep- resent the ratio of gross investment in stocks to "1Tobin considered the special case where cash with no return was the only riskless asset available. While he formally required that all assets be held in non-negative quantities (thereby ruling out short sales), and that the total value of risk assets held not be greater than the investment balance available without borrowing, these non-negativity and maximum value constraints were not introduced into his formal solution of the optimal investment mix, which in turn was used in proving the invariance property stated in the theorem. Our proof of the theorem is independent of the programming constraints neglec- ted in Tobin's proof. Later in this section we show that when short sales are properly and explicitly introduced into the set of possible portfolios, the resulting equations for the optimum portfolio mix are identical to those derived by Tobin, but that insistence on no short sales results in a somewhat more complex programming problem (when covariances are non-zero), which may however, be readily handled with computer programs now available. 12An alternative algebraic proof using utility functions explicitly is presented in the appendix, note I. 13 Lockwood Rainhard, Jr. has also independently developed and presented a similar proof of the theorem in an unpublished seminar paper. 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