THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS* John Lintner Introduction and Preview of Some Conclusions titive markets when utility functions are quad- HE effects of risk and uncertainty upon ratic or rates of return are multivariate normal.1 asset prices,upon rational decision rules We then note that the same conclusion follows for individuals and institutions to use in selecting from an earlier theorem of Roy's 19 without security portfolios,and upon the proper selection dependence on quadratic utilities or normality. of projects to include in corporate capital bud- The second section shows that if short sales are gets,have increasingly engaged the attention of permilted,the best portfolio-mix of risk assets professional economists and other students of the can be determined by the solution of a single capital markets and of business finance in recent simple set of simultaneous equations without years.The essential purpose of the present paper recourse to programming methods,and when is to push back the frontiers of our knowledge of covariances are zero,a still simpler ratio scheme the logical structure of these related issues,albeit gives the optimum,whether or not short sales under idealized conditions.The immediately are permitted.When covariances are not all following text describes the contents of the paper zero and short sales are excluded,a single quad- and summarizes some of the principal results. ratic programming solution is required,but The first two sections of this paper deal with sufficient. the problem of selecting optimal security port- Following these extensions of Tobin's classic folios by risk-averse investors who have the al- work,we concentrate on the set of risk assets ternative of investing in risk-free securities with held in risk averters'portfolios.In section III we a positive return(or borrowing at the same rate develop various significant equilibrium proper- of interest)and who can sell short if they wish. ties within the risk asset portfolio.In particular, The first gives alternative and hopefully more we establish conditions under which stocks will transparent proofs (under these more general be held long (short)in optimal portfolios even market conditions)for Tobin's important "sep-when "risk premiums"are negative (positive). aration theorem''that "..the proportion- We also develop expressions for different combi- ate composition of the non-cash assets is inde- nations of expected rate of return on a given pendent of their aggregate share of the invest- security,and its standard deviation,variance, ment balance..."(and hence of the optimal and /or covariances which will result in the same holding of cash)for risk averters in purely compe- relative holding of a stock,ceteris paribus.These "indifference functions"provide direct evidence "This paper is another in a series of interrelated theoretical and statistical studies of corporate financial and investment on the moot issue of the appropriate functional policies being made under grants from the Rockefeller Founda- relationships between"required rates of return" tion,and more recently the Ford Foundation,to the Harvard and relevant risk parameter(s)-and on the Business School.The generous support for this work is most gratefully acknowledged.The author is also much indebted related issue of how "risk classes"'of securities to his colleagues Professors Bishop,Christenson,Kahr,Raiffa, may best be delineated (if they are to be used).? and (especially)Schlaifer,for extensive discussion and com- Tobin [2,especially pp.82-85].Tobin assumed that mentary on an earlier draft of this paper;but responsibility for funds are to be a allocated only over "monetary assets"(risk- any errors or imperfections remains strictly his own. free cash and default-free bonds of uncertain resale price)and [Professor Sharpe's paper,"Capital Asset Prices:A Theory allowed no short sales or borrowing.See also footnote 24 be- of Market Equilibrium Under Conditions of Risk"(Journal of low.Other approaches are reviewed in Farrar [38). Finance,September 1964)appeared after this paper was in *It should be noted that the classic paper by Modigliani final form and on its way to the printers.My first section and Miller [16]was silent on these issues. Corporations were which parallels the first half of his paper(with corresponding assumed to be divided into homogeneous classes having the conclusions),sets the algebraic framework for sections II property that all shares of all corporations in any given class III and VI,(which have no counterpart in his paper)and for differed (at most)by a"scale factor,"and hence (a)were per- section IV on the equilibrium prices of risk assets,concerning fectly correlated with each other and(b)were perfect substi- which our results differ significantly for reasons which will be tutes for each other in perfect markets(p.266).No comment explored elsewhere.Sharpe does not take up the capital was made on the measure of risk or uncertainty (or other budgeting problem developed in section V below.] attributes)relevant to the identification of different "equiva- [13] 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
THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS * John Lintner Introduction and Preview of Some Conclusions T HE effects of risk and uncertainty upon asset prices, upon rational decision rules for individuals and institutions to use in selecting security portfolios, and upon the proper selection of projects to include in corporate capital bud- gets, have increasingly engaged the attention of professional economists and other students of the capital markets and of business finance in recent years. The essential purpose of the present paper is to push back the frontiers of our knowledge of the logical structure of these related issues, albeit under idealized conditions. The immediately following text describes the contents of the paper and summarizes some of the principal results. The first two sections of this paper deal with the problem of selecting optimal security port- folios by risk-averse investors who have the al- ternative of investing in risk-free securities with a positive return (or borrowing at the same rate of interest) and who can sell short if they wish. The first gives alternative and hopefully more transparent proofs (under these more general market conditions) for Tobin's important "sep- aration theorem" that ". . . the proportion- ate composition of the non-cash assets is inde- pendent of their aggregate share of the invest- ment balance . . " (and hence of the optimal holding of cash) for risk averters in purely compe- titive markets when utility functions are quad- ratic or rates of return are multivariate normal.' We then note that the same conclusion follows from an earlier theorem of Roy's 1191 without dependence on quadratic utilities or normality. The second section shows that if short sales are permitted, the best portfolio-mix of risk assets can be determined by the solution of a single simple set of simultaneous equations without recourse to programming methods, and when covariances are zero, a still simpler ratio scheme gives the optimum, whether or not short sales are permitted. When covariances are not all zero and short sales are excluded, a single quad- ratic programming solution is required, but sufficient. Following these extensions of Tobin's classic work, we concentrate on the set of risk assets held in risk. averters' portfolios. In section III we develop various significant equilibrium proper- ties within the risk asset portfolio. In particular, we establish conditions under which stocks will be held long (short) in optimal portfolios even when "risk premiums" are negative (positive). We also develop expressions for different combi- nations of expected rate of return on a given security, and its stand.ard deviation, variance, and/or covariances which will result in the same relative holding of a stock, ceteris paribus. These "indifference functions" provide direct evidence on the moot issue of the appropriate functional relationships between "required rates of return" and relevant risk parameter(s) - and on the related issue of how "risk classes" of securities may best be delineated (if they are to be used).2 *This paper is another in a series of interrelated theoretical and statistical studies of corporate financial and investment policies being made under grants from the Rockefeller Founda- tion, and more recently the Ford Foundation, to the Harvard Business School. The generous support for this work is most gratefully acknowledged. The author is also much indebted to his colleagues Professors Bishop, Christenson, Kahr, Raiffa, and (especially) Schlaifer, for extensive discussion and com- mentary on an earlier draft of this paper; but responsibility for any errors or imperfections remains strictly his own. [Professor Sharpe's paper, "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk" (Journal of Finance, September i964) appeared after this paper was in final form and on its way to the printers. My first section, which parallels the first half of his paper (with corresponding conclusions), sets the algebraic framework for sections II, III and VI, (which have no counterpart in his paper) and for section IV on the equilibrium prices of risk assets, concerning which our results differ significantly for reasons which will be explored elsewhere. Sharpe does not take up the capital budgeting problem developed in section V below.] 'Tobin [2I, especially pp. 82-85]. Tobin assumed that funds are to be a allocated only over "monetary assets" (risk- free cash and default-free bonds of uncertain resale price) and allowed no short sales or borrowing. See also footnote 24 be- low. Other approaches are reviewed in Farrar [38]. 2It should be noted that the classic paper by Modigliani and Miller [i6] was silent on these issues. Corporations were assumed to be divided into homogeneous classes having the property that all shares of all corporations in any given class differed (at most) by a "scale factor," and hence (a) were per- fectly correlated with each other and (b) were perfect substi- tutes for each other in perfect markets (p. 266). No comment was made on the measure of risk or uncertainty (or other attributes) relevant to the identification of different "equiva- [ 13 ] 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
14 THE REVIEW OF ECONOMICS AND STATISTICS There seems to be a general presumption among uncertainty per se(as distinct from the effects of economists that relative risks are best measured diverse expectations),and to derive further by the standard deviation (or coefficient of implications of such uncertainty.In particular, variation)of the rate of return,*but in the simp-the aggregate market value of any company's lest cases considered-specifically when all equity is equal to the capitalization at the risk- covariances are considered to be invariant (or free interest rate of a uniquely defined certainty- zero)-the indifference functions are shown to equivalent of the probability distribution of the be linear between expected rates of return and aggregate dollar returns to all holders of its stock. their variance,not standard deviation.(With For each company,this certainty equivalent is variances fixed,the indifference function between the expected value of these uncertain returns less the ith expected rate of return and its pooled an adjustment term which is proportional to covariance with other stocks is hyperbolic.)their aggregate risk.The factor of proportion- There is no simple relation between the expected ality is the same for all companies in equilibirum rate of return required to maintain an investor's and may be regarded as a market price of dollar relative holding of a stock and its standard devia- risk.The relevant risk of each company's stock tion.Specifically,when covariances are non- is measured,moreover,not by the standard de- zero and variable,the indifference functions are viation of its dollar returns,but by the sum of the complex and non-linear even if it is assumed that variance of its own aggregate dollar returns and the correlations between rates of return on differ-their total covariance with those of all other stocks ent securities are invariant. The next section considers some of the impli- To this point we follow Tobin [21]and Marko- cations of these results for the normative aspects witz14]in assuming that current security prices of the capital budgeting decisions of a company are given,and that each investor acts on his own whose stock is traded in the market.For sim (perhaps unique)probability distribution over plicity,we impose further assumptions required rates of return given these market prices.In the to make capital budgeting decisions independent rest of the paper,we assume that investors'of decisions on how the budget is financed.The joint probability distributions pertain to dollar capital budgeting problem becomes a quadratic returns rather than rates of returns,and for programming problem analogous to that intro- simplicity we assume that all investors assign duced earlier for the individual investor.This identical sets of means,variances,and covari- capital budgeting-portfolio problem is formula- ances to the distribution of these dollar returns.ted,its solution is given and some of its more However unrealisic the latter assumption may important properties examined.Specifically, be,it enables us,in section IV,to derive a set of the minimum expected return (in dollars of ex- (stable)equilibrium market prices which at pected present value)required to justify the least fully and explicitly reflect the presence of allocation of funds to a given risky project is lent return"classes.Both Propositions I(market value of firm shown to be an increasing function of each of the independent of capital structure)and II(the linear relation following factors:(i)the risk-free rate of return; between the expected return on equity shares and the debt- (ii)the "market price of (dollar)risk";(iii)the equity ratio for firms within a given class)are derived from the above assumptions(and the further assumption that cor- variance in the project's own present value return; porate bonds are riskless securities);they involve no inter- (iv)the project's aggregate present value re- class comparisons,"..nor do they involve any assertion as turn-covariance with assets already held by the to what is an adequate compensation to investors for assuming company,and ()its total covariance with other a given degree of risk...."(p.279). This is,for instance,the presumption of Hirschleifer projects concurrently included in the capital [8,p.I13],although he was careful not to commit himself to budget.All five factors are involved explicitly this measure alone in a paper primarily focussed on other is- in the corresponding (derived)formula for the sues.For an inductive argument in favor of the standard deviation of the rate of return as the best measure of risk,see minimum acceptable expected rale of return on an Gordon [5,especially pp.69 and 761.See also Dorfman in investment project.In this model,all means [3,p.I29 fi.]and Baumol [2]. Except in dominantly "short"portfolios,the constant We also assume that common stock portfolios are not term will be larger,and the slope lower,the higher the (fixed) "inferior goods,"that the value of all other common stocks is level of covariances of the given stocks with other stocks. invariant,and any effect of changes in capital budgets on the sThe dollar return in the period is the sum of the cash covariances between the values of different companies'stocks is dividend and the increase in market price during the period. ignored. 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
14 THE REVIEW OF ECONOMICS AND STATISTICS There seems to be a general presumption among economists that relative risks are best measured by the standard deviation (or coefficient of variation) of the rate of return, but in the simp- lest cases considered - specifically when all covariances are considered to be invariant (or zero) - the indifference functions are shown to be linear between expected rates of return and their variance, not standard deviation.4 (With variances fixed, the indifference function between the ith expected rate of return and its pooled covariance with other stocks is hyperbolic.) There is no simple relation between the expected rate of return required to maintain an investor's relative holding of a stock and its standard devia- tion. Specifically, when covariances are non- zero and variable, the indifference functions are complex and non-linear even if it is assumed that the correlations between rates of return on differ- ent securities are invariant. To this point we follow Tobin [211 and Marko- witz [ 141 in assuming that current security prices are given, and that each investor acts on his own (perhaps unique) probability distribution over rates of return given these market prices. In the rest of the paper, we assume that investors' joint probability distributions pertain to dollar returns rather than rates of return5, and for simplicity we assume that all investors assign identical sets of means, variances, and covari- ances to the distribution of these dollar returns. However unrealisic the latter assumption may be, it enables us, in section IV, to derive a set of (stable) equilibrium market prices which at least fully and explicitly reflect the presence of uncertainty per se (as distinct from the effects of diverse expectations), and to derive further implications of such uncertainty. In particular, the aggregate market value of any company's equity is equal to the capitalization at the risk- free interest rate of a uniquely defined certainty- equivalent of the probability distribution of the aggregate dollar returns to all holders of its stock. For each company, this certainty equivalent is the expected value of these uncertain returns less an adjustment term which is proportional to their aggregate risk. The factor of proportion- ality is the same for all companies in equilibirum, and may be regarded as a market price of dollar risk. The relevant risk of each company's stock is measured, moreover, not by the standard de- viation of its dollar returns, but by the sum of the variance of its own aggregate dollar returns and their total covariance with those of all other stocks. The next section considers some of the impli- cations of these results for the normative aspects of the capital budgeting decisions of a company whose stock is traded in the market. For sim- plicity, we impose further assumptions required to make capital budgeting decisions independent of decisions on how the budget is financed.6 The capital budgeting problem becomes a quadratic programming problem analogous to that intro- duced earlier for the individual investor. This capital budgeting-portfolio problem is fornmula- ted, its solution is given and some of its more important properties examined. Specifically, the minimum expected return (in dollars of ex- pected present value) required to justify the allocation of funds to a given risky project is shown to be an increasing function of each of the following factors: (i) the risk-free rate of return; (ii) the "market price of (dollar) risk"; (iii) the variance in the project's own presentvalue return; (iv) the project's aggregate present value re- turn-covariance with assets already held by the company, and (v) its total covariance with other projects concurrently included in the capital budget. All five factors are involved explicitly in the corresponding (derived) formula for the minimum acceptable expected rate of return on an investment project. In this model, all means 6We also assume that common stock portfolios are not "inferior goods," that the value of all other common stocks is invariant, and any effect of changes in capital budgets on the covariances between the values of different companies' stocks is ignored. lent return" classes. Both Propositions I (market value of firm independent of capital structure) and II (the linear relation between the expected return on equity shares and the debt- equity ratio for firms within a given class) are derived from the above assumptions (and the further assumption that cor- porate bonds are riskless securities); they involve no inter- class comparisons, ". . . nor do they involve any assertion as to what is an adequate compensation to investors for assuming a given degree of risk. . . ." (p. 279). 3This is, for instance, the presumption of Hirschleifer [8, p. II 31, although he was careful not to commit himself to this measure alone in a paper primarily focussed on other is- sues. For an inductive argument in favor of the standard deviation of the rate of return as the best measure of risk, see Gordon [5, especially pp. 69 and 76I. See also Dorfman in [3, p. I29 ff.] and Baumol [2]. 4Except in dominantly "short" portfolios, the constant term will be larger, and the slope lower, the higher the (fixed) level of covariances of the given stocks with other stocks. 5The dollar return in the period is the sum of the cash dividend and the increase in market price during the period. 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
VALUATION OF RISK ASSETS 15 and (co)variances of present values must be cept in the final section,we assume that the calculated at the riskless rate r*.We also show interest rate paid on such loans is the same as he that there can be no "risk-discount"'rate to be used would have received had he invested in risk-free in computing present values to accept or reject savings accounts,and that there is no limit on the individual projects.In particular,the "cost of amount he can borrow at this rate.Finally (5) capital"'as defined (for uncertainty)anywhere he makes all purchases and sales of securities and in the literature is not the appropriale rale to use all deposits and loans at discrete points in time, in these decisions even if all new projects have the so that in selecting his portfolio at any "trans- same"“risk”as existing assets. action point,"each investor will consider only The final section of the paper briefly examines (i)the cash throw-off (typically interest pay- the complications introduced by institutional ments and dividends received)within the period limits on amounts which either individuals or to the next transaction point and(i)changes in corporations may borrow at given rates,by rising the market prices of stocks during this same costs of borrowed funds,and certain other "real period.The return on any common stock is de- world"complications.It is emphasized that fined to be the sum of the cash dividends received the results of this paper are not being presented plus the change in its market price.The return as directly applicable to practical decisions,be-on any portfolio is measured in exactly the same cause many of the factors which matter very way,includinginterest received or paid. siginificantly in practice have had to be ignored or assumed away.The function of these sim- Assumptions Regarding Investors plifying assumptions has been to permit a (1)Since we posit the existence of assets rigorous development of theoretical relationships yielding posilive risk-free returns,we assume that and theorems which reorient much current each investor has already decided the fraction of theory (especially on capital budgeting)and pro- his total capital he wishes to hold in cash and vide a basis for further work.?More detailed non-interest bearing deposits for reasons of conclusions will be found emphasized at numerous liquidity or transactions requirements.10 Hence- points in the text. forth,we will speak of an investor's capital as the stock of funds he has available for profitable I-Portfolio Selection for an Individual Investor: investment after optimal cash holdings have been The Separation Theorem deducted.We also assume that(2)each investor Market Assumptions will have assigned a joint probability distribution We assume that (1)eack individual investor incorporating his best judgments regarding the can invest any part of his capital in certain risk- returns on all individual stocks,or at least will have specified an expected value and variance to free assels (e.g.deposits in insured savings ac- countss)all of which pay interest at a common every return and a covariance or correlation to positive rate,exogeneously determined;and that every pair of returns.All expected values of (2)he can invest any fraction of his capital in any returns are finite,all variances are non-zero and or all of a given finite set of risky securities which finite,and all correlations of returns are less than one in absolute value (i.e.the covariance matrix are (3)traded in a single purely competitive markel,free of transactions costs and taxes,at is positive-definite).The investor computes the given market prices,?which consequently do not expected value and variance of the total return depend on his investments or transactions.We on any possible porlfolio,or mix of any specified also assume that (4)any investor may,if he amounts of any or all of the individual stocks,by wishes,borrow funds to invest in risk assets.Ex- forming the appropriately weighted average or 7The relation between the results of this paper and the sum of these components expected returns, models which were used in [I]and [I2]is indicated at the end variances and covariances. of section V. 1These latter decisions are independent of the decisions sGovernment bonds of appropriate maturity provide regarding the allocation of remaining funds between risk-free another important example when their "yield"is substituted assets with positive return and risky stocks,which are of for the word "interest." direct concern in this paper,because the risk-free assets with Solely for convenience,we shall usually refer to all these positive returns clearly dominate those with no return once investments as common stocks.although the analysis is of liquidity and transactions requirements are satisfied at the course quite general. margin. 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
VALUATION OF RISK ASSETS 15 and (co)variances of present values must be calculated at the riskless rate r*. We also show that there can be no "risk-discount" rate to be used in computing present values to accept or reject individual projects. In particular, the "cost of capital" as defined (for uncertainty) anywhere in the literature is not the appropriate rate to use in these decisions even iJ all new projects have the same "risk" as existing assets. The final section of the paper briefly examines the complications introduced by institutional limits on amounts which either individuals or corporations may borrow at given rates, by rising costs of borrowed funds, and certain other "real world" complications. It is emphasized that the results of this paper are not being presented as directly applicable to practical decisions, be- cause many of the factors which matter very siginificantly in practice have had to be ignored or assumed away. The function of these sim- plifying assumptions has been to permiit a rigorous development of theoretical relationships and theorems which reorient much current theory (especially on capital budgeting) and pro- vide a basis for further work.7 More detailed conclusions will be found emphasized at numerous points in the text. I -Portfolio Selection for an Individual Investor: The Separation Theorem Market Assumptions We assume that (1) each individual investor can invest any part of his capital in certain risk- free assets (e. g. deposits in insured savings ac- counts8) all of which pay interest at a common positive rate, exogeneously determined; and that (2) he can invest any fraction of his capital in any or all of a given finite set of risky securities which are (3) traded in a single purely competitive market, free of transactions costs and taxes, at given market prices,9 which consequently do not depend on his investments or transactions. We also assume that (4) any investor may, if he wishes, borrow funds to invest in risk assets. Ex- cept in the final section, we assume that the interest rate paid on such loans is the same as he would have received had he invested in risk-free savings accounts, and that there is no limit on the amount he can borrow at this rate. Finally (5) he makes all purchases and sales of securities and all deposits and loans at discrete points in time, so that in selecting his portfolio at any "trans- action point," each investor will consider only (i) the cash throw-off (typically interest pay- ments and dividends received) within the period to the next transaction point and (ii) changes in the market prices of stocks during this same period. The return on any common stock is de- fined to be the sum of the cash dividends received plus the change in its market price. The return on any portfolio is measured in exactly the same way, including interest received or paid. Assumptions Regarding Investors (1) Since we posit the existence of assets yielding positive risk-free returns, we assume that each investor has already decided the fraction of his total capital he wishes to hold in cash and non-interest bearing deposits for reasons of liquidity or transactions requirements.'0 Hence- forth, we will speak of an investor's capital as the stock of funds he has available for profitable investnment after optimal cash holdings have been deducted. We also assume that (2) each investor will have assigned a joint probability distribution incorporating his best judgments regarding the returns on all individual stocks, or at least will have specified an expected value and variance to every return and a covariance or correlation to every pair of returns. All expected values of returns are finite, all variances are non-zero and finite, and all correlations of returns are less than one in absolute value (i. e. the covariance matrix is positive-definite). The investor computes the expected value and variance of the total return on any possible portfolio, or mix of any specified amounts of any or all of the individual stocks, by forming the appropriately weighted average or sum of these components expected returns, variances and covariances. '0These latter decisions are independent of the decisions regarding the allocation of remaining funds between risk-free assets with positive return and risky stocks, which are of direct concern in this paper, because the risk-free assets with positive returns clearly dominate those with no return once liquidity and transactions requirements are satisfied at the margin. 7The relation between the results of this paper and the models which were used in [ii] and [I 2] is indicated at the end of section V. 8 Government bonds of appropriate maturity provide another important example when their "yield" is substituted for the word "interest." 9Solely for convenience, we shall usually refer to all these investments as common stocks, although the analysis is of course quite general. 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
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/terms
16 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
VALUATION OF RISK ASSETS 17 total net investment (stock plus riskless assets value of w).Since any expected return y can be minus borrowing).Then the investor's net obtained from any stock mix,an investor adher- return per dollar of total net investment will be ing to our choice criterion will minimize the (1)夕=(1-0)r*+7=r*十（行r*);0≤w1 indicates that the investor borrows to buy variance associated with any y (and hence any w stocks on margin and pays interest amounting to value)the investor may prefer,and consequently, the absolute value of (1-w)r*.From (1)we is independent of y and w.This establishes the determine the mean and variance of the net re- separation theorem1,once we note that our turn per dollar of total net investment to be: assumptions regarding available portfolios15 in- (2a）=r*+2w(rr*),and sure the existence of a maximum 0. It is equally apparent that after determining (2b)c3y=02g2. the optimal stock portfolio (mix)by maximizing Finally,after eliminating w between these two 0,the investor can complete his choice of an equations,we find that the direct relation be- over-all investment position by substituting tween the expected value of the investor's net the of this optimal mix in(3)and decide which return per dollar of his total net investment and over-all investment position by substituting the risk parameters of his investment position is:of the available(,)pairs he prefers by refer- (3a）y=r*+fa,where ring to his own utility function.Substitution (3b)0=(行-y)/a. of this best y value in (2a)determines a unique In terms of any arbitrarily selected stock port- best value of the ratio w of gross investment in folio,therefore,the investor's net expected rate the optimal stock portfolio to his total net of return on his total net investment is related investment,and hence,the optimal amount of linearly to the risk of return on his total net investments in riskless savings deposits or the investment as measured by the standard deviation optimal amount of borrowing as well. of his return.Given any selected stock portfolio, This separation theorem thus has four immedi- this linear function corresponds to Fisher's ate corrolaries which can be stated: "market opportunity line";its intercept is the risk-free rate r*and its slope is given by 0,which (i)Given the assumptions about borrowing is determined by the parameters and ar of the and lending stated above,any investor whose particular stock portfolio being considered.We choices maximize the expectation of any particu- also see from(2a)that,by a suitable choice of w, lar utility function consistent with these condi- the investor can use any stock mix (and its asso- tions will make identical decisions regarding the ciated "market opportunity line")to obtain an proportionate composition of his stock (risk-asset) expected return,as high as he likes;but that, portfolio.This is true regardless of the particular because of (26 )and (36),as he increases his in- ulility function!whose expectation he maximizes. vestment w in the (tentatively chosen)mix,the (ii)Under these assumptions,only a single standard deviation ox(and hence the variance point on the Markowitz "Efficient Frontier"is of the return on his total investment also relevant to the investor's decision regarding his becomes proportionately greater. investments in risk assets.17 (The next section Now consider all possible stock portfolios. See also the appendix,note I for a different form of proof. Those portfolios having the same0 value will 15 Specifically,that the amount invested in any stock in lie on the same "market opportunity line,"but any stock mix is infinitely divisible,that all expected returns on individual stocks are finite,that all variances are positive those having different 0 values will offer differ- and finite,and that the variance-covariance matrixispositive- ent"market opportunity lines"(between expected definite. return and risk)for the investor to use.The in- 16When probability assessments are multivariate normal, the utility function may be polynomial,exponential,etc. vestor's problem is to choose which stock port- Even in the "non-normal"case when utility functions are folio-mix (or market opportunity line or0 value) quadratic,they may vary in its parameters.See also the reference to Roy's work in the text below. to use and how intensively to use it(the proper 17When the above conditions hold (see also final para- 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
VALUATION OF RISK ASSETS 17 total net investment (stock plus riskless assets minus borrowing). Then the investor's net return per dollar of total net investment will be (1) y =(1 -w)r*+wf =r*+w(f-r*); O? 1 indicates that the investor borrows to buy stocks on margin and pays interest amounting to the absolute value of (1-w)r*. From (1) we determine the mean and variance of the net re- turn per dollar of total net investment to be: (2a) y =r*+w(-r*), and (2b) a2y=W2.,2r. Finally, after eliminating w between these two equations, we find that the direct relation be- tween the expected value of the investor's net return per dollar of his total net investmnent and the risk parameters of his investment position is: (3a) y =r* +Ouy, where (3b) 0 = (r-S *) /0ru In terms of any arbitrarily selected stock port- folio, therefore, the investor's net expected rate of return on his total net investment is related linearly to the risk of return on his total net investment as measured by the standard deviation of his return. Given any selected stock portfolio, this linear function corresponds to Fisher's "market opportunity line"; its intercept is the risk-free rate r* and its slope is given by 0, which is determined by the parameters r and u,. of the particular stock portfolio being considered. We also see from (2a) that, by a suitable choice of w, the investor can use any stock mix (and its asso- ciated "market opportunity line") to obtain an expected return, y, as high as he likes; but that, because of (2b )and (3b), as he increases his in- vestment w in the (tentatively chosen) mix, the standard deviation oy (and hence the variance a2y) of the return on his total investment also becomes proportionately greater. Now consider all possible stock portfolios. Those portfolios having the same 0 value will lie on the same "market opportunity line," but those having different 0 values will ojjer dijTer- ent "market opportunity lines" (between expected return and risk) for the investor to use. The in- vestor's problem is to choose which stock port- folio-mix (or market opportunity line or o value) to use and how intensively to use it (the proper value of w). Since any expected return y can be obtained from any stock mix, an investor adher- ing to our choice criterion will minimize the variance of his over-all return o2y associated with any expected return he may choose by confining all his investment in stocks to the mix with the larges? 0 value. This portfolio minimizes the variance associated with any y (and hence any w value) the investor may prefer, and consequently, is independent of y and w. This establishes the separation theorem'4, once we note that our assumptions regarding available portfolios'5 in- sure the existence of a maximum 0. It is equally apparent that after determining the optimal stock portfolio (mix) by maximizing 0, the investor can complete his choice of an over-all investment position by substituting the 0 of this optimal mix in (3) and decide which over-all investment position by substituting of the available (y, ay) pairs he prefers by refer- ring to his own utility function. Substitution of this best y value in (2a) determines a unique best value of the ratio w of gross investment in the optimal stock portfolio to his total net investment, and hence, the optimal amount of investments in riskless savings deposits or the optimal amount of borrowing as well. This separation theorem thus has four immedi- ate corrolaries which can be stated: (i) Given the assumptions about borrowing and lending stated above, any investor whose choices maximize the expectation of any particu- lar utility function consistent with these condi- tions will make identical decisions regarding the proportionate composition of his stock (risk-asset) portfolio. This is true regardless of the particular utility functionl6 whose expectation he maximizes. (ii) Under these assumptions, only a single point on the Markowitz "Efficient Frontier" is relevant to the investor's decision regarding his investments in risk assets.17 (The next section 14See also the appendix, note I for a different form of proof. 15Specifically, that the amount invested in any stock in any stock mix is infinitely divisible, that all expected returns on individual stocks are finite, that all variances are positive and finite, and that the variance-covariance matrixispositive- definite. 16When probability assessments are multivariate normal, the utility function may be polynomial, exponential, etc. Even in the "non-normal" case when utility functions are quadratic, they may vary in its parameters. See also the reference to Roy's work in the text below. 17When the above conditions hold (see also final paraThis 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
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.) 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18 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.) 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VALUATION OF RISK ASSETS 19 its mathematical convenience,multivariate nor-of-is thus rigorously appropriate in the non- mality is doubtless also suspect,especially per-multivariate normal case for Safety-Firsters who haps in considering common stocks. minimax the stated upper bound of the chance It is,consequently,very relevant to note that of doing less well on portfolios including risk by using the Bienayme-Tchebycheff inequality,assets than they can do on riskless investments, Roy [19]has shown that investors operating on just as it is for concave-expected utility maxi- his "Safety First"'principle (i.e.make risky in-mizers in the "normal''case.On the basis vestments so as to minimize the upper bound of of the same probability judgments,these Safety- the probability that the realized outcome will fall Firsters will use the same proximate criterion below a pre-assigned "disaster level")should function(max )and will choose proportionately maximize the ratio of the excess expected port-the same risk asset portfolios as the more folio return (over the disaster level)to the orothodox"utility maximizers"we have hitherto standard deviation of the return on the port-considered. folio2-which is precisely our criterion of max 0 when his disaster level is equated to the risk-II-Portfolio Selection:The Optimal Stock Mix free rate r*.This result,of course,does not depend on multivariate normality,and uses a Before finding the optimal stock mix-the different argument and form of utility function. mix which maximizes 0 in (36)above-it is The Separation Theorem,and its Corrolaries necessary to express the return on any arbitrary (i)and (ii)above-and all the rest of our follow- mix in terms of the returns on individual stocks ing analysis which depends on the maximization included in the portfolio.Although short sales are excluded by assumption in most of the writings on portfolio optimization,this restric- tive assumption is arbitrary for some purposes at least,and we therefore broaden the analysis in this paper to include short sales whenever they Ui3 Ui2 Uil w事O are permitted. (化orrow) Computation of Returns on a Stock Mix,When (use r0 Short Sales are Permitted We assume that there are m different stocks in the market,denoted by i=1,2,...,m,and treat short sales as negative purchases.We shall use the following basic notation: Possiblel -The ratio of the gross investment in the ith stock (the market value of the amount bought or sold)to the gross FIGURE I investment in all stocks.A positive value of hi indicates a purchase,while This function also implausibly implies,as Pratt [I7]and a negative value indicates a short sale. Arrow [r]have noted,that the insurance premiums which peo- -The return per dollar invested in a ple would be willing to pay to hedge gioen risks rise progress- purchase of the ith stock (cash divi- ively with wealth or income.For a related result,see Hicks I6,P.8o2l. dends plus price appreciation) a Roy also notes that when judgmental distributions are -As above,the return per dollar inves- multivariate normal,maximization of this criterion minimises ted in a particular mix or porlfolio of the probability of "disaster"(failure to do better in stocks than savings deposits or government bonds held to maturity).It stocks. should be noted,however,minimization of the probability of short falls from "disaster"levels in this "normal''case is Consider now a gross investment in the entire strictly eguivalent to expected utility maximization under all mix,so that the actual investment in the i risk-averters'utility functions.The equivalence is not re- stock is equal to The returns on purchases stricted to the utility function of the form (o,I)(zero if "dis- aster"occurs,one if it doesn't),as claimed by Roy [1o,p.432) and short sales need to be considered separately. and Markowitz [14,p.293 and following.]. First,we see that if is invested in a pur- 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
VALUATION OF RISK ASSETS 19 its mathematical convenience, multivariate nor- mality is doubtless also suspect, especially per- haps in considering common stocks. It is, consequently, very relevant to note that by using the Bienayme-Tchebycheff inequality, Roy [19] has shown that investors operating on his "Safety First" principle (i.e. make risky in- vestments so as to minimize the upper bound of the probability that the realized outcome will fall below a pre-assigned "disaster level") should maximize the ratio of the excess expected port- folio return (over the disaster level) to the standard deviation of the return on the port- folio2l - which is precisely our criterion of max 0 when his disaster level is equated to the risk- free rate r*. This result, of course, does not depend on multivariate normality, and uses a different argument and form of utility function. The Separation Theorem, and its Corrolaries (i) and (ii) above - and all the rest of our follow- ing analysis which depends on the maximization Uj3 Uj2 U1" Ui3 Ui2 U w>O o /27< (borrow) w < 0 (use savings accounts) / Possible) Mixes) FIGURE I of 0 - is thus rigorously appropriate in the non- multivariate normal case for Safety-Firsters who minimax the stated upper bound of the chance of doing less well on portfolios including risk assets than they can do on riskless investments, just as it is for concave-expected utility maxi- mizers in the "normal" case. On the basis of the same probability judgments, these Safety- Firsters will use the same proximate criterion function (max o) and will choose proportionately the same risk asset portfolios as the more orothodox "utility maximizers" we have hitherto considered. II -Portfolio Selection: The Optimal Stock Mix Before finding the optimal stock mix - the mix which maximizes 0 in (3b) above -it is necessary to express the return on any arbitrary mix in terms of the returns on individual stocks included in the portfolio. Although short sales are excluded by assumption in most of the writings on portfolio optimization, this restric- tive assumption is arbitrary for some purposes at least, and we therefore broaden the analysis in this paper to include short sales whenever they are permitted. Computation of Returns on a Stock Mix, When Short Sales are Permitted We assume that there are m different stocks in the market, denoted by i = 1, 2, .. ., m, and treat short sales as negative purchases. We shall use the following basic notation: -hil The ratio of the gross investment in the Pth stock (the market value of the amount bought or sold) to the gross investment in all stocks. A positive value of hi indicates a purchase, while a negative value indicates a short sale. - The return per dollar invested in a purchase of the Pth stock (cash divi- dends plus price appreciation) - As above, the return per dollar inves- ted in a particular mix or portfolio of stocks. Consider now a gross investment in the entire mix, so that the actual investment in the ith stock is equal to Ihil. The returns on purchases and short sales need to be considered separately. First, we see that if 1hil is invested in a pur- This function also implausibly implies, as Pratt |I 7] and Arrow [i] have noted, that the insurance premiums which peo- ple would be willing to pay to hedge given risks rise progress- ively with wealth or income. For a related result, see Hicks [6, p. 802]. 2'Roy also notes that when judgmental distributions are multivariate normal, maximization of this criterion minimizes the probability of "disaster" (failure to do better in stocks than savings deposits or government bonds held to maturity). It should be noted, however, minimization of the probability of short falls from "disaster" levels in this "normal" case is strictly eqiuivalent to expected utility maximization under all risk-averters' utility functions. The equivalence is not re- stricted to the utility function of the form (o, i) (zero if "dis- aster" occurs, one if it doesn't), as claimed by Roy [I9, P. 432] and Markowitz [I4, P. 293 and following.]. 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
20 THE REVIEW OF ECONOMICS AND STATISTICS chase (0),the return will be simply hi.(5)=;[hr*)+r*] For reasons which will be clear immediately how- =r*十2h(行：一r*) ever,we write this in the form: because =I by the definition of. (4a）h:=h(f:-r*)+hlr* The expectation and variance of the return on Now suppose that is invested in a short any stock mix is consequently sale (ho exists if there is not also receive interest at the same rate on his one or more stocks with:not exactly equal to cash remittance to the lender of the stock.To r*.We assume throughout the rest of the paper facilitate the formal analysis,we assume that that such a portfolio exists. both interest components are always received by the short seller,and that margin requirements are Determination of the Optimal Stock Porlfolio Iooo.In this case,the short seller's relurn per As shown in the proof of the Separation dollar of his gross investment will be (2r*-r),Theorem above,the optimal stock portfolio is and if he invests in the short sale (<o),the one which maximizes 0 as defined in equa- its contribution to his portfolio return will be:tion (8).We,of course,wish to maximize this (46)(ar*)=(-r*)+r*.value subject to the constraint Since the right-hand sides of (4a)and (4b)are (9):=I, identical,the total return per dollar invested in which follows from the definition of But any stock mix can be written as: we observe from equation(8)that 0 is a homog- eneous function of order zero in the h::the value 2In recent years,it has become increasingly common for the short seller to waive interest on his deposit with the lender of e is unchanged by any proportionale change in of the security-in market parlance,for the borrowers of all k.Our problem thus reduces to the simpler stock to obtain it"flat"-and when the demand for borrowing one of finding a vector of values yielding the stock is large relative to the supply available for this purpose, the borrower may pay a cash premium to the lender of the unconstrained maximum of 0 in equation (8), stock.See Sidney M.Robbins,[18,pp.58-591.It will be after which we may scale these initial solution noted that these practices reduce the expected return of short values to satisfy the constraint. sales without changing the variance.The formal procedures developed below permit the identification of the appropriate stocks for short sale assuming the expected return is (2r*-f) The Optimum Portfolio When Short Sales If these stocks were to be borrowed "flat"or a premium paid, are Permitted it would be simply necessary to iterate the solution after replacing (-r)in (4b)and (5)for these stocks with the value () We first examine the partial derivatives of(8) and if,in addition,a premium is paid,the term ( with respect to thek;and find: should be substituted (where p:2 o is the premium (if any) 88 per dollar of sales price of the stock to be paid to lender of the (Io) =(c)-1[e:-λ(h+2h)], stock).With equal lending and borrowing rates,changes in oh: margin requirements will not affect the calculations.(I am where, indebted to Prof.Schlaifer for suggesting the use of absolute values in analyzing short sales.) (II）λ=/g2=2hc/2:2hhi 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
20 THE REVIEW OF ECONOMICS AND STATISTICS chase (hi > 0), the return will be simply hiri. For reasons which will be clear immediately how- ever, we write this in the form: (4a) h?ir = h( i-r*) + IhiI r*. Now suppose that Ihi I is invested in a short sale (hi > r* and hence with 0 > o exists if there is one or more stocks with r- not exactly equal to r*. We assume throughout the rest of the paper that such a portfolio exists. Determination of the Optimal Stock Portfolio As shown in the proof of the Separation Theorem above, the optimal stock portfolio is the one which maximizes 0 as defined in equa- tion (8). We, of course, wish to maximize this value subject to the constraint (9) 2i hiI =I, which follows from the definition of Ihi 1. But we observe from equation (8) that 0 is a homog- eneous function of order zero in the hi: the value of 0 is unchanged by any proportionate change in all hi. Our problem thus reduces to the simpler one of finding a vector of values yielding the unconstrained maximum of 0 in equation (8), after which we may scale these initial solution values to satisfy the constraint. The Optimum Portfolio When Short Sales are Permitted We first examine the partial derivatives of (8) with respect to the hi and find: (IO) ah = (0x-l [xi-(i + 2;htjx) ], aoi where, (I I) X = /C1 _x2_v / v 22In recent years, it has become increasingly common for the short seller to waive interest on his deposit with the lender of the security - in market parlance, for the borrowers of stock to obtain it "flat"- and when the demand for borrowing stock is large relative to the supply available for this purpose, the borrower may pay a cash premium to the lender of the stock. See Sidney M. Robbins, [i8, pp. 58-59]. It will be noted that these practices reduce the expected return of short sales without changing the variance. The formal procedures developed below permit the identification of the appropriate stocks for short sale assuming the expected return is (2r* - ft). If these stocks were to be borrowed "flat" or a premium paid, it would be simply necessary to iterate the solution after replacing (j- r*) in (4b) and (5) for these stocks with the value (ii) - and if, in addition, a premium pi is paid, the term (ri + pi) should be substituted (where pi > o is the premium (if any) per dollar of sales price of the stock to be paid to lender of the stock). With equal lending and borrowing rates, changes in margin requirements will not affect the calculations. (I am indebted to Prof. Schlaifer for suggesting the use of absolute values in analyzing short sales.) 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
VALUATION OF RISK ASSETS 21 The necessary and sufficient conditions on thethe sum of their absolute values.A comparison relative values of the h:for a stationary and the of equations (16)and (II)shows further that: unique (global)maximum23 are obtained by (18):=0=0/2; setting the derivatives in (Io)equal to zero, i.e.the sum of the absolute values of the which give the set of equations yields,as a byproduct,the value of the ratio of (12）8成十2=元，i=1,2,···,m; the expected excess rate of return on the optimal where we write portfolio to the variance of the return on this (I3)名=λh. best portfolio. It is also of interest to note that if we form the It will be noted the set of equations (12)- corresponding A-ratio of the expected excess which are identical to those Tobin derived by a return to its variance for each ith stock,we have different route24-are linear in the own-vari- at the optimum: ances,pooled covariances,and excess returns of the respective securities;and since the covariance (Ig）heo=(入/入)-2h,e/where matrix is positive definite and hence non- 入i=无/ci. singular,this system of equations has a unique The optimal fraction of each security in the best solution portfolio is equal to the ratio of its A;to that of the entire portfolio,less the ratio of its pooled (I4）80=2元； covariance with other securities to its own vari- where represents the ijth element of (ance.Consequently,if the investor were to act the inverse of the covariance matrix.Using on the assumption that all covariances were (13),(7),and (66),this solution may also be zero,he could pick his optimal portfolio mix written in terms of the primary variables of the very simply by determining the A:ratio of the problem in the form expected excess return元：=r:-r*of each (15)h0=(入)-12产f5-r*,alli. stock to its variance=,and setting each Moreover,since (13)implies h:=入/；for with no covariances,25zλt= (t6):z:|=λ2：lh:, A=2.With this simplifying assumption, the A:ratios of each stock suffice to determine Ao may readily be evaluated,after introducing the optimal mix by simple arithmetic;26 in the the constraint (9)as more general case with non-zero covariances,a (I7）2:z:0|=入2：h,0|=λ0 single set27 of linear equations must be solved in The optimal relative investments z:can conse- the usual way,but no (linear or non-linear) quently be scaled to the optimal proportions of programming is required and no more than one the stock portfolio h,by dividing each zr by point on the "efficient frontier"need ever be computed,given the assumptions under which 23It is clear from a comparison of equations (8)and (I), we are working. showing that sgn 0 sgn A,that only the vectors of values corresponding to>oare relevant to the maximization of 0. Moreover,since 6 as given in (8)and all its first partials shown The Optimum Portfolio When Short Sales in (Io)are continuous functions of the /it follows that when are not Permitted short sales are permitted,any maximum of 6 must be a station- The exclusion of short sales does not compli- ary value,and any stationary value is a maximum(rather than a minimum)when A>o because 0 is a convex function with a cate the above analysis if the investor is willing positive-definite quadratic torm in its denominator.For the to act on an assumption of no correlations same reason,any maximum of 6 is a unique(global)maximum. between the returns on different stocks.In this 2See Tobin,[21],equation (3.22),p.83.Tobin had,how- ever,formally required no short selling or borrowing,implying case,he finds his best portfolio of "long"holding that this set of equations is valid under these constraints [so by merely eliminating all securities whose A; long as there is a single riskless asset (pp.84-85);but the constraints were ignored in his derivation.We have shown 2s With no covariances,the set of equations (I2)reduces that this set of equations is valid when short sales are properly toλa=/ia=入i,and after summing over all i=I, included in the portfolio and borrowing is available in perfect 2...m,and using the constraint (9),we have immediately markets in unlimited amounts.The alternative set of equi- that|λo|=:|入，andλo>ofor max(instead of min) librium conditions required when short sales are ruled out is 24 Using a more restricted market setting,Hicks [6,p.8or] given immediately below.The complications introduced by has also reached an equivalent result when covariances are borrowing restrictions are examined in the final section of the zero (as he assumed throughout). paper. 27 See,however,footnote 22,above. 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
VALUATION OF RISK ASSETS 21 The necessary and sufficient conditions on the relative values of the hi for a stationary and the unique (global) maximum23 are obtained by setting the derivatives in (io) equal to zero, which give the set of equations (I2) Zitij + jZjaxij = i, i = I, 2, . . ,m; where we write (I3) zi = Xhi. It will be noted the set of equations (I 2)- which are identical to those Tobin derived by a different route24 -are linear in the own-vari- ances, pooled covariances, and excess returns of the respective securities; and since the covariance matrix xc is positive definite and hence non- singular, this system of equations has a unique solution (I4) zi? = 2jj where xii represents the i0th element of (x)-l the inverse of the covariance matrix. Using (I3), (7), and (6b), this solution may also be written in terms of the primary variables of the problem in the form (I5) hi? = (XO)-12jrii(j - r*), all i. Moreover, since (I3) implies (i6) 2zi I = X2h i, XO may readily be evaluated, after introducing the constraint (g) as (I 7) 2;i Izi? t i Ihi? I = ?o (~~) ~ = h0 The optimal relative investments zi? can conse- quently be scaled to the optimal proportions of the stock portfolio hi?, by dividing each zi? by the sum of their absolute values. A comparison of equations (i6) and (ii) shows further that: (i8) 2i -zi? I = ? = X?/a o2; i.e. the sum of the absolute values of the zi0 yields, as a byproduct, the value of the ratio of the expected excess rate of return on the optimal portfolio to the variance of the return on this best portfolio. It is also of interest to note that if we form the corresponding X-ratio of the expected excess return to its variance for each ith stock, we have at the optimum: (i9) hi? = (Xi/XO) - 2;j?xijiii where Xi = The optimal fraction of each security in the best portfolio is equal to the ratio of its Xi to that of the entire portfolio, less the ratio of its pooled covariance with other securities to its own vari- ance. Consequently, if the investor were to act on the assumption that all covariances were zero, he could pick his optimal portfolio mix very simply by determining the Xi ratio of the expected excess return xi = i- r* of each stock to its variance xij = rii, and setting each hi= Xi/2Xi; for with no covariances,25 2Ai = XO = 0/1ae2. With this simplifying assumption, the Xi ratios of each stock suffice to determine the optimal mix by simple arithmetic;26 in the more general case with non-zero covariances, a single set27 of linear equations must be solved in the usual way, but no (linear or non-linear) programming is required and no more than one point on the "efficient frontier" need ever be computed, given the assumptions under which we are working. The Optimum Portfolio When Short Sales are not Permitted The exclusion of short sales does not compli- cate the above analysis if the investor is willing to act on an assumption of no correlations between the returns on different stocks. In this case, he finds his best portfolio of "long" holding by merely eliminating all securities whose Xi- 231t is clear from a comparison of equations (8) and (ii), showing that sgn 0 = sgn X, that only the vectors of hi values corresponding to X > o are relevant to the maximization of 0. Moreover, since 0 as given in (8) and all its first partials shown in (io) are continuous functions of the hi, it follows that when short sales are permitted, any maximum of 0 must be a station- ary value, and any stationary value is a maximum (rather than a minimum) when X > o because 0 is a convex function with a positive-definite quadratic torm in its denominator. For the same reason, any maximum of 0 is a unique (global) maximum. 24See Tobin, [2I], equation (3.22), p. 83. Tobin had, how- ever, formally required no short selling or borrowing, implying that this set of equations is valid under these constraints [so long as there is a single riskless asset (pp. 84-85)]; but the constraints were ignored in his derivation. We have shown that this set of equations is valid when short sales are properly included in the portfolio and borrowing is available in perfect markets in unlimited amounts. The alternative set of equi- librium conditions required when short sales are ruled out is given immediately below. The complications introduced by borrowing restrictions are examined in the final section of the paper. 25 With no covariances, the set of equations (I2) reduceS to Xhs = = Xi, and after summing over all i =I, 2 ... m, and using the constraint (9), we have immediately that I X 0 = Ii I Xi 1, and X > o for max 0 (instead of min 0). 26 Using a more restricted market setting, Hicks [6, p. 8oi] has also reached an equivalent result when covariances are zero (as he assumed throughout). 27 See, however, footnote 22, above. 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
22 THE REVIEW OF ECONOMICS AND STATISTICS ratio is negative,and investing in the remaining (22a)=i=I,2,...m; issues in the proportionsk:=λ：/λ；in accord- where ance with the preceding paragraph. (22b-d)z0≥o,v:0≥o,z,,0-o. But in the more generally realistic cases when covariances are nonzero and short sales are not This system of equations can be expeditiously admitted,the solution of a single bilinear or solved by the Wilson Simplicial Algorithm [23] Now let m'denote the number of stocks with quadratic programming problem is required to determine the optimal portfolio.(All other strictly positive holdings z>o in (226),and points on the“efficient frontier,.”of course, renumber the entire set of stocks so that the continue to be irrelevant so long as there is a subset satisfying this strict inequality [and, riskless asset and a"perfect"borrowing market.) hence also,by (22d)v:=o]are denoted I, The optimal portfolio mix is now given by the 2,...,m'.Within this m'subset of stocks set of which maximize 0 in equation (8) found to belong in the optimal portfolio with posi- subject to the constraint that allo.As tive holdings,we consequently have,using the before,the (further)constraint that the sum of constraint (I9), the h;be unity (equation o)may be ignored in （红7a）2-1mz:0=λ02-1m'h,0=λ0 the initial solution for the relalive values of the so that the fraction of the optimal portfolio in- k:[because 0 in (8)is homogeneous of order vested in the ith stock (where i =I,2...m)is zerol.To find this optimum,we form the (23）h:0=8/入0=z,/Σ-1mz0. Lagrangian function Once again,using (I7a)and (II),the sum of the (2o)(九W）=0+24h zo within this set of stocks held yields as a by- which is to be maximized subject to k;o and product the ratio of the expected excess rate of 4;2o.Using (II),we have immediately return on the optimal portfolio to the variance 8中 of the return on this best portfolio: (2I) ah:≥o口-(ha+h,) (18a）2-1mz,0=λ0=0/g2. +a4,之o. Moreover,since z:>o in (22a and 226)strictly As in the previous cases,we also must have implies v,=o by virtue of(22c),equation (22a) A>o for a maximum (rather than a minimum) for the subset of positively held stocks i=I,2 of中，and we shall write z:=λh:andy:=au .m'is formally identical to equation (12). The necssary and sufficient conditions for the We can,consequently,use these equations to vector of relalive holdings z:which maximizes bring out certain significant properties of the 6 in (20)are consequently,28 using the Kuhn- security portfolios which will be held by risk- Tucker theorem [ol, averse investors trading in perfect markets.20In Equation(22a-22d)can readily be shown to satisfy the the rest of this paper,all statements with respect to six necessary and two further sufficient conditions of the "other stocks"will refer to other stocks included Kuhn-Tucker theorem.Apart from the constraintso within the porlfolio. and o which are automatically satisfied by the com- puting algorithm [conditions (22band 22c)]the four necessary conditions are: III Risk Premiums and Other Properties This condition is satisfied as a sri )「0φ7 of Stocks Held Long or Short in Optimal Portfolios equalily in our solutions by virtue of equation (22a)[See equation (2)].This strict equality also shows that, Since the covariances between most pairs of 2)k:0「8φ7 o=o,the first complementary slackness stocks will be positive,it is clear from equation condition is also satisfied. (Io)that stocks held long (>o)in a port- 3）[9e]●≥o.This coition ssisied because from folio will generally be those whose expected equation (20), The two additional suficiency conditions are of course oby virtue of equation ()This satisfied because the variance-covariance matrix is positive Ld」 definite,making中（飞，a concave function on and same equation shows that the second complementary 中(，w)a convex function of. slackness condition, 29 More precisely,the properties of portfolios when both may be writtenowhich is 4）aJ the investors and the markets satisfy the conditions stated at the outset of section I or,alternatively,when investors also satisfied because of equation (22c)since o. satisfy Roy's premises as noted previously. 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
22 THE REVIEW OF ECONOMICS AND STATISTICS ratio is negative, and investing in the remaining issues in the proportions hi = Xi/2Xi in accord- ance with the preceding paragraph. But in the more generally realistic cases when covariances are nonzero and short sales are not admitted, the solution of a single bilinear or quadratic programming problem is required to determine the optimal portfolio. (All other points on the "efficient frontier," of course, continue to be irrelevant so long as there is a riskless asset and a "perfect" borrowing market.) The optimal portfolio mix is now given by the set of hiz which maximize 0 in equation (8) subject to the constraint that all hi > o. As before, the (further) constraint that the sum of the hi be unity (equation 9) may be ignored in the initial solution for the relative values of the hi [because 0 in (8) is homogeneous of order zero]. To find this optimum, we form the Lagrangian function (20) 4(h, u) = 6 + 2iuihi which is to be maximized subject to hi > o and ui > o. Using (ii), we have immediately (2aI) il0 o for a maximum (rather than a minimum) of p, and we shall write zi = Xhi and vi = aui. The necssary and sufficient conditions for the vector of relative holdings zi0 which maximizes 0 in (20) are consequently,28 using the Kuhn- Tucker theorem [9], (22a) zi0tij + 2jzj -vij- = vi, i = I, 2, . . .m; where (22b-d) zi? > o, vi0 > o, z.0v.0 = a. This system of equations can be expeditiously solved by the Wilson Simplicial Algorithm [23]. Now let m' denote the number of stocks with strictly positive holdings zi? > o in (22b), and renumber the entire set of stocks so that the subset satisfying this strict inequality [and, hence also, by (22d) vi? = o] are denoted i, 2, . . ., i'. Within this m' subset of stocks found to belong in the optimal portfolio with posi- tive holdings, we consequently have, using the constraint (i9), (I7a) li=m'Zi? = o.1i=im'hio = ,o so that the fraction of the optimal portfolio in- vested in the ith stock (where i = I, 2 . .. im') is (23) hi? = zi-/x? = z 0/2i=jm,z1o . Once again, using (I7a) and (ii), the sum of the zio within this set of stocks held yields as a by- product the ratio of the expected excess rate of return on the optimal portfolio to the variance of the return on this best portfolio: (I8a) z ilm'Z 0 = X0 = X0/o Moreover, since zi? > o in (22a and 22b) strictly implies vi? = o by virtue of (22c), equation (22a) for the subset of positively held stocks i = I, 2 . . . m' is formally identical to equation (I2). We can, consequently, use these equations to bring out certain significant properties of the security portfolios which will be held by risk- averse investors trading in perfect markets.29 In the rest of this paper, all statements with respect to "other stocks" will refer to other stocks included within the portfolio. III Risk Premiums and Other Properties of Stocks Held Long or Short in Optimal Portfolios Since the covariances between most pairs of stocks will be positive, it is clear from equation (I9) that stocks held long (hi? > o) in a port- folio will generally be those whose expected 28 Equation (22a-22d) can readily be shown to satisfy the six necessary and two further sufficient conditions of the Kuhn-Tucker theorem. Apart from the constraints h _ o and u > o which are automatically satisfied by the com- puting algorithm [conditions (22b and 22c)] the four necessary conditions are: )o [?] 0 -o. This condition is satisfied as a strict Lhi equality in our solutions by virtue of equation (22a) [See equation (2i)]. This strict equality also shows that, L) h hi ? = -, the first complementary slackness condition is also satisfied. 3) 0 ] > o. This condition is satisfied because from equation (20), [aI = hi > o by virtue of equation (22b). This same equation shows that the second complementary slackness condition, 4) ui [ a]0 = a, may be written ui0 hi0 = o which is als iie also satisfied because of equation (22C) since a /- a. The two additional sufficiency conditions are of course satisfied because the variance-covariance matrix x is positive definite, making 4 (h, u0) a concave function on h and 0 (h0, u) a convex function of u. 29 More precisely, the properties of portfolios when both the investors and the markets satisfy the conditions stated at the outset of section I or, alternatively, when investors satisfy Roy's premises as noted previously. 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
