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/termsTHE 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