The valuation of risk assets and the selection of risky Investments in Stock Portfolios and Capital Budgets OR。 John Linne The review of Economics and Statistics, Vol 47, No. 1.(Feb, 1965), pp. 13-37 Stable url: ttp: //inks. istor. org/sici?sici=0034-6535%28196502%2947%3A1%3C13%3ATVORAA%3E2.0.C0%3B2-7 The Review of Economics and Statistics is currently published by The MIT press Your use of the jStoR archive indicates your acceptance of jSTOR's Terms and Conditions of Use, available at http://wwwistororglabout/termshtml.JstOr'STemsaofajournalormullpeprovidesinpartthatunlessyouhaveobtained the JStOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support(@jstor.org tMar1711:19:552007
The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets John Lintner The Review of Economics and Statistics, Vol. 47, No. 1. (Feb., 1965), pp. 13-37. Stable URL: http://links.jstor.org/sici?sici=0034-6535%28196502%2947%3A1%3C13%3ATVORAA%3E2.0.CO%3B2-7 The Review of Economics and Statistics is currently published by The MIT Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/mitpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Sat Mar 17 11:19:55 2007
THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS* John Lintner Introduction and Preview of Som lusions titive markets when utility functions are quad- HE effects of risk and uncertainty upon ratic or rates of return are multivariate normal 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 normalit of projects to include in corporate capital bud- The second section shows that if short sales are gets, have increasingly engaged the attention of permitted, 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 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 weithin 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 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 endent 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 is another in a series of interrelated theoretical on the moot issue of the appropriate functional relationships between"required rates of ret ion, and more recently the Ford Foundation, to the Harvard and relevant risk parameter(s)-and on the Buasietess ahoolowlede generhes support for th is work is most related issue of how"risk classes"of securitie ab his colleagues Professors Bishop, Christenson, Kahr, Raiffa, may best be delineated (if they are to be used,- (especially) Schlaifer, for extensive discussion and com mentary on an earlier draft of this. Tobin [2I, especially pp. 82-851. Tobin assumed that but responsibility for funds rfections rema y his own allocated only over"monetary assets"(risk ee cash and default-free bonds of Market Equilibrium Under Conditions of Risk"(Journal of low. Other approaches are reviewed in Farrar [38] Finance, September I964) appeared after this paper was in It should be noted that the classic paper by Modigl he printers. My first scction, and Miller [i6] was silent on these issues. Corporations were ich 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 Ill and (which have no counterpart in his paper) and for differed (at most) by a"scale factor, "and hence ection IV on the equilibrium prices of risk assets, concerning fectly correlated with each other and(6)were perfect substi- hich our results differ significantly for reasons which will be tutes for each other in perfect markets(p. 266). No comment plored elsewhere. Sharpe does not take up the capital was made on the measure of risk or dgeting problem developed in section V below.] attributes)relevant to the identification of different"equiva- [13]
THE REVIEW OF ECONOMICS AND STATISTICS There seems to be a general presumption among uncertainty per se(as distinct from the effects of y the standard deviation (or coefficient of implications of such uncertainty. In particulg economists that relative risks are best measured diverse expectations), and to derive further variation)of the rate of return, but in the simp- the aggregate market value of any companys t 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. 4(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 lotal 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 witz 14] 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. 6 The joint probability distributions pertain to dollar capital budgeting problem becomes a quadratic returns rather than rates of return 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 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 shown to be an increasing function of each of the lent return"classes. Both Propositions I(market value of firm 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 variance in the project's own presentvalue 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 anea e ruste compe ns, tio to investors for assuming company, and (o) its total covariance with other aThis is. for instance, the presumption of Hirschleifer projects concurrently included in the capital 18, p. 1131, although he was careful not to commit himself to budget. All five factors are involved explicitly his measure alone in a paper prim ir i fator st he stahe rd in the corresponding (derived)formula for the Gordon [s, especially pp 6g and 76r. See also Dorfman in investment project. In this model, all means he constant 6We also ass ommon stock portfolios are not term will be larger, and the er, the higher the( fixed) "inferior goods level of covariances of the gi ocks with invariant, and any effect of changes in capital budgets on the od is th the cash covariances between the values of different companies'stocks dividend and the increase in market price du
VALUATION OF RISK ASSETS 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-discountrale 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 appropriate rate 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 (ii) 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 retur as directly applicable to practical decisions, be- on any portfolio is measured in exactly the sar cause many of the factors which matter very way, including interest received or paid ctice have had to be 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. 7 More detailed non-interest bearing deposits for reasons of conclusions will be found emphasized at numerous liquidity or transactions requirements. 10Hence 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 deducted. We also assume that(2)each investor Market Assumptions will have assigned a joint probability distribution We assume that(1)each individual investor incorporating his best judgments regarding the can invest any part of his capital in certain risk- have specified an expected value and variance to returns on all individual stocks, or at least will free assets (e. g deposits in insured savings ac- every return and a covariance or correlation to counts)all of which pay interest at a common 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 are(3) traded in a single purely competitive one in absolute value(i.e the covariance matrix market, free of transactions costs and taxes, at is positive-definite). The investor computes the depend on his investments or transactions. We on any possible portfolio, 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 TThe relation between the results of this paper and the sum of these components expected returns models which were used in (rr] and [r2] is indicated at the end variances and covariances 10These latter decisions are independent of the decisions Government bonds of appropriate maturity regarding the another important example when their"yield"is urn a-aining funds between risk-free ' Solely for conveniet shall usually refer ominate those with no return once Investments as common though the analysis is of liquidity and requirements are satisfied at the
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 ra7〃ce 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 l1 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 investors utility function is con- ternative stock 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 y 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 parameter, (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 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 proof, 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 As a preliminary step, we need to establish the rowed funds are being assumed to be the same, relation between the investor's total investment G the gross amount invested in stocks, (i)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(i) 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 ber*, and the uncertain return(divi total nel 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 porfolio of stocks be r. Let u rep- ply requires finding the jointly optimal values resent the ratio of gross investment in stocks to for(1)the ratio of the gross investment in stocks aTobin considered the special case where cash with no to the total net investment, and(2) the ratio of required that all assets be held in non-l 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 that although the solution of(1)depends upon 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 separation Mfgis independent of the programming constraints neglec- e property stated in the theorem. Our proof of the theorem asserts that ted in Tobins 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 olio 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 ralio of the gross investment in stocks to the total net available. be readily handled with computer programs now to sub-problem 2 above)is independent of the may howev 1An alternative algebraic proof using utility functions invesiment ock wood Rainha Tobin proved this important separation theo- and presented a similar proof of the theorem in an unpublished em by deriving the detailed solution for the