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/termsVALUATION 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