¹ Journal of Finance, Vol. 8 (March, 1952)Google ScholarPubMed.

² Portfolio Selection: Cowles Foundation Monograph 16, (New York: John Wiley & Sons, 1959)Google Scholar.

³ Tobin, James, “Liquidity Preference as Behavior Toward Risk,” Review of Economic Studies, 25 (February, 1958), p. 73Google Scholar; Baumol, William J., “An Expected Gain-Confidence Limit Criterion for Portfolio Selection,” Management Science. Vol. 10(October, 1963), pp. 174–182Google Scholar; William F. Sharpe, “A Simplified Model for Portfolio Analysis,” ibid., Vol. 9 (January, 1963), pp. 277–90; Lintner, John, “Security Prices, Risk, and Maximal Gains from Diversification,” The Journal of Finance, Vol. 20 (12, 1965), pp. 587–615Google Scholar.

⁴ IBM's three portfolio selection programs are discussed by Milton Drandell in a paper presented at the Institute of Management Sciences/Operations Research Society of America Joint Western Regional Meeting, April, 1965.

⁵ Latané, Henry A., “Investment Criteria—A Three Asset Portfolio Balance Model,” The Review of Economics and Statistics, Vol. 45 (November, 1963), pp. 427–430Google Scholar; Hirshleifer, Jack, “Investment Decision Under Uncertainty: Application of the State-Preference Approach,” The Quarterly Journal of Economics, Vol. 80 (05, 1966), pp. 252–277Google Scholar.

⁶ The work of Tobin, Meltzer, Latané, and Chow all appear to have been influenced by the new approach to asset selection. For a more recent summary type article see, Chow, Gregory C., “On the Long-run and Short-Run Demand for Money,” Journal of Political Economy, Vol. 74 (April, 1966), pp. 111–129CrossRefGoogle Scholar.

⁷ If margin requirements are restrictive and the investment frontier for risk assets has so little curvature as to be almost parallel to the borrowing line, there may be some advantage in slightly tipping the borrowing line in favor of riskier assets so as to maximize the benefits from limited leverage. This can be accomplished by raising the borrowing rate in the manner of a shadow price to establish a higher point of tangency and then making a judgment as to whether the incremental increase in expected return for the portfolio as a whole is worth the incremental increase in portfolio variance, leverage the same.

⁸ Smith, Keith, “A Transition Model for Portfolio Revision,” Journal of Finance, (09, 1967, Forthcoming)Google Scholar.

⁹ This result is especially encouraging considering the lack of success that was experienced by Kalman Cohen and Jerry Pogue in their endeavor to improve upon the single-index model by developing multi-index models. See, “An Empirical Evaluation of Alternative Portfolio Selection Models,” Journal of Business (Forthcoming).

¹⁰ An arithmetic average, link-relative return index was also inserted into the Sharpe program. It produced a portfolio which was slightly more efficient than that produced by Standard and Poor's index, but not nearly as efficient as portfolio (VI). The main difference was the weight given to United Aircraft: six percent for portfolio (VI) but only 1.7 percent using the arithmetic average, link-relative return index. United Aircraft's index correlation was .358 for the ordinary Dow Jones index but jumped to .463 for the link-relative return index. The increase in the index correlation coefficient can probably be explained by stock dividends of 50 percent in 1955 and 20 percent in 1957 which, owing to the unique construction of the Dow Jones Industrial Average, effectively reduced the importance of United Aircraft in the ordinary Dow Jones Industrial Average index during subsequent years.

¹¹ Yule, George U. and Kendall, Maurice G., Introduction to the Theory of Statistics, 14th edition revised (London: Charles Griffin and Co., 1950), p. 150Google Scholar.

¹² Professor of Statistics, University of North Carolina.

¹³ Professor of Finance, University of Chicago.

¹⁴ Equation (13) has been derived by Latané in a more indirect manner by assuming a binomial distribution of equally probable good and bad returns and then making certain approximating assumptions which are essentially equivalent to the difference in results which are obtained when equations (1) and (3) are substituted into equation (9) rather than equation (11). See Latané, Henry, “Investment Criteria—A Three Asset Portfolio Balance Model,” The Review of Economics and Statistics, Vol. 45 (November, 1963), pp. 427–430CrossRefGoogle Scholar.

¹⁵ Equation (13) was derived from a formula which is only approximately correct. If 1 is equal to zero there will obviously be no income to offset capital depreciation. A large i, on the other hand, is likely to produce a portfolio which is overly conservative. The recommended procedure for maximizing income is to define returns as the ratio of expected wealth at the end of the period to wealth at the beginning of the period and then choose the lesser of p₁′ or p₁″, where X in equation (15) is set equal to one.

¹⁶ While one standard deviation provides almost no protection in a context of repeated investment, investors tend to be protected from mathematical bias of this sort by the existence of margin requirements. A value for X greater than or equal to the maintenance margin should be inserted into equation (15) if risk assets are purchased on margin and. there is any danger of receiving a margin call.

The purchase of risk assets on margin raises a fairly crucial question as to what is an appropriate investment interval. Professor Shelton has noted, in private conversations, that the expected return tends to increase over the return interval at an exponential rate while the standard deviation may only increase by the square root the return interval. This means, of course, that the longer the return interval, the higher the proportion of risk assets. When stocks are purchased on margin and subject to call there would appear to exist some danger of being sold out on the basis of a temporary dip in stock prices if the interval of analysis is too long. When risk estimates are obtained from past data one can guard against this unhappy event to some extent by choosing the risk and return interval which encompasses the most severe decline.

¹⁷ Baumol, William J., “An Expected Gain-Confidence Limit Criterion for Portfolio Selection,” Management Science, Vol. 10 (October, 1963), pp. 174–182CrossRefGoogle Scholar.

¹⁸ Macauley, Frederick R., The Movements of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856 (New York: National Bureau of Economic Research, 1938), pp. 147–148Google Scholar.

¹⁹ Fisher, Lawrence, “Some New Stock-Market Indexes,” Journal of Business, Vol. 39 (January, 1966), pp. 191–225CrossRefGoogle Scholar.

²⁰ Cohen, Kalman J. and Fitch, Bruce P., “The Average Investment Performance Index,” Management Science, Vol. 12 (February, 1966), B–195–215CrossRefGoogle Scholar.

²¹ Slatter, John, “Mutual Fund Liquidity, the Record Suggests, Often Calls the Turn”, Barron's, 02 6, 1967, p. 5Google Scholar.