In this paper, a general treatment of identifying the set of unbiased estimators of N-period mean returns is advanced and a new unbiased estimator, which promises near-minimum variance and minimal computation, is formulated. The new estimator is also equally applicable to other processes of compound growth.

The mean-variance capital asset pricing model (CAPM) of Sharpe and Lintner was extended by Brennan [3] to incorporate divergent borrowing and lending rates. He found that in equilibrium the security market line (SML) has the same structure as the SML under the single-rate CAPM of Sharpe and Lintner. That is, the expected return of a security or a portfolio remains linear in its systematic risk, with the intercept replaced by an equivalent risk-free return, which is an average of the divergent borrowing and lending rates weighted by the investors' taste parameters. The equivalent risk-free return is larger than the riskless lending rate and, hence, does not represent an inconsistency with the empirical findings by Friend and Blume [4] and by Black, Jensen and Scholes [1[ that the intercept of empirical SML estimated for the single-rate CAPM is larger than the riskless rate. Moreover, Brennan attempted to show that his construct can be extended to the extreme case where there are no riskless opportunities. The case of no riskless opportunities was of course investigated by Black [2], who generalized the CAPM and SML by inventing the concept of zero-beta port-folio to account for the same empirical problem encountered in the traditional SML tests of CAPM. Since the Sharpe-Lintner single-riskless-rate CAPM implies a perfect loan market, we may view the attempts by Black and Brennan as generalizing the CAPM by incorporating financial restrictions and loan market imperfections. Their primary motive, however, is empirical, i.e., to reconcile the results from the traditional SML tests with their generalized CAPM.

In so far as the concept of systematic risk is predicated on the Sharpe-Lintner theory of capital market equilibrium [5, 4], the time-horizon of systematic risk must conform with the time-horizon of market equilibrium. Since it has been suggested that market equilibrium is instantaneous [3, p. 188], it would follow that systematic risk should also be instantaneous. This paper is, therefore, concerned with the evaluation and measurement of instantaneous risk. Although Jensen [3] has made a similar attempt in a much larger study, we have reason to believe it is not satisfactory. We shall then begin in Section I by discussing Jensen's approach to the horizon problem. In Section II, an alternative procedure of evaluating systematic risk is suggested. Section III concludes the paper by comparing estimates of instantaneous risks based upon weekly returns of 30 Dow-Jones stocks. The motivation behind the paper is obvious. A correct formulation of instantaneous systematic risk is not only a logical extension of the capital market equilibrium theory but is also a yardstick for measuring portfolio performance in terms of risk and return.

The advent of the computer has permitted financial theorists to collect and analyze large amounts of financial data. In the field of investments some of the most important work has focused on historical rates of return in investments in common stocks. The classical study in this area is the Fisher-Lorie study [8, 9] in which intern al rates of return were calculated for every security listed on the New York Stock Exchange from 1926–1965. Other studies related to the area have been complicated by Herzog [10], Fisher [6, 7], Latané and Young [11], Soldofsky and Biderman [12], and Evans [3, 4].

This paper presents a method for solving the mean‐variance portfolio selection problem that is applicable to the case where the number of securities is nondenumerably infinite. Necessary conditions for the existence of an optimal portfolio density are obtained and an expression for the efficient frontier is derived. The conditions for the existence of an optimal portfolio of continuously maturing bonds when their covariance matrix is singular are used to derive an arbitrage‐free bond pricing equation. A method for estimating the covariance matrix and the associated efficient frontier is presented.

This paper presents a method for solving the mean-variance portfolio selection problem that is applicable to the case where the number of securities is nondenumerably infinite. Necessary conditions for the existence of an optimal portfolio density are obtained and an expression for the efficient frontier is derived. The conditions for the existence of an optimal portfolio of continuously maturing bonds when their covariance matrix is singular are used to derive an arbitrage-free bond pricing equation. A method for estimating the covariance matrix and the associated efficient frontier is presented.