This paper by Robert C. Merton, published in October 1970, provides an analytical derivation of the efficient portfolio frontier in the mean-variance sense. The author aims to explicitly derive the efficient portfolio frontiers and verify the characteristics claimed for these frontiers. The most significant implication, the separation theorem, is stated and proven within the context of a mutual fund theorem. The paper begins with an introduction where Merton discusses the existing literature on the efficient portfolio frontier, noting that for more than three assets, graphical methods are commonly used. He then derives the efficient portfolio set when all securities are risky, using Lagrange multipliers to solve the constrained minimization problem. The solution is expressed as a set of linear equations, leading to the equation for the variance of a portfolio as a function of its expected return, which is a parabola. Merton also proves the separation theorem, showing that any portfolio on the efficient frontier can be attained by a linear combination of two specific portfolios. These two portfolios, or "mutual funds," are unique up to a non-singular transformation and can be constructed based on the expected returns of the original assets. The paper further extends the analysis to include a riskless asset, demonstrating that the efficient frontier remains convex and that all efficient portfolios are perfectly correlated. Finally, Merton derives the security market line, which is a direct consequence of the efficient frontier when a riskless asset is included. The paper concludes with a discussion on the graphical solutions for different scenarios, emphasizing that the traditional approach of drawing tangents to the efficient frontier for risky assets only is incorrect when a riskless asset is present.This paper by Robert C. Merton, published in October 1970, provides an analytical derivation of the efficient portfolio frontier in the mean-variance sense. The author aims to explicitly derive the efficient portfolio frontiers and verify the characteristics claimed for these frontiers. The most significant implication, the separation theorem, is stated and proven within the context of a mutual fund theorem. The paper begins with an introduction where Merton discusses the existing literature on the efficient portfolio frontier, noting that for more than three assets, graphical methods are commonly used. He then derives the efficient portfolio set when all securities are risky, using Lagrange multipliers to solve the constrained minimization problem. The solution is expressed as a set of linear equations, leading to the equation for the variance of a portfolio as a function of its expected return, which is a parabola. Merton also proves the separation theorem, showing that any portfolio on the efficient frontier can be attained by a linear combination of two specific portfolios. These two portfolios, or "mutual funds," are unique up to a non-singular transformation and can be constructed based on the expected returns of the original assets. The paper further extends the analysis to include a riskless asset, demonstrating that the efficient frontier remains convex and that all efficient portfolios are perfectly correlated. Finally, Merton derives the security market line, which is a direct consequence of the efficient frontier when a riskless asset is included. The paper concludes with a discussion on the graphical solutions for different scenarios, emphasizing that the traditional approach of drawing tangents to the efficient frontier for risky assets only is incorrect when a riskless asset is present.