Robert C. Merton derived the efficient portfolio frontier in mean-variance space in 1970. He showed that the efficient frontier is a parabola, with the minimum-variance portfolio at the lowest point. The frontier represents the set of portfolios that offer the highest expected return for a given level of risk. Merton proved the separation theorem, which states that investors can separate their investment decisions into two parts: one involving a risk-free asset and the other involving a portfolio of risky assets. This theorem is central to modern portfolio theory. When all assets are risky, the efficient frontier can be derived analytically. The optimal portfolio weights are determined by the inverse of the variance-covariance matrix of returns. The efficient frontier is a parabola, and the minimum-variance portfolio is the point on the frontier with the lowest variance. The efficient frontier can also be represented in the mean-standard deviation plane, where it is a convex curve. Merton also proved a mutual fund theorem, which states that there exist two portfolios (mutual funds) such that all risk-averse investors will be indifferent between investing in these funds or in the original set of assets. These two funds are the minimum-variance portfolio and a portfolio that offers the highest return for a given level of risk. When one of the assets is risk-free, the efficient frontier changes. The risk-free asset allows investors to adjust their portfolios to achieve a desired level of risk and return. The efficient frontier in this case is a straight line that starts from the risk-free asset and touches the efficient frontier of risky assets. The mutual fund theorem is extended to this case, where one fund holds only the risk-free asset and the other fund holds only risky assets. Merton's work provides a foundation for modern portfolio theory and has important implications for investment decisions. The separation theorem and the mutual fund theorem are key results that help investors understand how to construct optimal portfolios. The efficient frontier is a crucial concept in finance, as it represents the best possible trade-off between risk and return.Robert C. Merton derived the efficient portfolio frontier in mean-variance space in 1970. He showed that the efficient frontier is a parabola, with the minimum-variance portfolio at the lowest point. The frontier represents the set of portfolios that offer the highest expected return for a given level of risk. Merton proved the separation theorem, which states that investors can separate their investment decisions into two parts: one involving a risk-free asset and the other involving a portfolio of risky assets. This theorem is central to modern portfolio theory. When all assets are risky, the efficient frontier can be derived analytically. The optimal portfolio weights are determined by the inverse of the variance-covariance matrix of returns. The efficient frontier is a parabola, and the minimum-variance portfolio is the point on the frontier with the lowest variance. The efficient frontier can also be represented in the mean-standard deviation plane, where it is a convex curve. Merton also proved a mutual fund theorem, which states that there exist two portfolios (mutual funds) such that all risk-averse investors will be indifferent between investing in these funds or in the original set of assets. These two funds are the minimum-variance portfolio and a portfolio that offers the highest return for a given level of risk. When one of the assets is risk-free, the efficient frontier changes. The risk-free asset allows investors to adjust their portfolios to achieve a desired level of risk and return. The efficient frontier in this case is a straight line that starts from the risk-free asset and touches the efficient frontier of risky assets. The mutual fund theorem is extended to this case, where one fund holds only the risk-free asset and the other fund holds only risky assets. Merton's work provides a foundation for modern portfolio theory and has important implications for investment decisions. The separation theorem and the mutual fund theorem are key results that help investors understand how to construct optimal portfolios. The efficient frontier is a crucial concept in finance, as it represents the best possible trade-off between risk and return.