The Efficient Frontier: The Math Behind "Optimal" Portfolios

This platform's lesson on modern portfolio theory dives into Harry Markowitz's key idea: combining the right assets lowers risk without sacrificing return. We’ll explore the efficient frontier, the mathematical object that comes from this concept. It’s a curve that shows the portfolios offering the highest expected return for each level of risk, or the lowest risk for any expected return. If you plot every possible combination of a given set of assets on a graph where risk is on the horizontal axis and expected return is on the vertical, the efficient frontier forms the upper-left boundary. This boundary represents the best trade-offs available. Other combinations will fall below or to the right, which means they’re worse in at least one respect.

A portfolio below the efficient frontier is sub-optimal. It takes on more risk than necessary for its expected return or achieves less return than possible for its level of risk. In simpler terms, some alternative combination of those same assets could improve one or both metrics without any trade-off. If you add a risk-free asset, like a Treasury bill, you get the Capital Market Line. This straight line shows the theoretical combinations of that risk-free asset and one specific optimal risky-asset combination. This point is called the tangency portfolio, where the Capital Market Line touches the efficient frontier. Together, they provide a better risk/return trade-off than the risky-asset frontier alone.