Efficient Frontier

Portfolio expected return is simply the weighted average of the component expected returns, but portfolio risk is not. Variance for a two-asset mix equals the first weight squared times the first variance, plus the second weight squared times the second variance, plus twice the product of the two weights, the two standard deviations and the correlation coefficient. Because that final term shrinks as correlation falls, any correlation below one produces a portfolio standard deviation lower than the weighted average of the individual standard deviations.

That gap is the diversification benefit, and it is what gives the frontier its concave, leftward-bowing shape. At perfect positive correlation the opportunity set collapses to a straight line between the two assets and no risk reduction occurs. As correlation approaches minus one the curve bows sharply toward the vertical axis, and in the limiting case a combination with zero variance becomes attainable.

The frontier is the solution to a constrained optimisation: minimise portfolio variance subject to achieving a specified expected return, with weights summing to one. Repeating that programme across a grid of target returns traces the minimum-variance frontier; discarding everything below the global minimum-variance portfolio leaves the efficient set. The input burden grows quickly, since an N-asset problem requires N expected returns, N variances and N times N minus one, divided by two, distinct covariances.

Constraints matter as much as inputs. Prohibiting short sales, capping single-issuer or single-sector weights, or imposing liquidity floors pulls the attainable frontier inward and below the unconstrained one. Where such constraints bind, the frontier is piecewise, and the points at which an asset enters or leaves the solution are the corner portfolios used in practical allocation work.

Introducing borrowing and lending at a risk-free rate changes the geometry entirely. Combinations of the risk-free asset and any risky portfolio plot along a straight line, because the risk-free asset has zero variance and zero covariance with everything else. The investor therefore seeks the steepest such line, which is the one tangent to the efficient frontier; its slope is the Sharpe ratio of the tangency portfolio, and it is called the capital allocation line.

Under the capital asset pricing model's assumption of homogeneous expectations, every investor identifies the same tangency portfolio, which must then be the market portfolio, and the line becomes the capital market line. Risk preference no longer affects which risky portfolio is held; it affects only the split between that portfolio and the risk-free asset. This separation of the investment decision from the financing decision is the two-fund separation theorem.

The frontier narrows the choice set but does not identify a single answer. The optimal portfolio for a given investor is where the highest attainable indifference curve touches the capital allocation line. Because more risk-averse investors have steeper indifference curves, they settle at lower-volatility points, lending part of their capital at the risk-free rate; less risk-averse investors sit beyond the tangency point, borrowing to lever the risky portfolio.

Candidates should be able to compute two-asset expected return and standard deviation, rank portfolios as efficient, inefficient or unattainable, and explain the impact of a correlation change. They should also state the model's weaknesses: outputs are highly sensitive to estimation error in expected returns, the framework is single-period, and variance treats upside and downside deviations identically, which understates risk when return distributions are skewed or fat-tailed.