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Set out what the minimum-variance frontier and the efficient frontier are

CFA® Level I · Portfolio Management · Portfolio Risk and Return: Part I · PM.2.b

The idea

The minimum-variance frontier plots, for each level of expected return, the portfolio with the lowest achievable variance. Its leftmost point is the global minimum-variance portfolio, and the portion above that point is the efficient frontier: no rational investor holds anything below it, because for every such portfolio there is another with the same risk and higher expected return. The frontier curves rather than running straight because imperfect correlation generates diversification, and the lower the correlations among available assets, the further left the frontier bows.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why does no rational investor hold a portfolio below the global minimum-variance point?

  1. Because such portfolios are unavailableWhy not: They are perfectly constructible; the objection is that each is dominated by one directly above it.
  2. Because another portfolio offers the same risk with higher expected return — correct
  3. Because they require leverageWhy not: Nothing about the lower half of the frontier involves borrowing. It is made up of ordinary long positions.

Those portfolios are dominated. The same standard deviation is available higher up the curve with more expected return.

A harder one

Why is the efficient frontier curved rather than a straight line?

  1. Because expected returns are non-linear in the weightsWhy not: Expected return is exactly linear in the weights. It is the risk axis that behaves non-linearly.
  2. Because short selling is prohibitedWhy not: Constraints on short selling change the frontier's extent, not the source of its curvature.
  3. Because risk rises more slowly than return — correct

Imperfect correlation means combined risk rises more slowly than a weighted average as assets are added. Only with perfect positive correlation does the combination line become straight; any correlation below one bends it to the left.

The hardest tier

An analyst adds an asset class with low correlation to an existing opportunity set. What happens to the efficient frontier?

  1. It shifts left and upward, offering more return for the same risk — correct
  2. It is unchanged unless the new asset has the highest expected returnWhy not: A low-correlation asset improves the risk side even with an unremarkable return.
  3. It shifts right, since more assets mean more riskWhy not: Adding an option can never worsen the opportunity set: the old portfolios remain available.

A genuinely diversifying asset improves attainable combinations even if its own expected return is unremarkable. That improvement, not the asset's standalone return, is the case for including it.

In the app: 9 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.

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