Charta is on Google Play: practise on the go. Get the app
Charta

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

CFA® Level I · Portfolio Construction · 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.

Formulas

Weights of the minimum-variance blend

w(asset 1) = (σ₂² − Cov) ÷ (σ₁² + σ₂² − 2 × Cov)

w(asset 1)
the weight in asset 1, the rest going in asset 2
σ₁², σ₂²
the variance of each asset
Cov
the covariance of the two returns

The blend with the least risk of any mix of the two. It can have a lower risk than either asset alone, and it is the left-most point of the minimum-variance frontier.

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. Such portfolios cannot be formed from risky assetsWhy not: They can be formed. They are simply inefficient, because a better one exists.
  2. Another portfolio has the same risk and more return - correct
  3. Such portfolios require leverageWhy not: Portfolios below the minimum-variance point do not need borrowing. They are inefficient for another reason.

The global minimum-variance portfolio is the point with the lowest risk of any mix of the assets. The efficient frontier curve bends back below it, but those lower points are not sensible choices. For each of them there is a point higher up the curve with the same risk (standard deviation) and a higher expected return. So the lower portfolios are dominated: a rational investor wants more return for the same risk and would never choose them. They are ordinary portfolios and need no borrowing. There is no formula.

A harder one

On a chart of expected return against standard deviation, why is the efficient frontier curved rather than a straight line?

  1. Because expected return changes by less than the weights as they shiftWhy not: Expected return is exactly linear in the weights. It is the risk axis that behaves non-linearly.
  2. Because investors face a limit on how much they can borrow to investWhy not: Borrowing limits change how far the frontier extends, not why it bends.
  3. Because mixing imperfectly correlated assets lowers risk below the weighted average - correct

The efficient frontier is the set of best portfolios: the most return for each level of risk. It bends because mixing assets reduces risk. A portfolio's expected return is just the weighted average of its assets' returns, so that part stays straight. Risk is different. Portfolio risk <= weighted average of the assets' risks, with equality only if correlation = 1 (correlation measures how closely two assets move together). Example: two assets, each with risk of 20% and correlation 0.5, held 50/50, give risk of about 17.3%, not 20%. That saving pulls the curve 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.

The efficient frontier shows the best return available for each level of risk. A low-correlation asset tends to move differently from your other holdings, so its ups and downs partly cancel theirs. Portfolio risk falls when assets are not perfectly correlated. Adding it only gives you more choices, since all the old portfolios are still available. So the frontier moves left and upward: the same risk can earn more, or the same return can come with less risk. Its own standalone return matters little.

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

Take the free diagnostic →

More in Portfolio Risk and Return: Part I