Minimum-Variance and Efficient Frontiers

Updated for the 2026-2027 CFA® Level I curriculum.

When you combine risky assets into a portfolio, some combinations give you more return for the same risk than others. The minimum-variance frontier maps every combination that minimizes risk for a given return. The efficient frontier keeps only the good half of that map.

CFA Level I tests whether you can tell the difference and pick the efficient portfolio from a list of choices.

Key Takeaways About Minimum-Variance and Efficient Frontiers

• The minimum-variance frontier includes both efficient and inefficient portfolios; the efficient frontier includes only the efficient half.

• The global minimum-variance portfolio (GMVP) sits at the leftmost point of the frontier, where risk is lowest.

• A portfolio is inefficient if another portfolio offers a higher return at the same or lower risk.

• Investors choose different points on the efficient frontier based on risk tolerance, not on a single "best" portfolio.

• Lower correlation between assets pulls the frontier left, reducing risk without changing expected returns.

• Level I exam questions usually test frontier classification, not frontier derivation.

What You Need to Know for CFA Level I

• Define the minimum-variance frontier and identify it on a graph.

• Locate the global minimum-variance portfolio and explain why it has no efficient portfolios below it.

• Classify given portfolios as efficient or inefficient using return and risk data.

• Explain that investor choice along the efficient frontier depends on risk aversion, not calculation.

• Explain how correlation between assets changes the shape and position of the frontier.

• Understand the two-asset minimum-variance weight formula well enough to interpret it, not just memorize it.

Where This Fits in Portfolio Risk and Return

This concept builds directly on portfolio standard deviation and diversification. Once you know how combining assets changes portfolio risk, the next step is plotting every possible combination on a risk-return graph. That plot is the feasible set. The minimum-variance frontier is the boundary of that feasible set. The efficient frontier is the useful part of that boundary.

What Is the Minimum-Variance Frontier?

Take every possible portfolio you can build from a group of risky assets. Plot each one by expected return (vertical axis) and standard deviation (horizontal axis). The result is a cloud of points called the feasible set. For each level of risk, one portfolio in that cloud has the lowest possible standard deviation. Connect those points and you get the minimum-variance frontier.

The frontier curves to the left because diversification reduces risk when assets are not perfectly correlated. No portfolio can plot to the left of this line. Every portfolio can plot on it or to the right of it.

What Is the Global Minimum-Variance Portfolio?

The global minimum-variance portfolio (GMVP) is the single point on the minimum-variance frontier with the lowest standard deviation of all feasible portfolios. It sits at the leftmost tip of the frontier curve.

The GMVP is not automatically the best portfolio for every investor. It only minimizes risk. It says nothing about maximizing return. An investor who cares about return as well as risk will look at portfolios above the GMVP, not just the GMVP itself.

Efficient vs. Inefficient Portfolios

The minimum-variance frontier splits into two parts at the GMVP.

Portfolios on the lower portion are dominated. For every point below the GMVP, there is a point above it with the same risk and a higher return. A rational investor never chooses a dominated portfolio, because a strictly better option exists at no extra risk.

Efficient portfolio. No other portfolio offers higher return at the same risk, or lower risk at the same return.

Inefficient portfolio. At least one portfolio exists with better return for the same risk, or lower risk for the same return.

How Investors Choose Along the Efficient Frontier

Every portfolio on the efficient frontier is a valid choice. None of them is universally optimal. A conservative investor picks a point near the GMVP, accepting lower expected return for lower risk. An aggressive investor picks a point further up the curve, accepting higher risk for higher expected return.

Level I does not ask you to derive the investor's exact optimal point. That calculation, which uses utility functions and indifference curves, belongs to a separate topic. For this note, remember only that investor choice happens along the efficient frontier, never below it.

How Correlation Shapes the Frontier

Correlation between assets determines how far left the frontier bends.

Lower correlation does not change any individual asset's expected return. It only changes how much risk you can remove by combining assets. This is a common trap: candidates sometimes assume correlation changes returns. It only changes the shape of the risk opportunity set.

Minimum-Variance Portfolio Formula (Two-Asset Case)

For a two-asset portfolio, the weight of Asset that minimizes portfolio variance is:

Notation legend:

• = weight of Asset in the minimum-variance portfolio

• = variance of Asset 's returns

• = variance of Asset 's returns

• = covariance between Asset and Asset returns

This formula finds the exact mix of two assets that produces the lowest possible portfolio variance. You will not need to derive this formula from scratch on the exam, but you should recognize it and understand that it depends on each asset's variance and their covariance, not their expected returns.

Worked Example

Anna is comparing four portfolios built from the same three asset classes.

Step 1: Compare portfolios with the same risk.

Portfolios W and X both have 8.0% standard deviation. X returns 7.0% versus W's 5.0%. W is dominated. W is inefficient.

Step 2: Compare portfolios with the same return.

Portfolios X and Y both return 7.0%. X has lower risk (8.0% versus 11.0%). Y is dominated. Y is inefficient.

Step 3: Check the remaining portfolios.

X and Z both survive the dominance test. X offers the lower-risk, lower-return combination. Z offers the higher-risk, higher-return combination. Neither dominates the other.

Portfolios X and Z sit on the efficient frontier. Portfolios W and Y are inefficient because a better alternative exists at the same risk or the same return. Anna would choose between X and Z based on her risk tolerance, not based on further calculation.

Common Exam Traps

Calling the entire minimum-variance frontier efficient. Only the portion at or above the global minimum-variance portfolio is efficient. The lower portion is dominated.

Confusing the GMVP with the optimal portfolio for every investor. The GMVP only minimizes risk. It ignores return preferences entirely.

Ignoring expected return when comparing portfolios. Classifying portfolios by standard deviation alone misses half the comparison. Dominance requires checking both return and risk.

Assuming correlation changes expected return. Correlation changes the risk opportunity set, not the return of the underlying assets.

Practice Question

An analyst reviews two portfolios built from the same asset universe:

• Portfolio 1: expected return 6.5%, standard deviation 9.0%

• Portfolio 2: expected return 6.5%, standard deviation 10.5%

Which statement is most accurate?

1. Portfolio 2 is efficient because it has a higher standard deviation.

2. Portfolio 1 dominates Portfolio 2 because it offers the same return at lower risk.

3. Both portfolios are efficient because they have the same expected return.

FAQs About Minimum-Variance and Efficient Frontiers