Strategic Financial Management · Portfolio Theory and Practice

Markowitz Efficient Frontier Theory Explained

Q: What is the efficient frontier in Markowitz theory?

It is the set of portfolios that give the highest expected return for each level of risk. It lies on the upper boundary of the feasible set, from the minimum variance portfolio upward.

Q: How do you find the minimum variance portfolio of two assets?

Use w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2Cov12) and w2 = 1 − w1. Then compute the portfolio variance with these weights.

Q: How is the optimal portfolio chosen?

The optimal portfolio is where your highest attainable indifference curve touches the efficient frontier. A more risk-averse investor picks a point with lower risk.

Q: Why does diversification reduce risk in this model?

Portfolio variance depends on covariances between assets. When correlation is below +1, the assets' fluctuations partly offset each other, so portfolio risk is less than the weighted average of individual risks.

Updated 11 October 2026 · Fact-checked

The Markowitz model picks portfolios using only expected return and variance. The feasible set holds all possible portfolios. The efficient frontier is the upper edge, starting at the minimum variance portfolio, giving the highest return for each risk level. Your indifference curves select the optimal point on it.

Understand Markowitz Efficient Frontier Theory

Harry Markowitz showed that you should judge a portfolio as a whole, not asset by asset. What matters is the portfolio's expected return and its risk, measured by variance or standard deviation. An asset that looks risky alone can lower total risk if it does not move in step with the others.

Assumptions of the model:
- Investors are rational and risk-averse. They prefer more return to less, and less risk to more.
- Investors decide only on expected return and variance (standard deviation) of the portfolio.
- All investors have the same single-period investment horizon.
- Assets are infinitely divisible, and there are no taxes or transaction costs.
- Investors have the same information, and return estimates and correlations are known.

The feasible set (opportunity set) is the region of all portfolios you can build from the available assets, plotted with risk on the x-axis and return on the y-axis. With two assets it is a curve. With many assets it is a solid area.

The efficient frontier is the upper-left boundary of the feasible set, starting from the global minimum variance portfolio and going up to the highest-return portfolio. A portfolio is efficient if no other portfolio gives a higher return for the same risk, or lower risk for the same return. Portfolios on the lower part of the boundary are inefficient, because a portfolio with the same risk and higher return exists above them.

The frontier is the same for all investors who share the same inputs. Where you stop on it depends on your risk attitude. An indifference curve joins combinations of risk and return that give you equal satisfaction. For a risk-averse investor it slopes upward and is convex. Curves higher and to the left are better. The optimal portfolio is where the highest reachable indifference curve just touches (is tangent to) the efficient frontier. A more risk-averse investor has steeper curves and picks a point nearer the minimum variance portfolio.

Key rules to remember

Portfolio expected return: Rp = w1R1 + w2R2 (in general, Rp = Σ wi Ri); Weights add up to 1. Return is a simple weighted average.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; The last term can also be written 2 w1 w2 Cov12. Take the square root for standard deviation.
Covariance and correlation: Cov12 = ρ12 × σ1 × σ2; ρ lies between -1 and +1. Diversification gain is largest when ρ is low.
Minimum variance weight (two assets): w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2 Cov12), and w2 = 1 − w1; Assumes short sales are allowed in the formula. Check that both weights are positive if short sales are barred.
Minimum variance when ρ = -1: w1 = σ2 ÷ (σ1 + σ2), giving σp = 0; Perfect negative correlation allows a risk-free portfolio.
Limits of portfolio risk: ρ = +1: σp = w1σ1 + w2σ2; ρ = -1: σp = |w1σ1 − w2σ2|; Portfolio risk is highest at ρ = +1, and the two-asset curve becomes a straight line.

How to solve Markowitz Efficient Frontier Theory questions

Use this method for any numerical or theory question on the Markowitz model.

1Write down the inputs: expected returns, standard deviations, correlation or covariance, and weights.
2Convert correlation to covariance with Cov = ρ × σ1 × σ2 if needed. Keep variances as σ², not σ.
3Compute the portfolio expected return as the weighted average.
4Compute the portfolio variance with the two-asset formula, then take the square root for standard deviation.
5If asked for the minimum variance portfolio, use the weight formula, then compute its return and risk.
6To judge efficiency, compare portfolios: one dominates another if it has a higher or equal return with lower or equal risk.
7For optimal choice, state that the portfolio is where the investor's highest indifference curve is tangent to the efficient frontier, and link it to risk aversion.
8Close with a clear conclusion in one line.

Quickest way: Minimum variance weights in under two minutes

When to use it: Use when the question gives σ1, σ2 and ρ (or covariance) and asks for the least risky mix of two assets.

Find Cov = ρσ1σ2.
Compute the numerator σ2² − Cov and the denominator σ1² + σ2² − 2Cov.
Divide to get w1, then w2 = 1 − w1.
Plug the weights into the return and variance formulas once, and check that the variance is no higher than the lower of σ1² and σ2². It is strictly lower only when ρ < σ1/σ2, taking σ1 as the lower-risk asset.

Common mistakes in Markowitz Efficient Frontier Theory

Using standard deviations instead of variances in the weight formula.
The question gives σ in percent, and students plug it straight in.
Fix: Square σ first. Only the ρ = -1 special case uses σ directly.
Forgetting the factor 2 in the covariance term.
Students memorise the formula halfway.
Fix: Write 2 × w1 × w2 × Cov every time, because the cross term appears twice.
Taking the weighted average of standard deviations as portfolio risk.
Return is a weighted average, so risk seems the same.
Fix: That is true only when ρ = +1. Otherwise use the variance formula.
Calling every point on the feasible set efficient.
Students confuse the feasible set with the efficient frontier.
Fix: Only the upper boundary from the minimum variance point upward is efficient.
Saying the optimal portfolio is the minimum variance portfolio for everyone.
Risk aversion is mixed up with risk minimisation.
Fix: The minimum variance point suits only an extremely risk-averse investor. Others sit further up the frontier at the tangency point.
Leaving the answer as variance when the question asks for risk.
Students stop after the long calculation.
Fix: Take the square root and report standard deviation in percent.

Worked examples

Example 1

Asset A has expected return 12% and standard deviation 20%. Asset B has expected return 18% and standard deviation 30%. Correlation is 0.4. Find the expected return and standard deviation of a portfolio with 60% in A and 40% in B.

Show the solution

Return = 0.6 × 12 + 0.4 × 18 = 7.2 + 7.2 = 14.4%.
Cov = 0.4 × 20 × 30 = 240.
Variance = (0.6)²(400) + (0.4)²(900) + 2(0.6)(0.4)(240).
= 0.36 × 400 + 0.16 × 900 + 0.48 × 240 = 144 + 144 + 115.2 = 403.2.
Standard deviation = √403.2 = 20.08% (approx).

Answer: Expected return is 14.4% and standard deviation is about 20.08%.

Example 2

Two securities have σ1 = 10% and σ2 = 20%, with correlation 0.5. Find the weights of the minimum variance portfolio and its standard deviation.

Show the solution

Cov = 0.5 × 10 × 20 = 100.
σ1² = 100 and σ2² = 400.
w1 = (400 − 100) ÷ (100 + 400 − 2 × 100) = 300 ÷ 300 = 1.
So w2 = 0. The minimum variance portfolio is wholly security 1.
Check: its standard deviation is 10%, below security 2's 20%. Any mix with security 2 raises variance, since 0.5 is high relative to the ratio σ1/σ2 = 0.5.
Standard deviation = √100 = 10%.

Answer: Invest 100% in security 1, with standard deviation 10%. Here diversification into security 2 cannot lower risk below that of security 1.

Exam tips

Show the covariance step separately. Marks are often given for each stage of the variance calculation.
For theory questions, list the assumptions as short bullets and then draw a neat sketch of the feasible set, frontier and indifference curves.
Always label the minimum variance portfolio and state that only the part above it is efficient.
Check that weights sum to 1 and variance is positive before moving on.
In MCQs, watch for the ρ = -1 and ρ = +1 special cases, which have shortcut results.

Practice questions from Portfolio Theory and Practice

Markowitz Efficient Frontier Theory in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Markowitz Efficient Frontier Theory: frequently asked questions

What is the efficient frontier in Markowitz theory?

It is the set of portfolios that give the highest expected return for each level of risk. It lies on the upper boundary of the feasible set, from the minimum variance portfolio upward.

How do you find the minimum variance portfolio of two assets?

Use w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2Cov12) and w2 = 1 − w1. Then compute the portfolio variance with these weights.

How is the optimal portfolio chosen?

The optimal portfolio is where your highest attainable indifference curve touches the efficient frontier. A more risk-averse investor picks a point with lower risk.

Why does diversification reduce risk in this model?

Portfolio variance depends on covariances between assets. When correlation is below +1, the assets' fluctuations partly offset each other, so portfolio risk is less than the weighted average of individual risks.