FRM Exam Part I · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

Markowitz Efficient Frontier and Optimal Portfolios Explained

Q: What is the minimum variance portfolio formula?

For two assets, the weight in asset 1 is (σ2² − Cov12) ÷ (σ1² + σ2² − 2·Cov12). The weight in asset 2 is one minus that. It gives the lowest possible portfolio variance from those two assets.

Q: What are the assumptions of mean-variance optimization?

Investors consider only expected return and variance over a single period. They are rational and risk averse and prefer more return for a given risk. Inputs are assumed known, and in practice are estimated with error.

Q: How do I find the optimal risky portfolio?

With a risk-free asset, choose the portfolio with the highest Sharpe ratio, which is the tangency portfolio. Without one, the best portfolio on the efficient frontier depends on the investor's risk aversion.

Q: Why does lower correlation improve the efficient frontier?

Lower correlation reduces the cross term in portfolio variance. The same expected return then comes with lower risk, so the frontier shifts left.

Updated 11 October 2026 · Fact-checked

The Markowitz efficient frontier is the set of portfolios that give the highest expected return for each level of risk, found by mean-variance optimization. The minimum variance portfolio is its leftmost point. To solve questions, compute portfolio variance from weights, volatilities and correlation, then compare portfolios or find the weights that minimize variance.

Understand Markowitz Efficient Frontier and Optimal Portfolios

Harry Markowitz showed that you should judge an asset by how it changes the risk of the whole portfolio, not by its own risk alone. Risk is measured by variance (or standard deviation) of returns. Return is measured by expected return.

Combine two risky assets and portfolio variance depends on their weights, their volatilities and their correlation. When correlation is below +1, the portfolio volatility is less than the weighted average of the two volatilities. This is the diversification benefit. The lower the correlation, the more the curve of possible portfolios bends to the left.

Plot every feasible portfolio with standard deviation on the horizontal axis and expected return on the vertical axis. The leftmost point is the minimum variance portfolio. Portfolios on the curve above it are efficient: no other portfolio has higher return for the same risk. Portfolios on the lower part of the curve are inefficient, because another portfolio offers more return for the same risk.

The frontier does not tell you which efficient portfolio to hold. That depends on investor preferences. A more risk-averse investor picks a point further left. A less risk-averse investor picks a point further right. Formally, the investor chooses the point where the highest attainable indifference curve touches the frontier.

Mean-variance optimization rests on assumptions. Investors care only about expected return and variance. They prefer more return and less risk. They are rational and risk averse. Inputs (expected returns, volatilities, correlations) are known and single-period. Returns are often assumed normal, or investors have quadratic utility. In practice the weights are very sensitive to input estimates, especially expected returns.

Key formulas to remember

Portfolio expected return: E(Rp) = w1·E(R1) + w2·E(R2); Weights sum to 1. Linear in weights.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ12·σ1·σ2; The last term uses covariance: Cov12 = ρ12·σ1·σ2.
Minimum variance weight (two assets): w1* = (σ2² − ρ12·σ1·σ2) ÷ (σ1² + σ2² − 2·ρ12·σ1·σ2); w2* = 1 − w1*. Equivalent form uses Cov12 in place of ρ12·σ1·σ2.
Perfect correlation limits: ρ = +1: σp = w1σ1 + w2σ2. ρ = −1: σp = |w1σ1 − w2σ2|; With ρ = −1 risk can be reduced to zero at a particular weight.
Optimal risky portfolio (tangency) criterion: Maximize [E(Rp) − Rf] ÷ σp; This is the Sharpe ratio. It applies when a risk-free asset exists.
Utility of a risk-averse investor: U = E(R) − 0.5·A·σ²; A is the risk aversion coefficient. Higher A means a point further left on the frontier.

How to solve Markowitz Efficient Frontier and Optimal Portfolios questions

Use this order for most frontier and optimization questions. Keep returns and volatilities as decimals.

1Identify what is asked: portfolio return, portfolio risk, minimum variance weights, or the best portfolio for an investor.
2Write down the weights, expected returns, standard deviations and correlation. Convert covariance to correlation or the reverse if needed.
3Compute expected return as the weighted average.
4Compute variance with the two-asset formula, including the 2·w1·w2·Cov term. Take the square root only at the end.
5For the minimum variance portfolio, use the weight formula, then check that the weights sum to 1.
6For the best portfolio, compare Sharpe ratios if a risk-free rate is given, or compare utility U = E(R) − 0.5·A·σ² if A is given.
7Sanity check: portfolio σ must be no higher than the weighted average of volatilities when ρ < 1.

Quickest way: Shortcut for minimum variance weights and option elimination

When to use it: Two-asset questions with four numeric options and limited time.

Compute Cov12 = ρ·σ1·σ2 once.
Use w1* = (σ2² − Cov) ÷ (σ1² + σ2² − 2·Cov). Do not solve the calculus.
The lower-variance asset should get the larger weight when ρ is low or zero. Eliminate options that contradict this.
If ρ = 0, use w1* = σ2² ÷ (σ1² + σ2²) directly.
If a weight is above 1 or below 0, the answer implies short selling. Check that the question allows it.
Use the calculator memory to store variances and covariance and avoid rounding early.

Common mistakes in Markowitz Efficient Frontier and Optimal Portfolios

Forgetting the covariance term or the factor of 2 in portfolio variance.
Students remember w1²σ1² + w2²σ2² and stop.
Fix: Write all three terms every time. The cross term is 2·w1·w2·ρ·σ1·σ2.
Averaging volatilities by weights to get portfolio risk.
Return is a weighted average, so risk seems to be one too.
Fix: That works only when ρ = +1. Otherwise compute variance, then take the square root.
Treating the whole curve as the efficient frontier.
The curve looks like one object.
Fix: Only the part from the minimum variance portfolio upward is efficient. The lower branch is dominated.
Reporting variance as standard deviation, or squaring a percentage wrongly.
Rushing and mixing units.
Fix: Use decimals, 0.20² = 0.04. Take the square root at the end and match the unit asked.
Thinking the minimum variance portfolio is the optimal portfolio for everyone.
It has the lowest risk, so it seems best.
Fix: The optimal point depends on risk aversion. Only an extremely risk-averse investor picks it.
Using the Sharpe-ratio criterion when no risk-free rate is given.
Mixing frontier selection with the tangency portfolio.
Fix: Without a risk-free asset, choice depends on preferences. With one, the tangency portfolio is the same for all investors.

Worked examples

Example 1

Asset A has σ = 20% and Asset B has σ = 30%. Their correlation is 0.25. What weight in A gives the minimum variance portfolio (no short selling restrictions)?

Show the solution

σA² = 0.04, σB² = 0.09.
Cov = 0.25 × 0.20 × 0.30 = 0.015.
Numerator: σB² − Cov = 0.09 − 0.015 = 0.075.
Denominator: 0.04 + 0.09 − 2 × 0.015 = 0.100.
wA = 0.075 ÷ 0.100 = 0.75.

Answer: 75% in A and 25% in B.

Example 2

Using the portfolio above with wA = 0.60 and wB = 0.40, and E(RA) = 8%, E(RB) = 12%, find the portfolio expected return and standard deviation.

Show the solution

Expected return = 0.60 × 8% + 0.40 × 12% = 4.8% + 4.8% = 9.6%.
Variance = 0.6² × 0.04 + 0.4² × 0.09 + 2 × 0.6 × 0.4 × 0.015.
= 0.36 × 0.04 + 0.16 × 0.09 + 0.0072.
= 0.0144 + 0.0144 + 0.0072 = 0.0360.
σp = √0.0360 = 0.1897, about 18.97%.

Answer: Expected return 9.6%, standard deviation about 18.97%.

Exam tips

Questions usually give correlation, not covariance. Convert before using the formulas.
Expect conceptual items: which part of the frontier is efficient, and what happens when correlation falls.
Know the limits ρ = +1 (straight line) and ρ = −1 (zero-risk portfolio possible).
Remember assumptions and weaknesses: sensitivity to input estimates, especially expected returns, and reliance on variance as the only risk measure.
Do the variance calculation on a financial calculator with memory and avoid rounding until the last step.

Practice questions from Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

Markowitz Efficient Frontier and Optimal Portfolios 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 and Optimal Portfolios: frequently asked questions

What is the minimum variance portfolio formula?

For two assets, the weight in asset 1 is (σ2² − Cov12) ÷ (σ1² + σ2² − 2·Cov12). The weight in asset 2 is one minus that. It gives the lowest possible portfolio variance from those two assets.

What are the assumptions of mean-variance optimization?

Investors consider only expected return and variance over a single period. They are rational and risk averse and prefer more return for a given risk. Inputs are assumed known, and in practice are estimated with error.

How do I find the optimal risky portfolio?

With a risk-free asset, choose the portfolio with the highest Sharpe ratio, which is the tangency portfolio. Without one, the best portfolio on the efficient frontier depends on the investor's risk aversion.

Why does lower correlation improve the efficient frontier?

Lower correlation reduces the cross term in portfolio variance. The same expected return then comes with lower risk, so the frontier shifts left.