Mean-Variance Portfolio Theory for CM2 Economic Modelling

What this chapter covers

Mean-variance portfolio theory: topics in the order to study them

1. 1Expected Return, Variance and Covariance of PortfoliosEvery later result uses portfolio mean and variance, so the formulas must be automatic first.
2. 2Risk Aversion and Utility AssumptionsThis explains why investors care only about mean and variance and what preferences are assumed.
3. 3Efficient Frontier and DiversificationWith the formulas and preferences in place, you can see which portfolios are efficient and why correlation matters.
4. 4Risk-Free Asset and Capital Market LineAdding a risk-free asset simplifies the frontier into a straight line and leads to the market portfolio.
5. 5CAPM, Beta and Security Market LineCAPM builds on the market portfolio and the CML, and splits risk into systematic and specific parts.
6. 6Limitations of Mean-Variance Theory and Alternative Risk MeasuresCriticism makes sense only after you know the model and its assumptions in full.

How to prepare Mean-variance portfolio theory

1. Write out the portfolio mean and variance formulas for two assets and for many assets, using weights, variances and covariances. Check the notation against your study material.
2. Practise computing portfolio variance with different correlations, including +1, 0 and −1, and note how the result changes.
3. Learn the assumptions behind mean-variance theory and the preference link to utility. Be ready to state each assumption in one clear sentence.
4. Sketch the efficient frontier, the CML and the SML by hand until you can label axes, intercepts and slopes without help.
5. Work through CAPM questions: find beta from covariance and market variance, then required return from the risk-free rate and market risk premium.
6. For the limitations topic, prepare a short list of weaknesses and match each with an alternative risk measure and what it fixes.
7. Finish with timed past-style questions, then repeat the calculations in R or Excel so you are ready for Paper B.

Common mistakes in Mean-variance portfolio theory

• Forgetting the covariance term or using weights that do not sum to 1 in the variance formula.

  Fix: Remember variance is not linear in weights. Write all squared and cross terms, and check the weights before you start.

• Mixing up variance and standard deviation, especially when drawing graphs or finding the CML slope.

  Fix: Label every quantity. The CML and frontier graphs use standard deviation on the horizontal axis, so take the square root of variance first.

• Confusing the CML with the SML.

  Fix: The CML has σ on the axis and applies to efficient portfolios only. The SML has β on the axis and applies to any asset or portfolio in CAPM.

• Stating CAPM results without the assumptions.

  Fix: Always add the key assumptions, such as homogeneous expectations and mean-variance investors, when asked to explain or comment.

• Saying diversification removes all risk.

  Fix: Say diversification reduces specific risk. Systematic risk remains, and risk only falls if correlation is below +1.

• Giving a list of limitations with no explanation or alternative measure.

  Fix: For each limitation, say what the model assumes, why that fails in practice, and which alternative measure addresses it.

Last-day revision: Mean-variance portfolio theory

• ✓Portfolio expected return = Σ wᵢ E(Rᵢ), where the weights sum to 1.
• ✓Two-asset variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁₂.
• ✓Covariance σ₁₂ = ρ₁₂ σ₁ σ₂, and correlation lies between −1 and +1.
• ✓Diversification lowers risk when correlation is below +1.
• ✓A risk-averse investor prefers higher return for the same risk and lower risk for the same return.
• ✓The efficient frontier holds portfolios with the highest return for each level of risk.
• ✓With a risk-free asset, the CML runs from the risk-free rate through the market portfolio.
• ✓CML slope = (E(R_M) − R_f) ÷ σ_M, which is the market price of risk.
• ✓CAPM: E(Rᵢ) = R_f + βᵢ (E(R_M) − R_f).
• ✓Beta = Cov(Rᵢ, R_M) ÷ Var(R_M).
• ✓The CML uses total risk; the SML uses beta.
• ✓Only systematic risk is rewarded in CAPM; specific risk can be diversified away.

Mean-variance portfolio theory practice questions

• An investor has quadratic utility U(w) = w - b w^2 with b > 0, defined where w < 1/(2b). Which statement about this utility function is corr…
• Two risky assets have a correlation coefficient of +1 between their returns. An investor forms portfolios by varying the proportions held in…
• Two assets have expected returns 8% and 14%, standard deviations 10% and 20%, and correlation 0.5. A portfolio holds 60% in the first and 40…
• An investor's one-year portfolio return is normally distributed with mean 12% and standard deviation 20%. The investor uses a 5% Value at Ri…
• In mean-variance portfolio theory, which assumption about investor preferences is sufficient to justify ranking portfolios using only expect…
• An investor has a utility function U(w) with U'(w) > 0 and U''(w) < 0 for all wealth levels w. Which statement about this investor is correc…
• Two securities A and B have expected returns of 8% and 14% respectively. An investor puts 60% of her wealth in A and 40% in B. What is the e…
• The risk-free rate is 6% p.a. The market portfolio has expected return 14% and standard deviation 16%. An investor wants a portfolio on the …

Mean-variance portfolio theory in other exams

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

• FRM Part IModern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)
• NISM CertificationsIntroduction to Modern Portfolio Theory
• CA FinalPortfolio Management
• CFA Level IPortfolio Risk and Return: Part I