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Set out the ideas of Modern Portfolio Theory and read the efficient frontier of Markowitz.

FRM® Part I · Foundations of Risk Management · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM) · FRM1.FR.5.a

The idea

Modern Portfolio Theory, due to Harry Markowitz, judges an asset by what it adds to a portfolio rather than by its standalone risk. Portfolio expected return is the weighted average of the asset returns, but portfolio variance also depends on covariances. For two assets with weights w1 and w2, volatilities σ1 and σ2 and correlation ρ, the formula is Portfolio variance = w1²σ1² + w2²σ2² + 2 × w1 × w2 × ρ × σ1 × σ2. With both weights positive and correlation below +1, portfolio volatility is below the weighted average of the volatilities, and that gap is the benefit of diversification. Plot expected return against standard deviation for every feasible portfolio of risky assets. The upper-left boundary, starting at the global minimum-variance portfolio and running upward, is the efficient frontier, giving the highest expected return for each level of risk. Any portfolio inside the region or on the lower branch is dominated. A common error is to think diversification removes all risk. In the standard result, a well-diversified portfolio keeps the systematic part, but real portfolios can retain residual risk from concentration, changing correlations and estimation error.

Formulas

Variance of a two-asset portfolio

portfolio variance = w₁² × σ₁² + w₂² × σ₂² + 2 × w₁ × w₂ × ρ × σ₁ × σ₂

w₁, w₂
the weight of each asset
σ₁, σ₂
the standard deviation of each asset's return
ρ
the correlation between the two returns

Only the last term carries correlation. With both weights positive and ρ below 1, portfolio volatility is below the weighted average of the volatilities.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement best describes the efficient frontier in Markowitz portfolio theory?

  1. The set of portfolios of risky assets that offer the highest expected return for each level of risk - correct
  2. The set of all portfolios that can be formed from a given group of risky assetsWhy not: Describes the full feasible set, not its efficient boundary; dominated portfolios lie inside it.
  3. The line that joins the risk-free asset to the market portfolioWhy not: Describes the capital market line, which only arises once a risk-free asset is added.
  4. The portfolios with the lowest possible standard deviation at every level of expected returnWhy not: Includes the lower branch of the minimum-variance set; efficiency needs the highest return for given risk.

The efficient frontier is the upper-left boundary of the feasible set of risky portfolios, starting from the global minimum-variance portfolio: no other portfolio offers more return for the same risk or less risk for the same return.

A harder one

Four portfolios of risky assets have these expected returns and standard deviations: A (5%, 8%), B (7%, 12%), C (6%, 12%), D (9%, 18%). Which portfolio cannot lie on the efficient frontier?

  1. AWhy not: Wrongly rejects the lowest-return portfolio; being low risk does not make it dominated.
  2. BWhy not: Mistakes the middle portfolio for the dominated one; it beats C at the same risk.
  3. DWhy not: Wrongly rejects the highest-risk portfolio; it can still be efficient because its higher risk is paid with more return.
  4. C - correct

C and B have the same risk of 12% but B offers 7% against 6%, so C is dominated. A, B and D each rise in both return and risk and none is beaten by another portfolio, so higher risk alone does not make a portfolio inefficient.

The hardest tier

Asset 1 has volatility 20% and asset 2 has volatility 30%; their correlation is 0.20. The global minimum-variance portfolio of the two holds what weight in asset 1?

  1. 73.6% - correct
  2. 69.2%Why not: Weights by inverse variance, which ignores the correlation and is correct only when it is zero.
  3. 60.0%Why not: Weights by inverse standard deviation rather than using the minimum-variance formula.
  4. 26.4%Why not: Swaps the weights: this is the weight for asset 2, because the numerator should use the other asset's variance.

w1 = (s2² - rho s1 s2) / (s1² + s2² - 2 rho s1 s2) = (0.09 - 0.012) / (0.04 + 0.09 - 0.024) = 0.078 / 0.106 = 0.7358. The less volatile asset gets the larger weight.

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