Lay out the efficient frontier and the mean-variance framework behind it.

FRM® Part I · Valuation and Risk Models · Measures of Financial Risk · FRM1.VR.1.a

The idea

The mean-variance framework describes each portfolio by two numbers, expected return and standard deviation, and assumes investors prefer more return and less risk. For two assets, the portfolio variance is given by Variance = w1² × σ1² + w2² × σ2² + 2 × w1 × w2 × ρ × σ1 × σ2. Portfolio volatility is below the weighted average of the volatilities whenever the correlation is below 1, for a fully invested portfolio with positive weights summing to 1 and non-zero volatilities. That gap is the diversification benefit. Plotting every feasible portfolio, the left edge is the minimum-variance frontier. Its upper part, from the global minimum-variance portfolio upward, is the efficient frontier: the highest expected return for each level of risk. Portfolios inside or below it are dominated. With a risk-free asset, the best risky mix is the tangency portfolio, which has the highest Sharpe ratio. Sharpe ratio = (E(R) − Rf) ÷ σ. Common mistakes are averaging volatilities, and treating the whole minimum-variance curve as efficient.

Formulas

Variance of a two-asset portfolio

σp² = 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

With a correlation below 1 and positive weights, portfolio volatility is below the weighted average of the volatilities. The trap is averaging the volatilities or dropping the 2 on the covariance term.

Sharpe ratio of a risky portfolio

Sharpe ratio = (E(R) − Rf) ÷ σ

E(R): the portfolio's expected return
Rf: the risk-free rate
σ: the standard deviation of the portfolio's return

The tangency portfolio is the risky mix with the highest Sharpe ratio. Ranking portfolios by return per unit of risk alone does not tell you which are efficient.

A practice question

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

In the mean-variance framework, which description of the efficient frontier is correct?

The portfolios offering the highest expected return at each level of risk, which form the upper part of the minimum-variance frontier above the global minimum-variance portfolio. - correct
The entire minimum-variance frontier, including the lower branch that lies below the global minimum-variance portfolio.Why not: Treats the whole minimum-variance curve as efficient; lower-branch portfolios are dominated by ones with the same risk and a higher return.
The straight line from the risk-free asset through the tangency portfolio, which is the best trade-off available to every investor.Why not: Confuses the efficient frontier of risky assets with the capital allocation line that only appears once a risk-free asset is added.
The single portfolio with the lowest variance, because mean-variance investors want to minimise risk above all.Why not: Equates the frontier with the global minimum-variance portfolio and ignores that investors also want higher expected return.

The minimum-variance frontier gives the lowest risk for each target return; its upper branch, starting at the global minimum-variance portfolio, gives the highest return for each risk and is the efficient frontier. Everything on the lower branch is dominated.

A harder one

A portfolio holds 60% in asset 1 (volatility 20%) and 40% in asset 2 (volatility 10%). The correlation between the two assets is 0.3. Portfolio volatility is closest to:

16.0%Why not: Weighted average of the two volatilities (0.6 x 20 + 0.4 x 10); this is the answer only if correlation is +1.
13.7% - correct
12.6%Why not: Sets the correlation to zero and drops the covariance term, which understates variance when correlation is positive.
13.2%Why not: Includes the covariance term but omits the factor of 2 on 2 x w1 x w2 x rho x s1 x s2.

Variance = (0.6 x 0.20)² + (0.4 x 0.10)² + 2 x 0.6 x 0.4 x 0.3 x 0.20 x 0.10 = 0.0144 + 0.0016 + 0.00288 = 0.01888, so volatility = sqrt(0.01888) = 13.74%. This is below the 16.0% weighted average because correlation is below 1.

The hardest tier

The risk-free rate is 3%. The tangency portfolio of risky assets has expected return 10% and volatility 15%. An investor holds only the risk-free asset and the tangency portfolio and wants a total portfolio volatility of 9%. The expected return of the investor's portfolio is closest to:

5.8%Why not: Puts 60% in the risk-free asset and 40% in the tangency portfolio; the volatility target fixes the weight in the risky portfolio at 60%, not 40%.
6.0%Why not: Multiplies the 60% weight by the tangency return of 10% and forgets the 40% held in the risk-free asset earning 3%.
7.2% - correct
9.0%Why not: Adds the full 10% tangency return times the weight on top of the risk-free rate, double counting the risk-free component of the risky holding.

The risk-free asset has zero volatility, so portfolio volatility = w x 15%. Setting this to 9% gives w = 0.6. Expected return = 3% + 0.6 x (10% - 3%) = 7.2%, which equals 0.4 x 3% + 0.6 x 10%.

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