Account for and compute both the position that minimizes risk and the position that gives the best ratio of expected return to risk.

FRM® Part II · Risk Management and Investment Management · Portfolio Risk: Analytical Methods · FRM2.IM.5.d

The idea

Two special positions are defined by the same marginal VaR machinery. The minimum-risk position is the mix of fully invested weights with the lowest portfolio volatility. At that point every position has the same marginal VaR, so shifting money between positions cannot reduce risk further. For two assets, let s1 and s2 be the volatilities and ρ the correlation. Then the weight in asset 1 is w1 = (s2² − ρ × s1 × s2) ÷ (s1² + s2² − 2 × ρ × s1 × s2). The best ratio of expected excess return to risk, the highest Sharpe ratio, is reached when each position's expected excess return over the risk-free rate divided by its marginal VaR is the same for all positions. Equivalently, the excess return of each position equals its beta times the portfolio's excess return. The common mistakes are mixing the two conditions and using individual VaR where marginal VaR is needed. Exam point: equal marginal VaRs mean minimum risk, and equal return per marginal VaR means the best Sharpe ratio.

Formulas

Minimum-risk weight in asset 1 of a fully invested two-asset portfolio

w₁ = (σ₂² − ρ × σ₁ × σ₂) ÷ (σ₁² + σ₂² − 2 × ρ × σ₁ × σ₂)

σ₁: the volatility of asset 1
σ₂: the volatility of asset 2
ρ: the correlation between the two assets

Ignores expected returns; asset 2 takes 1 − w₁. Adding the correlation term in the numerator instead of subtracting it, or weighting by inverse volatility, gives the wrong answer.

Required expected excess return for a position at the best-Sharpe portfolio

μᵢ − Rf = βᵢ × (μp − Rf)

μᵢ: the expected return of position i
Rf: the risk-free rate
βᵢ: the beta of position i to the portfolio
μp: the expected return of the portfolio

Holds when the weights are optimal, because marginal VaR is proportional to beta. Dividing the portfolio's excess return by beta instead of multiplying is the trap.

A practice question

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

A fully invested portfolio has the lowest possible risk with the weights it currently has. Which condition holds at this minimum-risk position?

All positions have the same component VaRWhy not: Equal component VaRs describe an equal-risk-contribution portfolio, which is a different allocation.
All positions have the same marginal VaR - correct
All positions have the same individual VaRWhy not: Individual VaRs ignore correlations and are not equal at the minimum.
All positions have the same expected return per unit of marginal VaRWhy not: This is the condition for the best Sharpe ratio, not for minimum risk.

If marginal VaRs differ, moving money from the high to the low marginal-VaR position lowers VaR. Risk is minimized when no such move helps, i.e. marginal VaRs are all equal.

A harder one

Asset 1 has volatility 15%, asset 2 has volatility 25% and their correlation is 0.3. For a fully invested two-asset portfolio, the weight in asset 1 that minimizes risk is closest to:

73.5%Why not: Sets the correlation to zero: 25² / (15² + 25²).
62.5%Why not: Weights in inverse proportion to volatility, which ignores correlation and the variances.
68.6%Why not: Adds the correlation term in the numerator instead of subtracting it: (625 + 0.3 x 375) / (225 + 625 + 2 x 0.3 x 375).
82.0% - correct

w1 = (s2² - rho x s1 x s2) / (s1² + s2² - 2 x rho x s1 x s2) = (0.0625 - 0.01125) / (0.0225 + 0.0625 - 0.0225) = 0.05125 / 0.0625 = 82.0%.

The hardest tier

The optimal portfolio has an expected excess return of 4.5% over the risk-free rate. Position Y has a beta of 1.5 to the portfolio. For Y's weight to be optimal, its expected excess return should be closest to:

4.50%Why not: The portfolio's own excess return; this would be right only if Y's beta were 1.
6.75% - correct
4.00%Why not: Y's expected excess return of 6.0% divided by beta; this is Y's return per unit of risk, not its required return.
3.00%Why not: Divides the portfolio excess return by the beta (4.5 / 1.5) instead of multiplying.

Marginal VaR is proportional to beta, so the optimum needs (mu_Y - r) / beta_Y = (mu_P - r). Required excess return = 1.5 x 4.5% = 6.75%.

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