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Show how correlation affects the risk of a portfolio.

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

The idea

Return correlation controls how much two positions offset each other, assuming normal or elliptical returns and VaRs with the same horizon and confidence level. For two positions with individual VaRs V1 and V2 and correlation ρ, the diversified VaR, written VaRd, is: VaRd = √(V1² + V2² + 2 × ρ × V1 × V2). At ρ = +1 this is V1 + V2, the undiversified VaR, with no benefit. At ρ = 0 it is the square root of V1² + V2². At ρ = −1 it is the absolute difference |V1 − V2|, so equal opposite positions cancel completely. Holding positions fixed, a higher correlation always raises portfolio VaR, and the diversification benefit (undiversified minus diversified VaR) shrinks. For N equal weights with common volatility σ and common correlation ρ, the portfolio volatility is: Vol = σ × √(1 ÷ N + (1 − 1 ÷ N) × ρ). For positive ρ it cannot fall below σ times the square root of ρ, which is the limit as N grows. Correlations that jump toward 1 in a crisis are why VaR can understate losses exactly when it matters.

Formulas

Diversified VaR of two positions

VaRd = √(V1² + V2² + 2 × ρ × V1 × V2)

V1, V2
the individual VaRs of the two positions
ρ
the correlation between their returns

Needs normal or elliptical returns and VaRs at the same horizon and confidence. At ρ = −1 it is |V1 − V2|.

Volatility of an equally weighted portfolio with a common correlation

Vol = σ × √(1 ÷ N + (1 − 1 ÷ N) × ρ)

σ
each asset's volatility
N
the number of assets
ρ
the common pairwise correlation

For positive ρ it cannot fall below σ × √ρ however large N becomes.

A practice question

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

Holding position sizes and volatilities fixed, what happens to the diversified VaR of a two-position portfolio as the correlation between them rises from 0 toward +1?

  1. It falls, because more correlated assets offset each otherWhy not: Reverses the effect: positive correlation means losses coincide, which raises VaR.
  2. It is unchanged, because the individual VaRs are unchangedWhy not: Individual VaRs do not depend on correlation, but diversified VaR contains the correlation term 2 x rho x V1 x V2.
  3. It rises and can exceed the sum of the individual VaRsWhy not: The sum of the individual VaRs is the ceiling, reached at rho = +1; diversified VaR cannot go above it.
  4. It rises and approaches the sum of the two individual VaRs - correct

Diversified VaR = sqrt(V1² + V2² + 2 x rho x V1 x V2) increases in rho and equals V1 + V2 at rho = +1.

A harder one

A portfolio has two positions with individual VaRs of USD 6 million and USD 8 million. The correlation falls from 0.6 to 0.2. By how much does the diversification benefit (undiversified VaR minus diversified VaR) increase, in USD million?

  1. About 3.08Why not: The benefit at the new correlation of 0.2 (14 - 10.92), not the increase over the benefit at 0.6.
  2. About 1.64 - correct
  3. About 1.45Why not: The benefit at the old correlation of 0.6 (14 - 12.55) rather than the change.
  4. About 5.60Why not: Multiplies the 0.4 drop in correlation by the undiversified VaR of 14, treating the benefit as linear in correlation.

At rho = 0.6: sqrt(36 + 64 + 2 x 0.6 x 48) = 12.55, benefit 1.45. At rho = 0.2: sqrt(100 + 19.2) = 10.92, benefit 3.08. The increase is 3.08 - 1.45 = 1.64.

The hardest tier

A desk holds USD 10 million in each of two assets with 2% daily volatility. Using a 99% one-day z of 2.326, the risk manager's model assumes a correlation of 0.2. In a stress, correlation rises to 0.8. The percentage increase in portfolio VaR is closest to:

  1. 22.5% - correct
  2. 50.0%Why not: Takes the ratio of (1 + 0.8) / (1 + 0.2) without the square root; that is the increase in variance, not in VaR.
  3. 300%Why not: Compares the correlations themselves (0.8 / 0.2 - 1), although VaR is not proportional to correlation.
  4. 60.0%Why not: Treats the 0.6 rise in correlation as a 60% rise in VaR.

Each individual VaR = 2.326 x 0.02 x 10 = 0.465. VaR = 0.465 x sqrt(2 + 2 x rho): 0.721 at rho = 0.2 and 0.883 at 0.8. Increase = sqrt(3.6 / 2.4) - 1 = 22.5%.

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