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Work out how varying the correlations among trading book assets changes the VaR capital charge.

FRM® Part II · Market Risk Measurement and Management · Correlation Basics: Definitions, Applications, and Terminology · FRM2.MR.8.e

The idea

Portfolio VaR combines the standalone VaRs with the correlation between them. Assume normal or other elliptical returns, and VaRs at the same horizon and confidence level. For two long positions with standalone VaRs VaR1 and VaR2 and correlation ρ, the portfolio VaR is given by VaRp = √(VaR1² + VaR2² + 2 × ρ × VaR1 × VaR2). It rises with correlation, from a diversified level at low or negative correlation up to the simple sum at ρ = 1, so the diversification benefit, the sum of standalone VaRs less portfolio VaR, shrinks. Under the VaR-based internal models approach, the capital charge is the higher of the previous day's VaR and a multiplier of at least 3 times the average VaR of the last 60 business days, using 10-day 99% VaR obtained by the square root of time. A correlation change therefore flows straight into capital. Caution: when one position is short, a higher correlation lowers VaR because the cross term changes sign. Correlations also tend to jump in crises.

Formulas

VaR of two long positions

VaRp = √(VaR1² + VaR2² + 2 × ρ × VaR1 × VaR2)

VaR1
the standalone VaR of the first position
VaR2
the standalone VaR of the second position
ρ
the correlation between the two positions' returns

Needs normal or elliptical returns and VaRs at the same horizon and confidence level. For a long-short pair the cross term is subtracted, so a higher correlation lowers VaR.

VaR-based capital charge for market risk

capital = max(VaR(t−1), mc × VaRavg)

VaR(t−1)
the previous day's 10-day 99% VaR
mc
the multiplier, at least 3
VaRavg
the average 10-day 99% VaR over the last 60 business days

The 10-day VaR comes from the 1-day figure × √10. Forgetting either the multiplier or the √10 scaling understates how much a correlation change moves capital.

A practice question

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

A portfolio's VaR is computed from the standalone VaRs of its long positions and the correlations among them. Holding the positions fixed, how does portfolio VaR vary with correlation?

  1. It rises with correlation, reaching the sum of standalone VaRs at a correlation of 1. - correct
  2. It falls as correlation rises, because co-movement signals stable markets.Why not: Reverses the relationship; the cross term 2 x rho x VaR1 x VaR2 adds to portfolio VaR.
  3. It does not depend on correlation, only on the position sizes.Why not: Correlation determines the diversification benefit, so portfolio VaR depends on it.
  4. It is highest at zero correlation and falls as correlation moves either way.Why not: Confuses correlation with a variance-type quantity; the relationship is monotonic for long positions.

VaR_p = sqrt(VaR1² + VaR2² + 2 x rho x VaR1 x VaR2) rises monotonically in rho for long positions, and equals the simple sum at rho = 1.

A harder one

Two positions have standalone 10-day 99% VaRs of USD 6 million and USD 8 million, and the correlation between their returns is 0.4. Portfolio VaR is closest to:

  1. USD 14.0 million.Why not: Adds the standalone VaRs, which assumes a correlation of one.
  2. USD 10.0 million.Why not: Ignores the correlation term, which assumes zero correlation.
  3. USD 11.8 million. - correct
  4. USD 7.8 million.Why not: Subtracts the correlation term instead of adding it.

VaR_p = sqrt(6² + 8² + 2 x 0.4 x 6 x 8) = sqrt(138.4) = 11.76, closest to USD 11.8 million.

The hardest tier

A desk is long USD 10 million of asset A (daily volatility 2.0%) and short USD 8 million of asset B (daily volatility 1.5%). The risk manager stresses the correlation between A and B from 0.2 to 0.8 and measures 1-day 99% VaR (z = 2.326). What happens to VaR?

  1. It rises from about USD 0.59 million to USD 0.71 million.Why not: Treats both positions as long, so the cross term is added instead of subtracted.
  2. It stays at about USD 0.54 million.Why not: Ignores correlation and uses the zero-correlation square root of the sum of squares.
  3. It stays at about USD 0.74 million.Why not: Adds the standalone VaRs, which would be correct only at perfect correlation for two long positions.
  4. It falls from about USD 0.49 million to USD 0.29 million. - correct

For a long-short pair, variance = (10 x 2%)² + (8 x 1.5%)² - 2 x rho x 0.2 x 0.12. At rho 0.2 VaR = 2.326 x 0.2117 = 0.49; at 0.8 it is 2.326 x 0.1265 = 0.29. Higher correlation lowers VaR for a hedged pair.

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