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Discuss what a breakdown in correlations means for a VaR or ES analysis.

FRM® Part I · Valuation and Risk Models · Calculating and Applying VaR · FRM1.VR.2.g

The idea

Correlation breakdown means that the correlations estimated in normal markets are not stable: in a crisis, correlations between risky assets tend to rise sharply toward one, and some hedging relationships fail altogether. Diversification then disappears just when it is needed. For two assets, with normal returns and zero means, portfolio VaR rises with the correlation. VaR = z × √(a² + b² + 2 × ρ × a × b). Here a and b are the stand-alone dollar standard deviations and ρ is the correlation. A higher ρ raises VaR and ES, with the sum of the stand-alone figures as the upper bound at ρ = 1. A model calibrated to calm periods therefore understates losses in stress. Risk managers respond by stressing correlations, using crisis-period estimates, and keeping the stressed matrix internally consistent. The error to avoid is assuming that correlation stays constant across market regimes.

Formulas

Two-asset portfolio VaR from stand-alone dollar risks

portfolio VaR = z × √(a² + b² + 2 × ρ × a × b)

z
the standard normal quantile
a
the stand-alone dollar standard deviation of the first position
b
the stand-alone dollar standard deviation of the second position
ρ
the correlation between them

Normal returns. VaR rises with ρ up to the sum of the stand-alone figures at ρ = 1, so calm-period correlations understate crisis risk.

A practice question

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

In the context of VaR and ES, what does a breakdown in correlations refer to?

  1. Correlations between assets fall to zero in a crisis so that all assets move independently.Why not: Wrong direction: crises typically push correlations up, not to zero.
  2. The correlation matrix becomes singular and cannot be inverted by the risk software.Why not: A numerical problem; the issue is the instability of estimated relationships over regimes.
  3. Correlations estimated in calm markets prove unstable and typically rise in stress, eroding diversification. - correct
  4. Volatilities rise while correlations remain unchanged, leaving diversification intact.Why not: Describes a different effect; the point of the breakdown is that correlations themselves change.

Estimated correlations describe average past behaviour. In a stress they often move toward one, so the diversification benefit built into the VaR or ES disappears when it is most needed.

A harder one

A portfolio has two equal positions of USD 5 million with daily volatilities of 1.2% and 1.8%. Using the 99% one-day VaR (z = 2.326), by how much does VaR rise if the correlation moves from 0.2 in normal markets to 0.9 in a crisis?

  1. About 24% - correct
  2. About 55%Why not: Compares the variances of the portfolio rather than the standard deviations.
  3. About 70%Why not: Reports the change in the correlation (0.9 - 0.2) as the percentage change in VaR.
  4. About 350%Why not: Uses the ratio of the correlations (0.9/0.2) as the change in VaR.

Dollar volatilities: 60,000 and 90,000. At rho = 0.2: sqrt(3.6e9 + 8.1e9 + 2 x 0.2 x 5.4e9) = 117,700, VaR = 273,900. At rho = 0.9: sqrt(11.7e9 + 9.72e9) = 146,400, VaR = 340,500. The increase is 340,500 / 273,900 - 1 = 24%.

The hardest tier

Two risk measures are stand-alone 99% one-day VaRs of USD 1.2 million (Position A) and USD 0.9 million (Position B). Their correlation in normal markets is 0.25 and is stressed to 0.85. Portfolio VaR = sqrt(a² + b² + 2 x rho x a x b). The stressed portfolio VaR is closest to:

  1. USD 1.67 millionWhy not: Uses the normal-market correlation of 0.25 rather than the stressed 0.85.
  2. USD 2.10 millionWhy not: Adds the stand-alone VaRs, which assumes a correlation of one rather than 0.85.
  3. USD 2.02 million - correct
  4. USD 1.78 millionWhy not: Leaves out the factor of 2 in the covariance term.

Variance = 1.44 + 0.81 + 2 x 0.85 x 1.2 x 0.9 = 2.25 + 1.836 = 4.086; the square root is 2.02. This is up from 1.67 under 0.25, and below the 2.10 upper bound at rho = 1.

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