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Define implied correlation in its two forms, the compound (tranche) version and the base version.

FRM® Part II · Credit Risk Measurement and Management · Credit Derivatives · FRM2.CR.13.g

The idea

Implied correlation is the Gaussian copula correlation that reproduces a quoted tranche price. Compound (tranche) correlation applies one flat correlation to a single tranche, such as 3 to 7 percent, and solves for the value that matches that tranche's market spread. Base correlation instead prices equity tranches from zero to each detachment point K. A mezzanine tranche from K1 to K2 is valued as the equity tranche 0 to K2 at its base correlation, minus the equity tranche 0 to K1 at its own base correlation. Compound correlation can be non-unique or absent for mezzanine tranches, because their value is not monotonic in correlation. Base correlation is well behaved and typically rises with detachment point, a pattern called a skew. For the equity tranche the two measures coincide.

Formulas

Expected loss on a mezzanine tranche from base correlation

L(K1 to K2) = (L2 × K2 − L1 × K1) ÷ (K2 − K1)

L1
the expected loss on the 0 to K1 equity tranche, as a share of its size, at its own base correlation
K1
the attachment point
L2
the expected loss on the 0 to K2 equity tranche, as a share of its size, at its own base correlation
K2
the detachment point

Convert to portfolio terms before differencing, then divide by the tranche width. Subtracting the percentages directly is wrong.

A practice question

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

Which description defines compound (tranche) correlation?

  1. The correlation that reproduces the price of the equity tranche from 0 to the detachment point, built up tranche by tranche.Why not: Describes base correlation, which is defined on equity tranches 0 to K.
  2. The historical correlation between the equity returns of the names in the reference portfolio.Why not: Compound correlation is implied from market tranche quotes, not estimated from historical equity returns.
  3. The average pairwise correlation implied by the spreads of all index tranches taken together.Why not: Compound correlation is solved tranche by tranche, producing a different value for each tranche.
  4. The single copula correlation that, applied to one tranche, reproduces that tranche's quoted market spread. - correct

Compound correlation is implied tranche by tranche: find the flat Gaussian copula correlation that makes the model spread equal the quoted spread for that tranche.

A harder one

The base-correlation equity tranche 0 to 3 percent has an expected loss of 80 percent of its size, and the equity tranche 0 to 7 percent has an expected loss of 55 percent of its size (each at its own base correlation). The expected loss of the 3 to 7 percent tranche, as a percentage of that tranche's size, is closest to:

  1. 1.45%Why not: Gives the difference in portfolio terms (1.45 percent of the portfolio) without rescaling by the tranche width of 4 percent.
  2. 67.5%Why not: Averages the two equity tranche loss percentages instead of differencing the expected losses in portfolio terms.
  3. 36.3% - correct
  4. 25.0%Why not: Subtracts the percentages directly without weighting by the sizes of 3 and 7 percent.

Equity 0-7: 0.55 x 7 = 3.85% of portfolio. Equity 0-3: 0.80 x 3 = 2.40%. Mezzanine loss = 1.45% of portfolio, divided by the 4% width = 36.3% of the tranche.

The hardest tier

Which statement about compound and base correlation is correct?

  1. For the 3 to 7 percent tranche the compound correlation equals the base correlation at the 7 percent detachment point.Why not: The base correlation at 7 percent prices the equity tranche 0 to 7 percent, a different tranche, so it differs from the 3 to 7 percent compound correlation.
  2. For the equity tranche (0 to 3 percent) the compound correlation and the base correlation are the same number. - correct
  3. Base correlations generally fall as the detachment point increases, while compound correlations rise.Why not: Base correlations typically rise with detachment point (a skew); the pattern attributed to them here is reversed.
  4. Both measures are derived from historical default data, so each is a physical rather than a market-implied correlation.Why not: Both are market-implied copula parameters calibrated to tranche quotes, not historical estimates.

The first base tranche is the equity tranche itself, so solving for its compound or base correlation is the same problem. For higher tranches the base measure uses a difference of two equity tranches and the two diverge.

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