Show what is meant by implied correlation and the way it can be gauged.
The idea
Implied correlation is the default correlation that, put into a pricing model such as the Gaussian copula, reproduces the observed market price or spread of a tranche, just as implied volatility reproduces an option price. It is found by root-finding: try a correlation, price the tranche, compare with the market price, adjust until they match. Compound correlation does this tranche by tranche, but a mezzanine tranche's price may not be monotonic in correlation, so it can yield two answers or none. Base correlation avoids this by treating each tranche as the difference of two equity tranches from zero, with a separate correlation for each detachment point. Let ELtr be the expected loss of the tranche from A to D, as a percentage of the pool. For a base correlation tranche, ELtr = EL(0 to D) − EL(0 to A). Different implied correlations across tranches (the correlation smile) show the single-correlation model is misspecified.
Formulas
Tranche expected loss under base correlation
tranche EL = [EL(0, D) − EL(0, A)] ÷ (D − A)
- EL(0, K)
- the expected loss of an equity tranche from zero to K, as a share of the pool, at the base correlation for K
- A
- the tranche's attachment point
- D
- the tranche's detachment point
The difference is a share of the pool; divide by the thickness for the loss on the tranche's own notional.
A practice question
The implied correlation of a CDO tranche is:
- the historical correlation of equity returns of the collateral issuers.Why not: That is an observed correlation of equity returns; implied correlation is backed out of tranche prices.
- the correlation between the tranche's coupon and the risk-free rate.Why not: This measures a rate dependency and not a default correlation input.
- the average rating of the tranches in the structure.Why not: Ratings are not a correlation measure.
- the correlation at which the model price equals the tranche's market price. - correct
Just as implied volatility is the volatility input matching an option's market price, implied correlation is the model input matching a tranche's market price.
A harder one
A base correlation approach prices a mezzanine tranche covering 3 to 7 percent of the pool as the difference between two equity tranches. The expected loss of the 0 to 3 percent tranche is 2.1 percent of the pool, and that of the 0 to 7 percent tranche is 3.2 percent of the pool. What is the mezzanine tranche's expected loss as a percentage of its own notional?
- 1.1 percentWhy not: Takes the difference in pool terms (3.2% - 2.1%) and does not divide by the tranche's 4 percent thickness.
- 27.5 percent - correct
- 45.7 percentWhy not: Uses the 0 to 7 percent tranche's loss as a percentage of its own notional (3.2 / 7), ignoring the part absorbed by the lower tranche.
- 5.3 percentWhy not: Adds the two equity tranche losses instead of subtracting them.
EL(3 to 7) = EL(0 to 7) - EL(0 to 3) = 3.2% - 2.1% = 1.1% of the pool. The tranche is 4% of the pool, so relative to its own notional the expected loss is 1.1 / 4 = 27.5 percent.
The hardest tier
For a mezzanine tranche, an analyst finds that the compound implied correlation can have two solutions for some prices and no solution for others. The best explanation is that:
- the market price is always arbitrage-free for mezzanine tranches, so only one root exists.Why not: Arbitrage-free prices do not guarantee a single root; the model price function is the issue.
- the Gaussian copula cannot price any tranche except equity.Why not: The Gaussian copula can price all tranches; the problem is non-monotonic dependence of mezzanine value on correlation.
- implied correlation is always independent of the tranche's attachment points.Why not: Implied correlations differ across attachment points, and that is the origin of the smile.
- mezzanine value is not monotonic in correlation, so there may be several roots or none. - correct
Equity value rises and senior value falls with correlation, so mezzanine can rise then fall. A non-monotonic pricing function makes the root-finding ambiguous, which is why base correlation was introduced.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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