Account for how tranche default sensitivities are gauged.
The idea
Tranche default sensitivities describe how a tranche's value or expected loss reacts to changes in the collateral's credit risk. They are measured by bump and revalue. First change one input, such as the default probability, the credit spread or correlation, by a small amount. Then re-run the pricing or simulation using the same random numbers. Finally take the change in tranche value or expected loss divided by the change in the input. Another form is the loss caused by one more default, which depends on whether the pool is already close to the tranche's attachment point. Per unit of notional, equity is the most sensitive and senior the least, so the sensitivities also give hedge ratios against index positions. The usual mistakes are assuming sensitivity is linear in the number of defaults and expressing it per pool rather than per tranche notional.
Formulas
Hedge ratio from tranche and index sensitivities
hedge notional = (tranche sensitivity ÷ index sensitivity) × index unit
- tranche sensitivity
- the tranche's value change per basis point of spread
- index sensitivity
- the index position's value change per basis point, for one unit of notional
- index unit
- the index notional to which the index sensitivity refers
Match sensitivities, not notionals. A tranche that loses as spreads widen is hedged by buying protection.
A practice question
The default sensitivity of a tranche measures:
- the proportion of the pool that must default before the tranche is wiped out.Why not: That is the tranche's detachment point, not a sensitivity.
- how the tranche's value or expected loss responds to a change in collateral default probability or spread. - correct
- the tranche's coupon divided by its rating-implied loss.Why not: This is a yield-to-risk ratio, not a measure of the response to credit changes.
- the probability that the servicer fails to forward collections.Why not: This is operational risk, not default sensitivity of the tranche.
Default sensitivities are partial derivatives of the tranche's value with respect to credit inputs, and are estimated by bumping the inputs and revaluing.
A harder one
A tranche of notional USD 50 million sits in a pool of notional USD 100 million. Revaluing after raising the pool's default probability by 0.5 percentage points raises the tranche's expected loss from 4.0 percent to 4.6 percent of tranche notional. Assuming a linear response, what is the increase in tranche expected loss, in USD million, if the pool's default probability rises by 1.5 percentage points?
- 0.3Why not: Uses a single 0.5 percentage point step instead of the three steps in 1.5 points.
- 1.8Why not: Applies the 1.8 percentage point increase in expected loss to the USD 100 million pool instead of the USD 50 million tranche.
- 2.3Why not: Multiplies the 4.6 percent level of expected loss by the tranche notional instead of the change.
- 0.9 - correct
Sensitivity = 0.6 percentage points per 0.5 percentage points, so 1.5 points gives 1.8 percentage points of tranche notional. 1.8% x USD 50 million = USD 0.9 million.
The hardest tier
A mezzanine tranche of notional USD 20 million loses USD 1,200 in value per 1 basis point widening of collateral spreads. A credit default swap index position of USD 10 million notional gains USD 400 in value per 1 basis point widening if the investor buys protection. Which hedge matches the tranche's spread sensitivity?
- Sell protection on USD 30 million notional of the index.Why not: Gets the direction wrong: the tranche loses when spreads widen, so the hedge must gain, which requires buying protection.
- Buy protection on USD 30 million notional of the index. - correct
- Buy protection on USD 3.3 million notional of the index.Why not: Inverts the ratio of sensitivities (400/1,200) when sizing the hedge.
- Buy protection on USD 20 million notional of the index.Why not: Hedges notional for notional, ignoring the tranche's different spread sensitivity.
Hedge ratio = tranche sensitivity / index sensitivity per unit = 1,200 / 400 = 3, so 3 x USD 10 million = USD 30 million of protection. The tranche's sensitivity is not proportional to its notional, so notional-matching does not work.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Structured Credit Risk
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- Show what is meant by implied correlation and the way it can be gauged.
- Name the motives behind the use of structured credit products.