Case 100Structured finance and securitisationHard

A trust holds five equal loans, each with a 10% default probability and no recovery, and its senior tranche is hit only if three or more default. Compute the senior loss probability with independent and with perfectly correlated defaults, and say which tranche gains from correlation.

The situation

Ashvara Loan Trust holds five loans of Rs 20 crore each, Rs 100 crore in all. Each loan has a 10% chance of defaulting over the life of the deal, and a default loses the whole loan. The trust has issued three tranches: an equity tranche of Rs 20 crore that takes the first default, a mezzanine tranche of Rs 20 crore that takes the second, and a senior tranche of Rs 60 crore that is hit only if three or more loans default.

The structurer's model assumes the five borrowers default independently. A rating analyst points out that all five are in the same industry and may default together.

Your task

What is the chance the senior tranche loses money under independence and under perfect correlation, and which tranche gains from correlation?

Quick check

With independent defaults, what is the chance the senior tranche takes a loss?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The senior tranche's loss probability is about 0.86% with independent defaults and 10% with perfectly correlated defaults, almost twelve times higher. Correlation leaves the pool's expected loss unchanged at Rs 10 crore but moves it up the stack. The equity tranche gains: its chance of any loss falls from 41.0% to 10% and its expected loss from Rs 8.19 crore to Rs 2.00 crore. Seniors lose, and a rating built on independence overstates their safety.

Step 1What is the senior loss probability if defaults are independent?

Count the ways. With independent loans the number of defaults follows a binomial pattern, like the number of heads in five tosses of a coin that lands heads one time in ten. Three defaults happen 10 ways, each with probability 0.1 cubed times 0.9 squared, 0.81% in all; four happen with 0.045% and five with 0.001%; so the senior tranche is hit 0.86% of the time. On those numbers it looks like a very safe note.

The relationship

P(\text{senior hit}) = \sum_{k=3}^{5} \binom{5}{k} (0.1)^k (0.9)^{5-k} = 0.81\% + 0.045\% + 0.001\% = 0.856\%

\binom{5}{k}	number of ways k of the five loans can default
0.1	each loan's default probability
k \geq 3	defaults needed to reach the senior tranche

What it says in wordsWith independent borrowers, three or more of five defaulting needs a run of bad luck, so the senior is hit less than one time in a hundred.

Step 2What changes if defaults are perfectly correlated?

Now the five loans behave as one. Either the industry holds up and none default, 90% of the time, or it fails and all five default, 10% of the time. The senior tranche is hit in every bad state, so its loss probability jumps from 0.86% to 10%, while the equity tranche's chance of any loss falls from 41.0% to 10%. Each loan's own 10% default chance has not changed; only how the defaults cluster has. That clustering is default correlationHow much more likely borrowers are to default together than they would be by chance, usually because they share a sector, region or economic driver..

With independent defaults Ashvara's pool almost never sees three or more defaults, so the senior tranche is hit 0.86% of the time, while with perfect correlation the pool either loses nothing or everything and the senior is hit 10% of the time.

Step 3Which tranche gains from correlation, and why?

Compare expected losses, which must add to the pool's Rs 10 crore either way. Under independence the equity tranche expects to lose Rs 8.19 crore, the mezzanine Rs 1.63 crore and the senior Rs 0.18 crore; under perfect correlation they lose Rs 2 crore, Rs 2 crore and Rs 6 crore. The equity holder gains because most scenarios now have no defaults at all; the senior holder loses because the scenarios that remain are catastrophic. The mezzanine sits in between and changes little.

Correlation leaves Ashvara's pool expected loss at Rs 10 crore but moves it up the stack: equity's expected loss falls from Rs 8.19 crore to Rs 2 crore while the senior's rises from Rs 0.18 crore to Rs 6 crore.

Close with what a rating analyst does with this. Real pools sit between the two extremes, so the senior's risk depends on a correlation that cannot be observed directly and has to be assumed, and five borrowers in one industry argue for a high one. A senior note rated on the independence assumption would be rated as if it failed less than once in a hundred deals, when the sector tells you it could fail one time in ten. Ask for the correlation assumption before accepting the rating.

Where candidates lose it

The usual error is assuming the senior tranche's risk depends only on each loan's default probability. The pool's expected loss is the same under both assumptions; what correlation changes is how that loss is shared, and the tail is where the senior lives.

The second is saying correlation hurts every tranche. The equity tranche benefits, which is why some investors have held equity against short senior positions as a correlation trade.

What the interviewer asks next

Recovery on default is 40% instead of zero. Redo the senior loss probability under independence.
How would you estimate default correlation for five borrowers in the same industry?
The senior attaches after four defaults instead of three. How much safer is it under each assumption?

Company names and figures are illustrative.