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The probability distribution used for mortgage backed securities divided them into tranches. If tranch#1 was worth 10 million USD, they got paid in full while the rest got paid partially. Most of them defaulted.

What was the probability distribution used? And what is the statistics behind what they did?

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    $\begingroup$ As this was way before my time, I kick it off with a comment, only. I think you are looking for David Li's paper "On Default Correlation" in the JoFI (working paper version: maths.lth.se/matstat/kurser/fmsn15masm23/default.pdf) There, he introduced copulas to the credit risk modelling, e.g. the Gauss copula. I do not know which copulas have been used effectively, back then though. $\endgroup$ Jan 13 at 10:32
  • $\begingroup$ Was the Gauss copula ever adopted on a wide scale for modeling stocks. If so, which paper justified it the way Li did for credit? $\endgroup$
    – develarist
    Jan 13 at 20:11
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    $\begingroup$ You might find the book Youssef Elouerkhaoui. Credit Correlation: Theory and Practice (2017) helpful. $\endgroup$ Feb 12 at 21:21
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I think you're asking about the different tranches in a multi-tranch mortgage securitization such as a Collateralized Mortgage Obligation (CMO) was sized, and what the math behind it was. This was how they did it:

  1. They used historical default statistics (probability of default and loss in the event of default) from prior episodes of high mortgage defaults, such as Texas in the early 1980's and New England in the early 1990's.
  2. Then they calculated a "stress scenario" that was 2x or 3x that level.
  3. Based on that, they calculated the cash flows of the pool, including the excess interest (mortgage note rate minus the coupon rates of the bonds issued) and subordination.
  4. From the excess cash flow, they sized the tranches so that a single A-rated tranche would get paid back in the event of the stress scenario. (Note: Not sure if that's exactly where the single A subordination was. Please check the Mortgage Rating Criteria from the rating agencies to verify.)
  5. The expected losses of each of the rated classes was supposed to be similar to the expected loss of a similarly rated corporate bond, based on their long-term default experiences.

And this is where they went wrong:

  1. In their models, they always assumed that diversification of collateral pool would reduce overall losses, because up until 2008, there was never a national (or global) real estate recession.
  2. The loss experiences in 2008 were much worse than they ever expected. As a point of reference, look at Fannie Mae's historical loss experiences, and Fannie Mae collateral was (relatively) better quality.
  3. With re-securitizations, their model was calculating the expected loss. When you re-securitize a subordinated bond (BBB) into tranches of AAA and lower-rated bonds, you assumed that there would be more defaults but the lower-rated bonds would protect the higher rated ones. But you were also creating a leveraged bet on defaults and losses in the event of default. Once the underlying mortgages defaulted at far higher rates, the subordination of the BBB bonds were wiped out much more quickly, leading to losses on the AAA bonds.
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  • $\begingroup$ Great response! Are there any references for the points that you mentioned above? $\endgroup$
    – AK88
    Oct 12 at 16:01
  • $\begingroup$ @AK88 Thanks. Try looking for the Mortgage Rating Criteria or mortgage default studies from S&P, Moody's, and Fitch. See if you could find some from before 2008 to compare to the current ones. After 2008 there have been studies by a lot of academics and the Federal Reserve that might be helpful. $\endgroup$
    – Si Chen
    Oct 13 at 15:51
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There was no one single method for RMBSs, but S&P's LEVELS model or equivalent was often used with structuring. A common, vanilla structuring was into a "six pack" plus I/O strip, the six pack being AAA, AA, and so forth. In other words, seven tranches. There were many structures, however.

Moody's Mortgage Metrics (M3) was another common model used. Fitch ResiLogic was another.

By way of example, here's an archived prospectus from 2006:

https://www.sec.gov/Archives/edgar/data/1366182/000112528206003776/b413822_424b.txt

In this case we have five senior tranches (each AAA) and several mezzanine tranches (AA+ down to BBB-).

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