Chart of the day: Credit convexity (ultrawonky)

By Felix Salmon
May 6, 2009


This chart comes from a blog entry by Ann Rutledge, which eventually formed some of the basis for a big National Journal cover story by Corine Hegland. It’s not easy to understand, but essentially the action is in the top right hand corner, which I’ve annotated for the sake of comprehension.

The x-axis, along the top, essentially shows the degree of subordination of a tranche of an asset-backed bond. At the far right is 99%, which means that the lowest tranche accounts for just 1% of the total face value; at the far left is 78%, which means that the lowest tranche is much thicker, accounting for 22% of face value. The y-axis, down the left hand side, is the amount of loss that a bond investor experiences, in basis points. And each line represents the proportion of the pool which goes into default.

What we see in the chart is something pretty interesting. Expected default rates on these structures were always pretty low, in single digits, and at those levels no one ever takes any losses; the holders of the junior tranches make the most money, because they’re getting the highest coupons.

Eventually, when default rates rise, losses rise — and generally, the thinner the tranche, the higher the losses. That’s why the lines generally point down and to the right. (Ignore the horizontal lines along the bottom, for these purposes.) Intuitively, most people looking at the securitization market think that when you have a highly subordinated (highly leveraged) tranche, then it can get wiped out quite quickly once default rates start rising.

But in fact it doesn’t always work that way, and that’s where the convexity comes in. Check out the light-brown line corresponding to a 31% default rate: at the far right hand side of the graph, it actually goes up and to the right. If the lowest (equity) tranche had just 2% of the face value, then it would lose nothing, while if it had 7% of face value, it would lose quite a lot. The reason is that the extra leverage gooses the yield on the tranche so much that the extra income more than makes up for the default losses. As Rutledge puts it in a presentation she gave to the Japan Society:


When you have a thin, high-yield tranche, then, you actually benefit from increased leverage at pretty high default rates. Until, suddenly, it falls apart.

Look what happens when the default rate ticks up from 31% to 32% or 33% or 34%: suddenly the advantage of leverage massively backfires, and losses start skyrocketing. Those kind of default rates were never built in to the models being used to rate and value these tranches, though, and ignorance was bliss: the demand for these securities only ever rose, because people ignored the tail risk on the other side of that 31% barrier.

Because the models said the bonds were so safe — look how well they perform even at a 31% default rate! — they themselves became popular instruments to securitize, in the form of CDO-squareds* and the like. Lots of yield, no risk — what’s not to love? Of course, the problem was the tail risk — and because CDO-squareds were made up only of highly-leveraged tranches, even the most senior tranches ended up going to zero when those default rates ticked over the magic line into the murky world of extreme tail risk.

More generally, however, it just takes one glance at this chart to see that all manner of weird stuff is going on there — that there are artifacts of the securitization process which were not at all intended. Writes Rutledge:

The sensitivity of value to default risk and structure, credit convexity, is an intermediate to advanced level problem in fixed income mathematics that, as far as we know, is not taught in any academic finance program other than ours…

Usually when my students (who typically have five to ten years of deal experience) make these diagrams, they gasp in astonishment at the clarity and stark simplicity of what they have never seen before.

What’s quite clear is that the people buying these tranches — often banks playing the regulatory-arbitrage game — generally had no idea what they were letting themselves in for. They knew that they were getting a high yield, and they knew that the Basel rules allowed them to allocate relatively low levels of capital against these securities. Which was fine, because the securities in question (often triple-A tranches of CDO-squareds) had high credit ratings, bestowed by ratings agencies wielding models they didn’t really understand.

And then it all blew up.

*Update: Or even just CDOs. As Corine Hegland emailed to me:

What the heck is the difference between a CDO and a CDO-squared? Does the financial world understand that when it uses pieces of structured securities to build a CDO, instead of using old-fashioned corporate bonds, that it’s basically building a CDO-squared to begin with?

Neither SIFMA, nor other industry materials, nor the rating agencies maintain this distinction, which makes me think that it gets lost, but it’s important. With mortgages, for example, the first securitization technically created residential mortgage-backed securities, or RMBS, which functionally behave like CDOs; the mezz tranches of the RMBS then went into CDOs, which functionally behaved like CDOs-squared. (and the mezz tranches of THOSE CDOs then went into CDOs-squared, creating CDOs-quadrupled? or just tripled?)


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