I came across an excellent working paper thanks to a post on Sebastian Pokutta's blog: "The Economics of Structured Finance" by Joshua Coval, Jakub Jurek and Erik Stafford, available on Harvard Business School's website. In particular, the authors "show how modest imprecision in the parameter estimates can lead to variation in the default risk of the structured finance securities [instruments pooling together bonds, mortgages or other assets, such as collateralised debt obligations (CDO)] which is sufficient, for example, to cause a security rated AAA to default with reasonable likelihood."
The article caught my attention because I am teaching financial optimization again this coming semester (a Master-level course that is required of the students in the Analytical Finance program and is also widely taken by other Master students in the department); last year the structuring of asset-backed securities via dynamic programming was one of the very last topics I had time to cover, using the Optimization Methods in Finance textbook by Gerard Cornuejols and Reha Tutuncu, before the semester ended. This year I am planning to spend more time on dynamic and stochastic programming, and in particular on CDO design.
The HBS working paper contains an excellent introduction to CDOs that should be within grasp of anyone with a basic knowledge of probabilities, as well as numerical simulations to justify their conclusions. The two fundamental steps in CDO design are known as pooling and tranching. First, risky assets such as mortgages are pooled together; later, securities with different cash flow risks are manufactured by assigning them different priorities, which is known as tranching.
The three main types of tranches are junior, mezzanine and senior. Junior tranches default if only a small number of the underlying assets default; in exchange for this higher level of risk, investors receive higher interest rates. Senior tranches are the last ones to default, and carry a much lower interest rate. Mezzanine tranches are in the middle, between junior and senior tranches. (There are some additional subtleties - for instance, the tranches have different maturities, with the senior tranches being paid first and the junior tranches being paid last, which explains why the junior tranches are the riskiest. All tranches receive interest payment at first but only the senior tranches receive payment of principal. Lower-priority tranches start receiving principal payments once the senior tranches have been retired, i.e., paid in full.)
Such prioritization of claims is fundamental in creating CDOs with much lower estimated risk of default than the underlying assets, and in particular, creating AAA-rated tranches out of bonds or mortgages that were not AAA-rated. Coval and his co-authors give a great example (pp.6-7) to explain how this works. Consider two bonds bundled together, each with the same probability of default, paying $0 in case of default and $1 otherwise. The junior tranche bears the first $1 of losses, which means an investor buying it gets nothing if either of the bonds defaults. The senior tranche bears the second $1 of losses, i.e., both bonds need to default in order for the investor owning the senior tranche. Therefore, the risk associated with the senior tranche is determined by the joint probability of the bonds defaulting (that is, the probability that they default at the same time). This notoriously difficult task has been the subject of many articles in the media following the financial crisis, including Wired's "Recipe for Disaster: the Formula that Killed Wall Street" by Felix Salmon, about which I wrote here and here. If the defaults are uncorrelated and the probabilities of each bond defaulting are both equal to 0.1, the senior tranche will have a default probability of 0.01, making it much safer than the underlying assets.
Coval and his co-authors also extend this example to a case with three $1 bonds and three tranches (see p.7). The junior tranche defaults if at least one bond defaults, the mezzanine tranche defaults if at least two bonds default, and the senior tranche defaults if all three bonds default. Using similar assumptions as above, the probabilities of default of the junior, mezzanine and senior tranches can be computed as, respectively: 1-(0.9)^3=0.271 for the junior tranche (probability of at least one bond defaulting is one minus the probability of no bond defaulting, the probability of no default for one bond is 1-0.1=0.9 and the defaults are independent), 1-(0.9)^3-3*(0.1)*(0.9)^2=0.028 for the mezzanine tranche (3*(0.1)*(0.9)^2 comes from picking which bond, out of 3, will not default [this happening with probability 0.1] in the case where exactly two default) and (0.1)^3=0.001 for the senior tranche. The message conveyed by the authors is that "by including a third bond in the pool, two-thirds of the capital - as measured by the tranche notional values - can be repackaged into claims that are less risky than the underlying bonds."
The paper is full of insights on the mechanism of structured finance and makes a convincing case that the ranges of default likelihoods determining investment-grade ratings such as AAA or AA were too narrow; interestingly, the ranges for speculative-grade bonds are wider (see Table 1 p.32). Thus, imprecisions in estimating probabilities will often result in marked changes in a CDO rating. The authors do not offer any guidelines for future improvements on the rating system, but I think it would be reasonable to either offer a range of ratings for structured finance products, to convey the much greater difficulty in analyzing the cash flow risk of CDOs (than of traditional instruments), or a different rating scale, say from 0 to 10, to make sure investors realize a top-rated CDO has a different risk profile than a top-rated company bond.
The authors point out: "[M]ost securities could only have received high credit ratings if the rating agencies were extraordinarily confident about their ability to estimate the underlying securities' default risks, and how likely defaults were to be correlated." They also state: "The structure of collateralized debt obligations magnifies the effect of imprecise estimates of default likelihoods [...] These problems are accentuated [... in the case of] collateralised debt obligations of CDO tranches, commonly known as CDO-squared. (CDO^2)" They argue that "CDOs of mortgage-backed securities are effectively CDO^2's", which might explain the severity of the recent housing crash.
This paper will be a valuable read for anyone interested in structured finance.