First, I analyze the case of two institutions who already defaulted.
At the day of Bear Stearns collapse default probability for Bear Stearns reached 26% according to the model, while Lehman kept what is basically a triple-A (very low) default level. On the day Lehman collapsed, LEH default probability reached 84%. The contrast with Lehman's CDS-based probability level around BSC default is striking. In this case, the probability of default for Lehman jumped right after Bear Stearns problems surfaced, it stayed at extremely high level during the crisis and then went back to “normal” up till the final collapse with the exception of substantial liquidity related bump in August 2008. Structural model was not a good predictor because asset/liabilities dynamics is not observed directly. Empirical volatility simply won’t work and we need to model volatility to get reasonable values out of structural model.
Probability of default according to the model on the following days:
03/14/2008, Bear Stearns collapsed
Lehman: 1.49E-28
Bear Stearns: 26%
09/14/2008, Lehman collapsed
Lehman: 84%
Now, I am going to the banks that are still alive at the time of this post.
The probabilities of default range from reasonably small based on the structural model (.05%-.5%) to escalated level (~25-30%) if we are to trust market signals (CDS spreads) more. I do believe that CDS spreads provide a better indicator of default than the ones coming from a model. The biggest problem right now is still the condition of bank balance sheets and that's precisely what any structural model would have to take as given. We simply can't estimate what is the real value of assets since equity layer for banks (in March) is practically zero. The only reason for banks equity to be above zero is government sponsored securities with implicit conversion price and other explicit/implicit guaranties. Not knowing what is the value of assets we really don't know how far the banks are from default point and that's why any structural model is not as good an indicator as the one coming from informed beliefs on the market about the condition of banks.
It is interesting to note, that combining Wells Fargo and Wachovia balance sheets decreased default probability for Wells Fargo. This is reflected in both CDS spreads and in the structural model. At the same time getting Lehman to default immediately shocked all banks into condition close to default. This also might indicate the consolidation of assets&liabilities into single, transparent pool as one of the possible solutions for financial crisis. That would help to reduce uncertainty and information asymmetry that is causing tumult on the markets.
Probability of default according to the model on the following days:
03/14/2008, Bear Stearns collapsed, no Wachovia for Wells Fargo
Citigroup: 1.08E-28
Wells Fargo: 8.74E-08
09/14/2008, Lehman collapsed, Wachovia and Wells Fargo combined
Citigroup: 1.51E-14
Wells Fargo: 5.92E-10
03/11/2009, most recent default state, Wachovia and Wells Fargo combined
Citigroup: 0.5%
Wells Fargo: 5.92E-10
Now, defaults are by nature not an isolated events. Generally, there is some correlation and results would critically depend on the model used to describe time-dependence of correlation matrix.
This is the most controversial part because anyone who is doing stress-testing would need to find a common factor that drives such rare events as defaults for the companies. There are methods using correlation in equity (stock prices) as a proxy for this factor, it is also possibility to build different explicit single-factor models for defaults and try to fit them with data. The structural model with economically sound estimator based on important observable risk factors for the company is a better choice here. It is, however, getting really complicated very quickly. Nor matter what you do, the results are still highly unreliable and in practice stress-testing would have to consider a variety of different companies/economy specific scenarios. Separate question is indeed how you'd communicate these results to public...If you leave really scary scenarios out no one would believe in your tests, if you put them in there could be a wide spread panic.
Anyway, here we do a very rough estimate for our two sample banks – Citigroup and Wells Fargo.
I use standard Gaussian copula with correlation to model correlated defaults. I do loss modeling step by step for each year with losses counted for each year using marginal default probabilities.
The mean joint loss timeline is as follows:
Year1 Year2 Year 3 Year 4 Year 5
4.9% 8.1% 10.5% 13.3% 15.7%
If we get a lower correlation into the model our mean loss doesn’t change much, but our distribution tails get thinner. That is with lower default correlation we have BOTH lower chance of losing a lot AND lower chance of not losing anything.
This is extremely important to keep in mind every time when people are talking about institutions that are "too big to default" institutions and "systemic risk" posed by certain financial intermediaries. Your perspective on that would really be determined by the slice of equity/debt you are holding. People who hold equity slice would indeed prefer that correlation be smaller and risks to be well-diversified since they are first in line for all losses. They also surprisingly are on the same side with senior debt holder here, since those are also less likely to lose the principal. Who does not like such arrangement? Well, it is clear - the people who hold so-called mezzanine layer, most of them are really important and sophisticated investors. Those investors would prefer correlation to be strong, so that all fall (or live) together. Those are the same people who form very tight and impenetrable circle of Wall Street insiders. Those are the people who actually on the same side with the government, against equity holders and senior debt holders. The government likes such arrangement because it is easier to control, the Wall Street likes it because it creates barriers for entry of new players, who can expose their incompetency and take a share of their profit.
More on that later.
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