In a note last week, we discussed some key issues about sovereign defaults as we approach another Congressional battle on the debt ceiling for the United States of America. In this note, we compare the data, techniques and accuracy of sovereign and public firm default models. Modeling corporate default is much easier, at least in the short run, than modeling sovereign defaults. This note explains why in a statistical sense. From a practical perspective, Kamakura senior research fellow Prof. Jens Hilscher (Brandeis University) put it simply: "Corporates default because they have to. Sovereigns default because they want to." As the U.S. approaches its all-too-frequent congressional confrontations on the debt ceiling, it is clear that the U.S. will default in 2013 only if a majority of one house of Congress "wants to."

This has major implications for a long list of exchange traded funds:

Long-duration Treasury Exchange-Traded Funds: (NYSEARCA:TLH), (NYSEARCA:TLT), (NYSEARCA:IEF), (NASDAQ:DTYL), (NASDAQ:DLBL), (NYSEARCA:ILTB), (NYSEARCA:TENZ), (NYSEARCA:ITE), (NYSEARCA:TLO), (NYSEARCA:EDV), (NASDAQ:VGIT), (NASDAQ:VGLT), (NYSEARCA:TMF), (NYSEARCA:TYD), (NYSEARCA:LBND), (NYSEARCA:UBT), (NYSEARCA:UST), (NYSEARCA:TMV), (NYSEARCA:TYO), (NYSEARCA:DSTJ), (NYSEARCA:DSXJ), (NYSEARCA:SBND), (NYSEARCA:PST), (NYSEARCA:TBT), (NASDAQ:DTYS), (NASDAQ:DLBS), (NYSEARCA:TBF), (NYSEARCA:TTT), (NYSEARCA:TYNS), (NYSEARCA:TYBS), (NYSEARCA:TBX)

Shorter-duration Treasury Exchange-Traded Funds: (NYSEARCA:SHY), (NYSEARCA:SHV), (NYSEARCA:IEI), (NYSEARCA:BIL), (NYSEARCA:TUZ), (NYSEARCA:FIVZ), (NASDAQ:DTUL), (NASDAQ:VGSH), (NASDAQ:DTUS), (NASDAQ:DFVS), (NASDAQ:DFVL), (NYSEARCA:SST), (NYSEARCA:ISTB), (NYSEARCA:TBZ).

**Differences in Data Quantity and Default Rates**

In this note we compare the public firm and sovereign default data bases maintained by Kamakura Risk Information Services to reach some conclusions about the relative difficulty of modeling sovereign and corporate defaults. The key points are summarized in the following table:

The first observation is an obvious one. The sovereign data set is much smaller than Kamakura's comparable public firm data set. For the version 5.0 public firm model, Kamakura's data set included 1.76 million observations and 2,046 default events. Currently 37,000 public firm default probabilities are updated daily. For sovereigns, however, the data set is much smaller even though the sovereign data set covers a longer time span: January 1980 to December 2007, compared to the KRIS version 5.0 corporate data set, which spans a 19 year period. There are only 28,101 monthly observations in the sovereign data base. The principal reason for the difference in the length of the time series is daily stock price information. The University of Chicago's Center for Research in Security Prices maintains a much longer stock price time series, going back much farther in time than January 1990, but the license agreement for that data prohibits the development of commercial products based on that data. In the case of sovereigns, there is no such data limitation.

The other key fact summarized in the table is that the number of sovereign default events, as defined in previous note, number only 160, and only 123 of these have all of the relevant explanatory variables available. In the case of corporates, there were 2,046 default events in the KRIS data base. The annualized default rate for sovereigns was a shocking 5.128%, compared to an average annualized default rate for public firms of 1.383%. That's right, sovereigns on average are much more risky than public firms, much to the surprise of many.

**Differences in the Difficulty of Modeling Sovereign and Corporate Default Rates**

In many publications, Professor Robert A. Jarrow and I have advocated the use of naïve one-variable credit models as a way of setting an accuracy standard that helps to measure "how good is good enough" in default modeling. In the KRIS version 5.0 public firm data base, the best performing naïve model has an ROC accuracy ratio of 90% over the 1.76 million observations. The ROC accuracy ratio is 100% for a model which rates every defaulting observation as more risky than every non-defaulting observation. The ROC accuracy ratio is 50% for a model with no explanatory power, because one can score a 50% accuracy in comparing one defaulter with one non-defaulter by simply flipping a coin.

On the sovereign data set, the best performing naïve model has only a 73% accuracy, a full 17% lower than one finds on public firm data set. From the outset, we can expect a sovereign default model to be harder to build.

**Is Domestic Currency Default a Possibility? Why Don't Sovereigns Just Print Money?**

Unlike corporate default, where cross default clauses make a default on yen borrowings the same as a default on dollar borrowings, the currency matters with sovereigns. It matters for two reasons. First, cross default provisions are not as all pervasive. Secondly, the sovereign in theory has a right to just print money, as this Zimbabwean bill indicates.

This bill, presented to me by good friend Pieter Strydom at Ernst & Young in Johannesburg in 2008, was about 75% less valuable by the time I showed it to a friend a week later. As of early 2009, 1 U.S. dollar was convertible into 37,456,777 Zimbabwean dollars, according to oanda.com, before the Zimbabwe dollar was effectively abandoned on April 12, 2009. Clearly, there are practical limits as to how much money can be printed before the government effectively ceases to function.

For that reason, in this note and our note of last week, we treat "failure" as if cross default clauses were the norm and ignore differences in the currency of the borrowing for this reason.

**Modeling the Term Structure of Default for Corporates and Sovereigns**

The standard way of modeling with a monthly default data base, whether it's for sovereigns or for corporates, is to fit a logistic regression which models default risk over the coming month, since the "sovereign failure flag" is set to 1 (from 0) if the sovereign fails in the following 30 days.

A different flag can be set as follows. Of all of the sovereigns who survive at least one month, define the "sovereign failure flag" as 0 if they ALSO survive the second month. Set the flag equal to 1 if they failed in the second month after surviving the first month. If we fit a model to this definition of sovereign failure, we are deriving an estimate of the CONDITIONAL probability of default in the second month, given that the sovereign survived the first month.

For both sovereigns and corporates, we do this model fitting for each month out to sixty months. There are a total of sixty logistic regressions that are used to estimate the next sixty months of default probabilities for each sovereign and public company in the KRIS data base. We call the unannualized monthly default probability for the first month P[1], for the second month (conditional on surviving the first month) P[2], and the nth month (conditional on surviving the first n-1 months) P[n].

How are these default probabilities combined to form the term structure of default probabilities? If one has a time horizon of N months, the probability that a sovereign will NOT go bankrupt during those N months is

Q[N]=(1-P[1])(1-P[2])(1-P[3])…(1-P[N-1])(1-P[N])

The probability that the sovereign WILL go bankrupt by the end of the Nth month is

P*[N]=1-Q[N]

This is the cumulative probability of default for an N month time horizon. Since credit spreads and credit default swaps are all annualized, we can annualize the cumulative probability of default for an N month horizon. The annualized default probability for an N month horizon is

A[N]=1-(1-P*[N])12/N

The process is exactly the same for corporates and sovereigns, and we can compare their accuracy for each of these 60 spot and forward default probability estimates. The graph below compares the term structure of the United States of America with that of large regional bank Regions Financial Group (NYSE:RF). It may surprise many that the United States has higher default probabilities than Regions in the short run and about the same default probabilities at 5 years (0.32% for the United States, 0.33% for Regions on an annualized basis). The credit default swap indications for the United States at 5 years are 0.20%. You won't find predictions of bankruptcy for Regions as soon as October 18 if you google the bank, but you will find such predictions for the United States of America.

**What is the Difference in Accuracy Between a Sovereign and Corporate Default Model?**

When one makes an apples to apples comparison of the spot and forward accuracy of the corporate and sovereign default models, the results are very interesting. For the first 21 months, the sovereign model is considerably less accurate than modern public firm "reduced form" default models built using a sophisticated logistic regression. Instead, the sovereign logistic regression model has about the same accuracy in the short run as the Merton model of risky debt does for corporations.

As the time horizon lengthens, however, the sovereign model at about month 21 becomes more accurate than any corporate model for the same forward default point. By month 60, the sovereign model is about 8 percentage points more accurate (in terms of ROC Accuracy Ratio) than the best reduced form corporate model and a full 20 percentage points more accurate than the Merton model for risky debt for public companies.

In short, the process of building a sovereign default model parallels that for public firms quite closely. Data differences create challenges, but at the end of the day, the sovereign modeling effort pays big dividends in terms of accuracy.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article. Kamakura Corporation has business relationships with a number of firms mentioned in the article.