The derivatives industry has gone through some dramatic changes in its views of the links between credit default swap spreads and default probabilities. One of the difficulties in understanding these links is the power of “conventional wisdom,” i.e. the perception of what others think, to obscure important strengths and weaknesses of the conventional approach. We saw this most clearly in the 2007-2010 credit crisis, when the widely used copula method resulted in massive losses because it ignored the impact of the movement of home prices on CDO valuation.

In this blog post, we point out the many factors proven to drive the linkages between CDS spreads and default probabilities and how to use them to more accurately estimate default probabilities from observable CDS spreads.

This blog is a summary of remarks made to the Global Market Solutions conference in Tokyo, July 14, 2010. We are grateful to the participants in that session for many helpful comments. We are also grateful to our colleague Professor Robert Jarrow for 15 years of helpful research and conversations on this topic. We start with one example of a financial journalist’s use of conventional Wall Street CDS calculations. The article below, written on June 16, 2010, claims that BP plc had a “39 percent chance of defaulting” over the next five years:

On the same day, Kamakura Risk Information Services estimated that the 5 year cumulative default probability for BP plc was only 0.32%. This estimate, using version 5 of the KRIS reduced form or “Jarrow-Chava” model, was based on 1.76 million observations of public firms and 2,046 defaults. On what was the 39% estimate of the cumulative probability of default for BP based? On indicated BP CDS pricing and a standard formula developed by ISDA, the International Swaps and Derivatives Association. This formula is partially documented on the website www.cdsmodel.com.

This model documents another step in the change in the conventional wisdom about the links between CDS spreads and default risk. We can summarize those developments in three specific eras:

2003 “Why do I need a default probability model?

I know CDS=(1 – recovery rate)(probability of default)”Early in the development of the CDS market, the conventional wisdom parodied in this quote believed that the credit default swap spread was a simple fraction of the true default probability (more precisely, the annualized matched maturity default probability) and that this fraction was determined solely by the forecasted recovery rate on the CDS contract.

2005 “Why would I provide credit protection if all I got paid was the expected loss?”Later, this quote from a fixed income trader at one of the world’s most sophisticated pension funds makes it obvious that the 2003 quotation was too simple. Indeed, as we explained in this blog entry,

Robert A. Jarrow, Li Li, Mark Mesler, and Donald R. van Deventer, “The Determinants of Corporate Credit Spreads: An Update,” Kamakura blog, www.kamakuraco.com, September 23, 2009. Redistributed on www.riskcenter.com on September 24, 2009.

The standard Economics 1 assumption that a provider of credit gets a lower and lower spread as he tries to increase his market share with Company ABC, all other things (like the default probability) being equal, means that marginal revenue should equal marginal cost and the equilibrium credit spread should be well above the product of (1 – recovery rate)*(default probability).

In 2007, we showed that a wide variety of factors drove this equilibrium of supply and demand in this companion paper to the blog entry above:

Jarrow, Robert A., Li Li, Mark Mesler, and Donald R. van Deventer, “The Determination of Corporate Credit Spreads,” RISK Magazine, September, 2007.

We come back to that paper below.

2009 ISDA adopts standard converter from CDS spread basis to upfront basis, but this says nothing about the empirical probability of defaultBy 2009, the move by ISDA to emphasize more upfront payments in the credit default swap market created the need for a standard tool to calculate settlement terms for contracts traded either on an upfront basis or a spread basis. This formula has become the new “conventional wisdom” on the links between swap spreads and the underlying default probabilities. It was undoubtedly the source of the BP plc 39% default risk quotation above.

In this blog, it’s important to point out the similarities in purpose between the standard “yield to maturity” calculation that has been common in the bond markets for 80 years and the ISDA Standard CDS Converter partially documented on www.cdsmodel.com. Both formulas are highly simplified calculations designed as a slang or short-hand for discussing price. In a similar way, even though the weaknesses of the Black-Scholes options model are well known, the “implied volatility” is a short cut for talking about the price of an option.

For that usage, the standard ISDA Standard CDS Converter serves its purpose well. Its assumptions are simple and clear, and the ISDA Standard CDS Converter is distributed free. The following documentation is available on www.cdsmodel.com:

There are two purposes listed for the ISDA Standard CDS Converter:

- To convert back and forth between upfront fee-based pricing to spread-based pricing and vice versa
- To calculate settlement amounts

To put the ISDA CDS converter in perspective, we paraphrase Kamakura Managing Director for Research Professor Robert Jarrow. As Prof. Jarrow describes it, there are two ways to understand the linkages between CDS spreads and default probabilities:

**Bottoms up approach:** Build a model, assume it’s true, and solve for continuous default probability (“risk neutral”) that matches observable pricing

**Top down approach:** Determine the factors driving the intersection of supply and demand and use an econometric approach to derive (empirical) default probabilities

The ISDA Standard CDS Converter is a “bottoms up” approach that makes a number of simplifying assumptions, something very much in keeping with the “yield to maturity” analogy:

- Forward interest rates are step-wise constant, not a smooth continuous curve such as those we have discussed in our many blogs on yield curve smoothing
- Default intensity, the continuous time probability of default, is constant over the life of the credit default swap contract. Similarly, the yield to maturity formula assumes that interest rates are constant over the life of the bond in question.
- The only relevant factors to consider are the instantaneous probability of default, the recovery rate, and interest rate levels
- The counterparty on the CDS contract will not default

There is nothing wrong with this approach–as long as one is aware of its limitations. The yield to maturity formula, for example, is deeply embedded in bond markets but it is well known that it is not best practice for valuation or risk management. In the same way, the ISDA Standard CDS Converter is embedded in the mechanics of settlement but it is too simple for accurate valuation and risk management.

The reason the ISDA Standard CDS Converter is too simple is that it assumes only three factors affect the link between CDS spreads and default probabilities:

- Interest rates
- Recovery rates
- The constant continuous time default intensity

We now turn to the “top down” approach to show why the ISDA Standard CDS Converter is too simple for accurately describing the links between spreads and default probabilities. Instead of assuming the (ISDA) theory is true, we assume the market data is true and derive insights from it. In this section, we summarize our findings in the RISK 2007 publication and companion blog entry listed above. For data on default probabilities, we used the then-current reduced form or “Jarrow-Chava” default probabilities from Kamakura Risk Information Services. The current version of the model, version 5.0, is based on 1.76 million monthly observations and data on 2,046 public firm defaults.

For credit default swap information, we used time-stamped data from broker GFI:

- Daily data from January 2, 2004 to November 3, 2005
- More than 500,000 credit default swap bid, offered, and traded price observations.
- 223,006 observations of bid prices
- 203,695 observations of offered
- 19,822 observations of traded prices.

There are a number of specifications for the link between spreads and default probabilities. It is common in academia to use a linear specification, but as shown below, this specification can lead to predictions of negative spreads versus the risk free curve:

Instead, a cumulative probability function would guarantee that predicted spreads would never be outside the range from 0% to 100%. While there are a number of choices one could make, we selected the logistic formula for our empirical work:

In this example, a CDS spread quote of 600 basis points is written as 0.06. There are n explanatory variables and one or more default probabilities used as explanatory variables. For ease of exposition, we assume here that there is only one empirical default probability used as an explanatory variable. One of the reasons for the choice of the logistic function is its common use in default probability estimation, as in the Kamakura Risk Information Services default probabilities. Jarrow and Chava (*Journal of Banking and Finance,* 2004) prove that the logistic formula is the maximum likelihood estimator for a 0/1 problem like forecasting a flag for default/no default. In our usage here, we are attempting to predict a CDS spread. The coefficients of the explanatory variables X and the default probability can be determined most elegantly using general linear methods. Alternatively, one can transform the equation so that the linear function in the exponent in the denominator can be fitted by ordinary least squares. While not best academic practice, we have found minimal differences between the coefficients determined by ordinary least squares and those determined by general linear methods.

In the RISK 2007 paper, we find that the following variables are statistically significant predictors of the CDS spread:

- 7 maturities of KRIS version 3.0 Jarrow – Chava default probabilities
- KRIS version 3.0 Merton default probabilities
- 17 dummy variables for each rating category
- A dummy variable indicating whether or not the company is a Japanese company
- 10 company specific financial and equity ratios
- 6 dummy variables for each CDS maturity
- A dummy variable if the CDS is providing protection on senior debt
- 2 dummy variables for the restructuring language of the CDS contract
- Selected macro-economic factors

See Jarrow, Li, Matz, Mesler, and van Deventer Kamakura Technical Guide, version 4.1, Appendix D 2006 for details.

Two of these variables are surprising in their statistical significance:

- The Japanese dummy variable was statistically significant with a t-score equivalent of 133 standard deviations, indicating that CDS spreads on Japanese names are much narrower than they are on the otherwise equivalent non-Japanese name.
- Company size had a t-score equivalent of 100 standard deviations from zero, implying that CDS spreads are narrower for large companies, everything else constant.

These variables in total explain more than 80% of the variation in the transformed CDS spread using ordinary least squares regression, and they explain a much higher percentage when looking at the absolute levels of the CDS spread.

Since the data used in establishing predicted spreads was from the benign 2004-2005 period, a predicted CDS spread using this relationship is the CDS spread that would be “normal” given the attributes of the company at the time, defining the 2004-2005 period as “normal.” As an example, we show the predicted spread from Kamakura Risk Information Services for the highly troubled company YRC Worldwide as of July 6, 2010:

The predicted spread, on an annualized basis, is a very high 4200 basis points at the common 5 year maturity.

This feature allows spreads to be estimated on the 29,200 public firms in 33 countries covered by Kamakura Risk Information Services as of the date of this blog.

Using the Model in Reverse to Estimate Default Probabilities from CDS Spreads

The same model can be used in reverse, to derive empirical default probabilities from observable CDS spreads. This produces a much more accurate assessment of empirical default risk than assuming the ISDA model is true and implying a default probability that fits market prices for CDS.

There are just a few steps in making this obvious. First we start with the logistic regression and the explanatory variables and coefficients we have derived. For exposition, we assume that there was only one default probability in the model:

Recall that the CDS spread in this formula is expressed as a decimal, so 600 basis points is written as 0.06. We then transform the equation so that we have the linear exponent alone on the right hand side of the equation:

We then solve for the default probability as a function of the CDS spread and all of the other attributes of the company that were statistically significant in linking CDS spreads and default probabilities:

This formula is much simpler and much more accurate than the ISDA Standard CDS Converter in predicting the default probability given observable pricing on the CDS contract. Why?

- It assumes the market pricing is accurate, not that a given theory like the 3-factor ISDA Standard CDS Converter is true. For example, it produces more reasonable results for cases like the quote on BP’s 39% cumulative default when the best available estimate of BP’s 5 year cumulative default probabilities on KRIS 5.0 was 32 basis points
- It implicitly recognizes trader risk aversion and lack of CDS market liquidity, instead of assuming those factors don’t exist.

We give 2 examples of errors that result from the use of the ISDA Standard CDS converter from the very statistically significant company size and Japanese company variables discussed above. Consider two companies, Company A and Company B, which have identical default probabilities but which differ only in nationality or size:

**For a Japanese company**

The ISDA formula will implicitly assume that the lower CDS spread on Japanese Company A means it has a lower default probability than an otherwise identical non-Japanese Company B. This is an error. In fact the two companies have identical default risk but in the market place investors who focus on Japanese companies are simply willing to take less reward for that default risk than investors who invest in non-Japanese companies.

**For a very large company**

The ISDA formula will implicitly assume that the lower CDS spread on very large Company A means that it has a lower default probability than an otherwise identical small Company B. This is another error. Again, the two companies have identical default risk but in the market place investors are willing to provide credit protection on large companies for less compensation than they would demand on a smaller company with similar default risk

**Using the Top Down Approach to Derive Default Probabilities from CDS Spreads**

Kamakura Risk Information Services includes all of the coefficients and explanatory variables for the formula given above. Inputs are updated daily and made available both via the web site www.kris-online.com and via bulk download from the Kamakura Risk Information Services data server. For a subscription to this KRIS data, please contact us at info@kamakuraco.com.

Donald R. van Deventer

Kamakura Corporation

Honolulu, July 14, 2010