One of the most persistently used formulas in fixed income markets is the relationship

**Credit Spread = (1 – Recovery Rate)(Default Probability)**

This simple formula asserts that the credit spread on a credit default swap or bond is simply the product of the issuer’s or reference name’s default probability times one minus the recovery rate on the transaction. The persuasive belief that this formula, or at least a simple variation on it, is true has led to a wide array of models implying default probabilities from credit spreads. In the popular press, these models are frequently invoked in headlines like “ BP Swaps Put Odds of Default at 39% ,” a June 16, 2010 forecast during the Gulf Oil spill.

In this note, we ask two questions. First, what are the implications of this formula if it is true? Second, are the implications consistent with the facts? This is the essence of basic model validation.

**The Implications of the Formula**

The simple credit spread formula has been most often invoked in the early days of the credit default swap market. It has a number of implications if we take it literally.

- Only two factors drive credit spreads, the default probability and the recovery rate.
- Since the default probability and recovery rate can vary by maturity, at any point in time the formula determines the full term structure of the credit spread.
- Since the recovery rate can only vary from 0% to 100%, in no case should the credit spread be a larger number than the default probability.

We follow Jarrow, van Deventer, and Wang’s paper “ A Robust Test of Merton’s Structural Model of Credit Risk ” in this note. We have enumerated a short list of important implications of the model. We test these implications against observable data. If the data is inconsistent with the implications of the model, we reject the model. This is an essential series of model validation procedures in many areas of risk management. We start with the third implication.

**The Credit Spread Must Be Less Than or Equal to the Default Probability**

In this section we employ trade-weighted bond price data from the TRACE system for August 5, 2014. We assemble all bond issues which meet the following criterion:

Seniority of debt: Senior

Maturity: 1 year or more

Coupon: Fixed rate, with semi-annual payments

Callability: Non-call, except for “make whole” calls

Trade volume: At least $5 million in transaction volume

There were 202 bond issues that met these criterion. For each bond, we assembled the matched maturity U.S. Treasury yield from theH15 statistical release from the Federal Reserve, for which the U.S. Department of the Treasury is the original source. We also assembled the matched maturity modern reduced form default probabilities from Kamakura Corporation, which are described below. We calculated the ratio of credit spread (traded-weighted average yield minus the matched maturity default probability) to default probability for all 202 issues, and ranked them from lowest spread to default ratio to highest. What were the results? Only 4 of 202 bond issues had credit spreads less than or equal to the default probability. We show the 50 lowest credit spread to default probability ratios in this chart:

The full distribution of the credit spread to default probabilities is shown in this histogram:

When we order the credit spread to default probability ratios from lowest default probabilities to highest default probabilities, we get the following graph. The black line is a model-independent “median spline” connecting grouped data points:

The graph shows clearly that there is a wide variation of credit spread to default probability ratios. The ratio of spread to default probability declines as default risk increases. The reason for this is obvious. When default in the near term is highly likely, the bonds will trade near their anticipated recovery value, and the credit spread loses its relevancy. The median credit spread to default probability ratio on August 5 was 10.6, mush larger than the ratio of 1.0 or less predicted by the model. The average credit spread to default probability ratio was 16.7.

Obviously, the third implication of the model above is false and the model is rejected by any normal probability standard. We now turn to the other implications of the model.

**Only 2 Explanatory Variables Determine the Entire Term Structure of the Credit Spread**

Again, following Jarrow, van Deventer, and Wang, we test this implication in a model independent fashion. First, we use the same data from August 5, 2014 for all issues with trade volumes over $5 million. If the model above is true, the explanatory power of the model should be near 100% and there should be no constant term in an expression of the form credit spread = a + b(default probability). The latter comment stems from the fact that the simple model above implies a credit spread of zero when the default probability is zero. The parameter “b” would be the implied estimate of one minus the recovery rate. Here are the results:

The constant term is statistically significant and implies a credit spread of 1.18% even if the default probability is zero. The linear expression explains only 3.66% of the variation in the model. Since recovery rates on non-defaulted bonds are unobservable, adding a recovery rate to the model is not an option. The coefficient of the default probability, 0.209, is the fitted estimate of (1 minus recovery rate), so the implied recovery rate is 79.1%.

If we eliminate the constant term to be consistent with the implication that zero default probability should imply zero spread, the explanatory power is a bit better at an adjusted r-squared of 15.32% and an implied recovery rate of 40%.

This econometric exercise alone should lead to a rejection of the model because of two inconsistencies with the model: a non-zero credit spread when default probabilities are zero and low explanatory power inconsistent with the implication that two factors explain all credit spread movements. Still, we give the model one more try.

**Another Test of the Implications**

Perhaps credit spread data is so noisy that explanatory power is low even if the default probability and recovery rate are the only statistically significant variables that predict credit spreads. We test the implication that no variable, other than these two, is statistically significant in predicting credit spreads. We assemble a data base of credit default swap bids, offered, and traded levels from GFI Group. We then add variables to a statistical model of credit spreads above and beyond the default probability and the recovery rate using the Kamakura Risk Information Services credit spread data base. We used 2,389,309 observations in an econometric process documented in the Kamakura Risk Information Services Technical Guide, Version 6.0, Appendix D, Implied Credit Default Swap Spreads, June 24, 2013. This technical guide is available to regulators and clients of Kamakura Corporation via info@kamakuraco.com.

Were there any incremental variables that added explanatory power to a spread prediction model beyond default probability and (implied) recovery rate? More than a few. A total of 67 variables, including 3 maturities of default probability, were statistically significant in predicting credit spread over these 2.4 million observations. The accuracy of the resulting formula was five times greater than the 15% adjusted r-squared that we found above. In short, it would be nearly impossible for any careful analyst NOT to improve on the simple credit spread formula that says credit spread equals one minus the recovery rate times the default probability.

In short, the implications of the simple credit spread model above are rejected in full as inaccurate. The model is too simple to provide more than marginal understanding of the credit spread.

**A Finance Theory Perspective**

Prof. Robert Jarrow, Managing Director of Research at Kamakura Corporation, makes the same points much more powerfully and precisely from a finance theory point of view. The principal paper in this regard is “ Problems with Using CDS to Imply Default Probabilities” in the *Journal of Fixed Income*, Spring, 2012. A companion piece provocatively titled “All Your CDS Models are Wrong” appeared in Creditflux, November 2012.

**Conclusion**

The simple and popular formula which says credit spreads equal one minus the recovery rate times the default probability is dramatically wrong. It implies that no credit spreads should be higher than the firm’s default probability, but on August 5, 2014, 98% of credit spreads were higher than the default probabilities on 202 heavily traded bond issues. The formula implies that credit spreads will be zero for issuers with zero default probabilities, but fitting a linear credit spread model shows that the constant term is statistically significant and positive, not zero. The formula implies that only two factors explain the entire term structure of credit spreads over all time periods, but in fact they explained only 15% of credit spread variation on August 5. A richer econometric specification found that 64 financial variables, along with three different maturities of default probabilities, had explanatory power for credit spreads five times higher than the simple model provides.

We conclude that the simple credit spread model is simply wrong and recommend that it be avoided in all professions from finance and risk management to journalism.

**Appendix
**

**Background on the Kamakura Public Firm Default Probability Models**

The Kamakura Risk Information Services version 5.0 Jarrow-Chava reduced form default probability model (abbreviated KDP-jc5) makes default predictions using a sophisticated combination of financial ratios, stock price history, and macro-economic factors. The version 5.0 model was estimated over the period from 1990 to 2008, and includes the insights of the worst part of the recent credit crisis. Kamakura default probabilities are based on 1.76 million observations and more than 2000 defaults. The term structure of default is constructed by using a related series of econometric relationships estimated on this data base. KRIS covers 35,000 firms in 61 countries, updated daily. Free trials are available at Info@Kamakuraco.com. An overview of the full suite of Kamakura default probability models is available here.

**Using Default Probabilities in Asset Selection
**We recommend this introduction to the use of default probabilities in fixed income strategy by J.P. Morgan Asset Management.

General Background on Reduced Form Models

For a general introduction to reduced form credit models, Hilscher, Jarrow and van Deventer (2008) is a good place to begin. Hilscher and Wilson (2013) have shown that reduced form default probabilities are more accurate than legacy credit ratings by a substantial amount. Van Deventer (2012) explains the benefits and the process for replacing legacy credit ratings with reduced form default probabilities in the credit risk management process. The theoretical basis for reduced form credit models was established by Jarrow and Turnbull (1995) and extended by Jarrow (2001). Shumway (2001) was one of the first researchers to employ logistic regression to estimate reduced form default probabilities. Chava and Jarrow (2004) applied logistic regression to a monthly database of public firms. Campbell, Hilscher and Szilagyi (2008) demonstrated that the reduced form approach to default modeling was substantially more accurate than the Merton model of risky debt. Bharath and Shumway (2008), working completely independently, reached the same conclusions. A follow-on paper by Campbell, Hilscher and Szilagyi (2011) confirmed their earlier conclusions in a paper that was awarded the Markowitz Prize for best paper in the *Journal of Investment Management* by a judging panel that included Prof. Robert Merton.

**Author’s Note** **
**Regular readers of these notes are aware that we generally do not list the major news headlines relevant to the firm in question. We believe that other authors on SeekingAlpha, Yahoo, at

*The New York Times*,

*The Financial Times*, and the

*Wall Street Journal*do a fine job of this. Our omission of those headlines is intentional. Similarly, to argue that a specific news event is more important than all other news events in the outlook for the firm is something we again believe is inappropriate for this author. Our focus is on current bond prices, credit spreads, and default probabilities, key statistics that we feel are critical for both fixed income and equity investors.