In an article in August of 2014, we focused on one of the most persistently used formulas in fixed income markets:

Credit Spread = (1 - Recovery Rate)(Default Probability)

One is barraged on a daily basis with press and internet commentary using default probabilities "implied" from credit spreads. 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. In our August 2014 note, we asked 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. In our August 2014 analysis, we used the example of one day's data to prove this simple formula is a very poor description of the relationship between credit spreads and default probabilities. We repeat the exercise in this note using 5.9 million observations from the TRACE bond data in Kamakura Corporation's KRIS service and reach the same conclusions using all bond trades from January 1, 2007 through January 3, 2017.

**The Implications of the Formula: A Review**

As we noted in our earlier article, the simple relationship above 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 again 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 can 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. Since the consistency of the third implication with observable data is easily observable, we focus on whether or not that implication is true.

**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 embedded in Kamakura Corporation's KRIS default probability and bond information service for all observations from January 1, 2007 through January 3, 2017 for which data is available. 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

Trade volume:Sufficient to trigger TRACE price reporting

There were 5,919,282 observations that met these criteria. For each observation on each bond, we assembled the matched maturity U.S. Treasury yield from the H15 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 observations. Only 215,069 of 5,919,282 bond issues had credit spreads less than or equal to the default probability. The chart below shows the large number of observations for which the credit spread is greater than the default probability (those in red), 96.37% of the total. Only 3.63% of the observations (those in blue) have a credit spread that is less than or equal to the default probability as implied by our simple formula:

**Why is This Simple Formula So Wrong?**

Many readers are surprised that a simple formula accepted by so many analysts can be so wrong. My colleague Suresh Sankaran and I recently pointed out that one of the main reasons the formula is wrong is that the calculation of the credit spread itself is riddled with false assumptions:

The credit spread calculation assumes the corporate bond will pay its full principal amount in all scenarios.

The credit spread calculation assumes that the full principal amount will be paid at maturity in all scenarios.

The credit spread calculation assumes that all interest coupons will be paid in all scenarios.

The credit spread calculation assumes that bonds of different maturities and coupons have different cash flows in all scenarios. This is of course false in the default scenarios: all bonds of the same seniority have identical cash flows upon default: remaining interest payments are zero and the principal that will be paid is the recovery amount; and the payment date is the date [or series of dates] that recovery payments are made after the bankruptcy is resolved in court.

Credit spreads are assumed constant for all periods prior to maturity of bond k. Credit spreads for bond k are different from bond j if they have different maturities, but these constant spreads are inconsistent from time zero to years to maturity = min(j,k). To give a specific Lehman example on September 15, 2008, the credit spread formula implies that the credit spread is 16.96% for the Lehman 2027 bond but 45.23% for the Lehman bond due in January 2012. In short, for the period from September 2008 to January 2012, the spread formula implies that the coupons for the 2012 bond have a spread that is almost 30 percentage points higher than the 16.96% spread that applies to coupons covering the same time period on the bond due in 2027. This inconsistency is nonsense.

The risk free yield is constant for all periods until the risk-free bond's maturity (false, this is a well-known problem with the yield to maturity calculation). Even for the risk free curve, the yield to maturity for bonds of different maturities implies different discount rates during the overlapping period when both bonds are outstanding.

Given that the credit spread calculation contains so many false assumptions, it should surprise no one that its relationship with the default probability is complex.

**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 from January 1, 2007 through January 3, 2017, 96.37% of 5.9 million observations had credit spreads higher than the firm's default probability. This graphic of the credit spread history (in blue) versus the 1-year (in orange) and 10- year (in green) default probabilities for Canadian Natural Resources (NYSE:CNQ) bonds due 2018 shows a dramatic change in the relationship between spread and default probability over time. This is typical, not an exceptional example.

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.

**Disclosure:** I/we 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.