Credit Spreads and Default Probabilities: A Simple Model Validation Example

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.