Risk-Neutral Default Probability from a Zero-Coupon Bond Price
Summary
The note derives a simple risk-neutral default probability for a zero-coupon bond using its current price, maturity payment, and assumed recovery value. It sets the probability-weighted payoff relative to the purchase price to zero, with one payoff for default and another for repayment, then rearranges the equation to solve for default probability. A numerical example illustrates the calculation using a bond priced below its maturity payment and a lower recovery amount.
The result is an implied risk-neutral probability, not an estimate of real-world default frequency. The answer says it can overstate physical default likelihood because investors require compensation for bearing credit risk. It presents the calculation as an introductory simplification and points readers toward credit derivatives material for hazard rates from credit default swaps and the CDS-bond basis. It does not address discounting, term structures, changing recovery assumptions, or other complexities needed for practical credit pricing.
Key ideas
- A simple zero-coupon bond model equates expected payoff relative to current price to zero.
- The implied default probability depends on bond price, maturity payment, and assumed recovery.
- Risk-neutral default probabilities can differ from real-world default probabilities because they include compensation for risk.
- The calculation is an introductory simplification and omits richer credit-market modeling details.
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# Risk-Neutral Pricing Formula for Zero-coupon bonds with Default Risk # Risk-Neutral Pricing Formula for Zero-coupon bonds with Default Risk I am looking for the equations or papers showing the risk-neutral pricing for zero-coupon bonds including default risk. I already tried Googling and searching SSRN and Jstor. ## Answer by RWP - Down by the Bay (score 1, accepted) https://quant.stackexchange.com/a/53041 A brief educational note and then where you can find the info... As a first step, set the expected payoff equal to 0 where prob_D = probability of default, cur_Px = current price, mat_Px = maturity payment, and R = recovery. Therefore prob_D * (recovery - cur_Px) + (1 - prob_D) * (mat_Px - cur_Px) = 0 results in prob_D = (cur_Px - mat_Px) / (R - mat_Px) As an example, say a zero is trading at 75, matures at 100 and would recover 25. Then: prob_D * (25-75) + (1 - prob_D) * (100-75) = 0 solves to prob_D = 1/3 You can think of the first half of the above equation, the part prior to the plus sign, as your return (which will be negative) if there is a default prior to maturity. The second half of the equation can be thought of as your return if there is no default prior to maturity. You probability weight those returns and they sum to 0. Of course that's not a real world probability of default as there should be some risk for which the holder is being compensated, which means that calculated prob_D is an overstatement of the real world probability of default. And yes, this is vastly simplified, but just understanding the above will help as you read more about this in the 2004 JPM credit derivatives handbook. Directions: Google "Morgan Stanley Credit Derivatives Handbook" In the first few results you will see old (2004-2008) versions of credit handbooks from both MS and JPM. Find the JPM one. There is way more in there than you need, but they are great. Pages 15-23 and then from 45-54. Pay special attention to the examples on pgs 50 and 51. If you can understand how to calculate hazard rates from CDS and then how to calculate CDS-bond basis, you'll be able to answer your question. You can understand those things by looking at the above. This is not easy stuff to wrap your head around for the first time.
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