Handling Negative Credit Spreads in Default Probability Curves
Summary
The document considers negative spreads produced by sector spread estimates or stressed credit curves, and whether they should be converted to zero default probability. It starts from a simple spread-to-default-probability approximation based on recovery, then notes that implied probabilities should remain within valid bounds. Negative spreads can arise in scenario perturbations, even when observed market spreads are positive.
The answer emphasizes that curve construction often fits hazard rates rather than default probabilities directly. A small negative hazard rate may occur during perturbations; the suggested practice is to record and monitor such cases while continuing calculations, whereas an excessively negative rate should trigger an error. It also warns that nonnegative pointwise default probabilities can still produce an inconsistent term structure, such as declining cumulative default probability over time.
This is practitioner guidance, not a detailed derivation or empirical study. The approach treats small negative hazards as tolerable for calculations, so implementation should distinguish scenario artifacts from persistent calibration or arbitrage inconsistencies.
Key ideas
- A negative spread can produce an invalid negative default probability under a simple spread-based approximation.
- Credit curve fitting often uses hazard rates rather than fitting default probabilities directly.
- The answer suggests logging and monitoring small negative hazard rates while rejecting excessively negative values.
- A term structure can be inconsistent even if each quoted default probability is nonnegative.
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Full text
# PDs for negative credit spreads
# PDs for negative credit spreads
My question is about credit spreads and the corresponding probability of default (PD). One of the most simple relations between credit spreads and PDs is (see e.g. ch7 in Malz(2011)) $$ PD \approx \frac{s}{1 - RR}, $$ where PD is the one year PD, $s$ is the 1-year credit spread and $RR$ is the recovery rate.
I wanted to ask for common market practices in case that $s$ is negative. Clearly, if $s$ is derived from CDSs the spread is non-negative. But if $s$ is derived from sector spreads (e.g. via numerical methods) $s$ could be negative. Is it to careless to simply assume that $s < 0$ implies $PD = 0$. Does anyone have experience or can point to literature?
Thank you in advance.
References:
Malz, Allan M. Financial risk management: models, history, and institutions. Vol. 538. John Wiley & Sons, 2011.
## Answer by Dimitri Vulis (score 1)
https://quant.stackexchange.com/a/51290
I was actually asked this (or something very similar) at a job interview for a credit quant job about 20 years ago. My answer actually hasn't changed much!
$PD$ is a risk-neutral probability that depends on the choice of recovery assumption $RR$ (no term structure). It still should not be <0 or > 1 irrespective of the choice of $RR$. But if it is, it's somehow not as jarring as physical negative probability.
Negative spreads (and other inconsistencies seemingly admitting arbitrage) easily arise when you perturb a credit curve under risk scenarios. For example, the observed credit spread for some government agency might be 5 bps, and you're trying to compute the P&L impact of the spread tightening 10 bps, i.e. spread -5 bps and negative $PD$. Just as badly, $PD$ can be non-negative, but still imply risk-free arbitrage with the probability of default decreasing with time. $PD=0$ at $t=0$ and $PD=-0.1$ at $t=1$ is bad, but no worse than $PD=.2$ at $t=1$ and $PD=.1$ at $t=2$.
Most curve fitters solve actually not for $PD$, but for the hazard rate, assumed to be constant between the nodes. You should throw if the hazard rate is too large a negative number. If the hazard rate is a small negative number (or a too large positive number), you should log this, and look at this log periodically as part of your ongoin performance monitoring. But after logging the warnings, you should proceed, because the formulas still kind-of work.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.