# 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?

References:

Malz, Allan M. Financial risk management: models, history, and institutions. Vol. 538. John Wiley & Sons, 2011.

Quantitative Finance Asked by Cettt on November 15, 2021

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.

Answered by Dimitri Vulis on November 15, 2021

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