# 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.

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.

• I feel this skips over the reasons you can get a negative hazard rate - it is simply saying that the company you are looking at is less risky that the risk measure/country it resides within. I would estimate that if you look at a developing nation there are many companies with negative credit spreads – will Feb 23 '20 at 3:36
• I'd be very curious to know more about your estimate & to see some examples of companies trading negative credit spreads (as opposed to risk scenarios). For example, AMXLMM (Mexican telco) is usually ~30 bps tigher than UMS sovereign. So? Negative hazard rates can arise, for example, if you are solving for hazard rates using recovery assumption 40% (and so seem to see decreasing probability of default, admitting arbitrage), and the person who publishes the quotes assumes, maybe 70% recovery if default happens in year 1 and 50% if it happens in year 3, and that happens not to admit arbitrage. – Dimitri Vulis Feb 23 '20 at 5:57
• if you're assuming a different recovery rate though, you're of course going to get a different result, the same way that you get different implied volatilities if you use a different method to calculate the time to maturity. The example i'm talking about is if you have a company issuing bonds in a currency where the company is percieved as less likely to default than the country - exactly like your AMXLMM example. I do not think it comes entirely from using different recovery assumptions. – will Feb 23 '20 at 13:42
• Different recovery assumptions is a non-exhaustive example of reasons why hazard rate may come out negative. But I'm not sure I can agree that corporates can be significantly less likely to default than their sovereign. Historically whenever Argentina or Venezuela sovereigns defaulted, almost all their corporates and provinces took the opportunity to default too. If AMXLMM or Codelco (Chilean copper miner) or Petrobras can borrow at lower yields than their respective sovereigns (all currencies including local), it may be because the market expects the recovery to be higher in default. – Dimitri Vulis Feb 23 '20 at 14:12
• The probability of the sovereign defaulting on local-law local fiat currency debt is hard to gauge because it happens so infrequently. I can only think of 2 examples - Peru and Russia in 1998. Usually the government can just print more fiat currency (devaluing it) and repay the debts. But I'm still very curious to see your examples of negative credit spreads please. – Dimitri Vulis Feb 23 '20 at 14:16