# Hazard process and affine term structure

How can I extrapolate the hazard processes and calibrate an affine term-structure model from the historical series of curves (1y, 2y, ..., 10y tenors) of the CDs spreads of different entities?

I once had a 50y CDS, on a reference entity that had a somewhat liquid 5y CDS quote, and a not at all liquid 10Y CDS that sometimes appeared on broker screens but not often enough to have IHS Markit consensus number.

I will answer from the point of view of two different practical problems.

Mark to market. If you assume constant hazard rate between the dates of the CDS quotes that you mark (spreads or upfront fees) until the last CDS, then the only extrapolation past the last CDS quote that is likely to be acceptable to your model validation / product control / regulators is to assume that the last hazard rate (between the penultimate and the last CDS) will continue to be constant until the end of time.

Using the constant hazard extrapolation, you are likely to get probabilities of default and CDS quotes at 20y, 30y, 50y that intuitively don't feel right. Practically, just mark 7y / 10y to levels that seem to make economic sense to you when extrapolated using your library.

Strategy/Rich-cheap. Suppose a credit only had a 5-year bond or 5Y CDS observable, and you're trying to figure out a 10Y or 15Y CDS (or, the same queston phrased more commonly, if a new bond were issued at par for that maturity, what should be its coupon/Z-spread).

Empirically, when a credit has a lot of points, the shape of the curve (not so much the level) depends on the rating. IG curves have a certain regime-dependent shape. HY curves have a different shape. XO are in between. So I got "average" curve shapes (disregarding the levels) by rating from consensus CDS spreads from IHS Markit and from bond Z-spreads.

I assumed constant recovery and log-logistic survival $$S(t)=\frac{1}{1+{\left(\lambda t\right)}^p}$$(see, for example, these class notes). Optimizer found curve parameters $$p$$ and $$\lambda$$, the objective function trying to minimize both the distance from the observed data and from a curve shape consistent with the issuer's rating. Among other tests, the resulting curves were very good at predicting the Z-spread of new bonds, both IG and HY.