Hi
@RiskQuant
I don't have realistic information on mapping in the direction of PD --> Rating. Sorry

It seems like a slightly unusual direction to go from ratio-based PD to ordinal-based level of measurement, the motivation isn't obvious to me (sorry, I am surely ignorant about some usage). My exposure is rather to mapping ratings (A, B, ...) to risk ratings numbers (e.g., 10, 9, ...). The real art and science is
validation. Although, analytically, if I were going to do it, there seem like so many ways but I would probably do something simple like take a historical database of companies with the credit ratings and simply calculate the defaults rates (with intervals); e.g., companies in this database that were rated "AA" experienced actual defaults of X% with standard deviation of Y, so we will define an "AA" rating the
historically-informed band = X +/- Y*a, with some rule for arbitrating overlaps. I could imagine other methods ; e.g., ranking the PDs and binning the results.
It just seems to me to be an exercise wherein you would either (i) borrow from the credit rating agencies since you are, for some reason, "backing into" (backing down to a lower level of measurement) anyways, so why not utilize their definitions, or (ii), if not, use the opportunity to a priori define the rating categories. I don't know the motivation ...
In case it's helpful, here is Moody's
Public Firm Expected Default Frequency (EDF) Credit Measures: Methodology, Performance, and Model Extensions:
http://trtl.bz/29iNqp4
This is the updated KMV and, in fact, this document is just an update of KMV's original method document (I did some work for the author of the original KMV document, Peter Crosbie). It discusses the mapping of the DD to EDF, which I realize it's maybe not what you are looking for, but just in case. As I mention in my OP above under
Variation #2: KMV (Merton but with two adjustments), the N(-DD) produced by a lognormal asset return assumption is not a reliable PD, so Moody's (KMV) mapped the PD to an EDF (page 13,
emphasis mine):
3.4 Moving from DD to EDF
"The Distance-to-Default provides an effective rank ordering statistic to distinguish firms likely to default from those less likely to default. We have verified its effectiveness by observing a strong empirical relationship between DDs and observed default rates: firms with larger DDs are less likely to default. However, one still needs to take a further step to derive PD estimates.
In the basic structural credit risk model DDs are normally distributed as a result of the geometric Brownian motion assumption used to model the dynamics of asset values. However, actual default experience departs significantly from the predictions of normally distributed DDs. For example, when a firm’s DD is greater than 4, a normal distribution predicts that default will occur 6 in 100,000 times. Given that the median DD of the entire sample of firms in the EDF dataset is not far from 4, this would lead to about one half of actual firms being essentially default risk-free. This is highly improbable.
Instead of approximating the distribution of DDs with a standard parametric distributional function, the EDF model constructs the DD-to-PD mapping based on the empirical relationship (i.e., the relationship evidenced by historical data) between DDs and observed default rates. Moody’s Analytics maintains the industry’s leading default database, with over 8,600 defaults as of the end of 2011. The process for deriving the DD-to-EDF empirical mapping begins with the construction of a calibration sample – large North American corporate firms – for which we have the most reliable default data. It is reliable in the sense that “hidden” defaults – defaults that occurred, but that were neither reported nor observed – are relatively less likely to cause estimation errors. The DD-to-EDF mapping is created by grouping the calibration sample into buckets according to the firms’ DD levels, and fitting a nonlinear function to the relationship between DDs and observed default frequencies for each bucket. A stylized version of the resulting DD-to-EDF mapping is plotted in Figure 8 in green, along with the DD-to-PD mapping (the orange line) implied by a normal distribution of DDs."
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