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The following shows link how to map a PD to a S&P rating:

S&P Rating PD range [%]
AAA [0-0.05)
AA [0.05-0.09)
A [0.09-0.23)
BBB [0.23-1.16)
BB [1.16-5.44)
B [5.44-4.21)
CCC [14.21-)

I know that this mapping only is what the writer of this paper have come up with, but it shows a general trend I have observed by working with credit risk for many years.

Bad risk grades do most often have wider PD ranges than better risk grades. In this table the PD range in B is from 5.44 % to 14.21 % but it for A is 0.09 % to 0.23 %.

Is there a mathematical/statistical reason for that, or it is only practical. With practical I mean that there probably is not so big difference between a PD of 7 or 10 % when you are going to invest in a corporate.

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    $\begingroup$ One of the challenges of bulding any conordance between grades used by different rating agencies and internally by institutons is that some grading scales may consider with different weights LGD in addition to PD; some may consider PD over different time horizons (1Y, 5Y, 10Y); etc. I rather doubt that any mapping to PD like the one you cited can mean much. $\endgroup$
    – Dimitri Vulis
    Commented Jul 16, 2022 at 13:39
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    $\begingroup$ The letter grades (AAA, AA, ...) were invented a long time ago (1908?) by non-quant ppl and we don't know the exact thinking behind them, but you are right (and it is an important obs.) that the difference betwen high grades (say AAA vs AA) is much smaller than the difference between lower grades. Possibly Mr. Moody thought that AAA should be very very rare/strictly defined and the lower grades have increasingly lax/imprecise criteria. CCC then would be a broad range of diverse companies which he didn't want to spend time to analyze into finer equal sized categories. He focused on the top. $\endgroup$
    – nbbo2
    Commented Jul 17, 2022 at 7:51

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Rating grades are normally based on a logarithmic scale. If we develop a rating scale for a bank's internal use it is common that pd doubles or increase by 50% from rating class to rating class.

This is not the case for the scale you are referring to, but it might have been like this when the model was first developped. During recalibration they might have stuck with the logic for assigning rating classes and simply updated the corresponding pds.

Having rating classes on a logistic scale makes sense for multiple reasons.

  • We want to have a fairly uniform distribution over the rating classes, i.e. if all observations fall into one rating class it is not very helpful. A logistic scale makes this much more likely than a uniform scale.
  • Normally grade assignement involves some form of logistic regression, so a logistic scale is natural.
  • It is much easer to distinguish someone with a PD of 0.1% from someone with a 5% PD than 95% to 99.9%. Classes reflect the level of information/ certainty which is higher for lower PDs.
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