In the odds and score definition you quote, a higher score means a worse situation.

There is also a symmetry around whether one considers “default” or “non-default” to be the target event.

I opined that “Odds are generally taken to be the Good:Bad odds” (rather than the Bad:Good odds) without knowing what most of the Credit Risk industry does, so it may be from your sample of quite a few credit risk books that my convention is actually in the minority.

A convenience factor attached to this convention is that PDs are usually small (.001 to .1), and this leads to a convenient range of numbers for odds (999 to 9). In the convention you mention, the range for those “odds” would be fractions (0.001 to 0.11).

Thank you for raising the point and it would be interesting to hear which convention enjoys the most support.

I understand odds definition in the context of event defined as ‘Default’ as PD/(1-PD) i.e prob (event) / prob (non-event). If we consider Log (Odds) = Score, then the above relationship gives result as PD = 1/(1+exp(-score)). This relationship is explained in quite a few credit risk books. Why is then your odds definition provided as (1-PD)/PD. What is correct??

Any feedback on commonly adopted score scaling choices would be welcome - especially from the specialist scorecard building companies - tell us what the commonest choices are so that this blog can maximise familiarity.

Alternatives I have seen include banging in the location anchor at a point representing, say, 50:1 odds. Such a point is likely to be not too far from the middle point of your range since it is close to PD=2%. OTOH 1:1 odds has the minor convenience of log_odds=0 which makes the scaling formula slightly simpler. Also, this “location anchor” may be chosen at 200, 600, whatever; somewhere where scores are extremely unlikely to ever go below 0 (or below 100, which produces a 2-digit score) or above 999.

It’s all about convenience and amenability for non-technical users so as to minimise opportunities for transcription errors and misunderstandings so it may as well be done thoughtfully.

I’ve seen PDO choices of 20 and of 15. Even 10 would give a decent granularity for most purposes, but, why skimp, take 20.

PDO=12 would be a quirky choice because the scores would then be analogous to musical intervals on the Western system of 12 semitones per octave, so scores would map neatly to piano keys. Credit risk committees could then debate lifting the score cut-off from B-flat to F-sharp. The rest of this scenario is left to the reader’s imagination.

Note that the Basel floor of PD=0.03% equates to odds of 3332 and log_odds=8.11; this then represents a sensible limit on axes on graphs for example. I typically choose log_odds=-2 (PD=88%) to represent minus infinity and log_odds=8 (PD=0.03%) to represent positive infinity, which gives 10 “octaves” of the PD scale encompasing all but the most extreme of values. Sticking to these limits on all log_odds graphs then gives a uniform “frame” which will enhance interpretation and comparison.

I have also sometimes used these infinity bounds to “Winsorise” or truncate calculated odds values that go crazy because of zero division or log of zero or small sample effects.

You may prefer your own estimates of infinity.

Whatever your choice of scoring scale, it is helpful to determine these two “infinities” as a guide: e.g. if you use 600@1:1 and PDO=20 you will get minus infinity at score=542 and plus infinity at score=831. Scores outside this range are possible but extreme, and in the case of >831 they exceed the Basel limit.