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Odds provide a useful frame for considering that important business question: where to set the score cut-off.

The basic business logic is to set the cut-off at the score where marginal profitability equals zero – i.e. if you moved the cut-off any lower you would be losing money on each additional applicant so approved, whereas if you set the cut-off higher you would be leaving money on the table. Easy to say, but not so easy to do, because the concept of marginal costs is a movable feast depending on accounting treatments and assumptions about fixed and variable costs, as well as the context within the current business strategy.

But anyway, the odds allow one to frame the question in an easy-to-grasp way: how many goods does it take to offset one bad? If the answer is 15, it means that your tipping point is at 15:1 odds, which can be converted to the score as per previous post. This would then be the cut-off. This post assumes a simple automatic accept/decline score, ignoring ‘refer’ bands and contested decisions and overrides etc.

To arrive at “15” would involve a full revenue/cost modelling through the product cycle (lifetime customer value?), for 15 goods versus 1 bad. Naturally the “cost” that dominates here is the credit loss of principal (LGD) for the default.

Don’t pay any attention to the example value “15” used above – it’s going to make a lot of difference what product is involved, secured vs unsecured, limits, etc.

The credit risk world likes to work with ‘odds’ and related quantities so these are covered today.

You could just do everything in terms of probability, i.e. PD, which is unambiguous. PD lies in [0,1] and a *small* number (like 0.002) is a *better*
customer than a bigger number (like 0.013). In typical modelling
situations (in Australia, in the good times..), a lot of PDs would have
one or two or even three leading zeroes and these numbers are not handy
for transcription or to quickly convey which zones they lie in.

It goes without saying that it often more palatable to format a PD as a percentage, e.g. PD = 0.013 as PD = 1.3%.

‘Odds’ have a special status because they are intimately linked with logistic regression, the main PD-modelling statistical tool. Odds can be worked out from the PD, and vice versa, as follows:

- odds = 1/PD – 1
- PD = 1/(1 + odds)

For example, odds = 8 means exactly the same thing as PD = 1/9 = 0.1111..

Odds are generally taken to be the Good:Bad odds; thus a *bigger* number for odds is a *better*
situation. I have seen analysts using Odds the other way up i.e. the
Bad:Good odds. You can come out alive but it will confuse your
colleagues; +/- changes of sign will cascade through and
graphs will tilt the opposite way.

One step closer to the logistic zone is to transform to “log_odds”.

- log_odds = ln(odds)
- odds = exp(log_odds)

‘ln’ means natural logs, i.e. to the base ‘e’. Actually, mathematicians *always*
mean natural logs when they say log and as a matter of pride would
never mention the base, or contemplate a base other than ‘e’ unless it
was a neat way to summarise a problem that had structure particular to
integral bases. Ambiguity can arise: computer systems that are
tech-oriented, like SAS or MATLAB, assume ‘log’ means ln, whereas those
that are business-oriented, like MS/Excel, assume that ‘log’ means
log_to_base_10. It also doesn’t help that ‘ln’ is not comfortable
in speech.

By ‘log’ I always mean natural log, and I use log10 or log2 to mean logs to base 10 or 2. For the meantime, the terminology ‘log_odds’ will be used, which is easy in speech, but if anyone can suggest better nomenclature they are welcome to put it forward.

If we’ve taken the right choices so far, a *bigger* number for log_odds is a *better* situation. Note that log_odds can be negative (when odds < 1 which is when PD > 0.5).

To make the numbers more convenient to handle, it is common practice
to convert the log_odds to a ‘score’ on a user-friendly scale that
wouldn’t involve negatives or decimal places. For the first time in this
chain of transformation, arbitrary scaling constants are involved in
this choice: one for *location* and one for *scale* (spread). A typical approach is illustrated below:

- for location: bang a stake in the ground at the point that will
represent odds of 1 (== log_odds of zero == PD of 0.5): so, for example,
choose
*a score of 500*to represent this point (which BTW would be a lousy customer) - for scale: this is normally done by specifying
*how many points it takes to double the odds*(PDO). A comfortable choice would be PDO=20, which says that a score of 520 <=> odds=2, 540 <=> odds=4, 560 <=> odds=8 etc.

Because log_odds is a logarithmic scale, the above choices work out and amount to a linear transformation of log_odds to score. The two scaling parameters, and hence the transformations from log_odds to score and back, will depend on these fairly arbitrary choices.

PDO=20 gives a nice granularity to the scores, which will mostly land
in the 500-800 zone and you won’t feel the need to use decimal points
i.e. whole-number scores suffice. As long as PDO is chosen to be
positive, it will still be the case that a *bigger* score is a *better* situation.

All the above transformations are absolute arithmetic ones that always apply, irrespective of context such as outcome window, default definition, calibration, closed goods in/out, etc. If you find you disagree with someone via these calcs, it means you started from different contexts and therein lies the entire explanation for your disagreement.

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