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# Default Rate Methodology

#### sitingbull

##### New Member
Hello Guys,

I am a bit confused on the methodology to calculate a default rate on a number of loans :

Say we have a portfolio of 27 lines of loans there are 8 loans who defaulted, normally the default rate would be 8 / 27 but on some papers I have seen that they calculate it as 8 / 19, are there any reasons or references we can refer to to justify this methodology?

Thank you very much

SB

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @sitingbull No, I don't think we (the FRM) uses such a metric. My hunch is that's an impression created by the Giacomo DeLaurentis reading (is that the author?), which is error-prone and imprecise. He has a measure called the "forward PD" which is what should be called the conditional probability such that, for example: If we have a portfolio of 27 loans at the beginning of the year (aka, beginning of period) and 8 default during the year (so that 19 survive), the conditional PD = 8/27 = 29.6%. Then going forward to the next year, the year begins with 19 survivors such that, if 8 default during this next year, its conditional PD would be 8/19 = 42.1%.

If there are 27 viable loans at the beginning of the period and 8 default during the period, there is a viable metric given by 8/(27-8) = 42.1% which is called the odds ratio (aka, just "odds" is a synonym I think); the wikipedia over-explains it here https://en.wikipedia.org/wiki/Odds_ratio ... but we don't use it for basic default probability purpose because it's not a probability. During the year, we indicate a probability of default given by 8/27 (aka, default rate) and a probability of survival given by 19/27 and, as such, the sum of these outcome is 100.0% (probabilities need to sum to 100%). Thanks,

#### sitingbull

##### New Member
Thank you David. I would really appreciate if you could shed some light on this piece of literature which I am trying to understand, here is the extract : This piece is from Bill Yang's Paper titled " Point-in-time PD term structure models for multi-period scenario loss projection: Methodologies and implementations for IFRS 9 ECL and CCAR stress testing". It was also published in RISK magazine.

I am being confused with this formula, as i am pretty convinced this should give a PD of 8/19. The example is as follows, its a Fresh Start no portfolio before start date, we are on the 1st of Jan 2016, and during that year, some companies considered as SICR(significant increase in credit risk) are put in that portfolio, which is 27(not in one go, but in a random manner), at the end of that period let say dec- 16, we accounted for a number of default of 8 companies. Calculation is done a bit after December, from my understanding we need to have 8/19, instead of 8/27? Am i doing a mistake in the reading of this formula? Because for me, in this example, there is k-1 and 0 are the same.

Thank you

SB

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi SB (@sitingbull ) Well I read the image-text with header "2.1 Forward probability of default" as fully consistent with the conventional definition of a conditional probability. The formula, too. The denominator, n(ik)(t(k)), "denotes the number of borrows who survived" the period immediately before the period (t(k-1), t(k)); so that denominator is the number of surviving borrowers at the beginning of the period, before any have defaulted. And the use of the word "within" is consistent with conventional definition that divides by the total number of surviviors at the beginning of the period, not the number after (i.e., net of) defaults.

Another way to look at this is that the default rate (expected default frequency, EDF), if representative and useful, also represents a default probability, and a (CDF) probability must lie between 0 and 100%. If we define EDF as default(during period)/survivors(after defaults), then notice under this approach that if 20 out of 27 default, then our default rate would be 20/7 which is greater than 100% and not a probability. We need 20/27, and under the conventional definition, any default rate must lie with (0%, 100%) consistent with its potential utility as a default probability. I admit that I don't think I've seen the conditional PD called a forward PD (unconditional PD is forward, too!) but that doesn't trouble me. But I read the text and formula as perfectly consistent with the conventional definition. I hope that's helpful!

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