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# General query on probability of default.

#### kausthub

##### New Member
@David Harper CFA FRM
I completed my FRM Part 1 on may 18th, and I am glad to inform you that I have faired well in the exam all thanks to your material and question sets.
I was just reading the notes on Economic capital in banks, and I was wondering how the probability of default is calculated. I understand it is assumed to be have a binomial distribution. But say for a retail personal loan with no collateral put forward, what would be used to model the probability of default? Please give me your suggestions, and any other material I could refer to.

Thanks again for the help.

-kausthub

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @kausthub Really glad to hear the exam went well for you! Of course you are correct that the FRM syllabus assumes PD is a binomial. See below assigned Schroeck for his thoughts on this. Realistically, I think credit spreads are often used to estimate or proxy default probabilities. In regard to this, Hull's Chapter 24 (Credit Risk) discusses the primary decision around estimation based on historical data versus implied from forward-looking credit spreads. Keep in mind that if we have a dataset, we don't need to specify a parametric (aka, analytical) distribution: we can use an empirical distribution. Not unrelated, when I've practiced data science on this topic, it's less about distributions than simulations; or even some/most of the machine-learning techniques don't really assume any distribution. I hope that's a bit helpful!

Here is Schroeck Chapter 5, page 177:
"Note that, since default is a Bernoulli variable with a binomial B(1;PD)-distribution: σ^2(PD) = PD*(1-PD). Since it is typically difficult in practice to measure the variance of the loss rate σ^2(LR) due to the lack of sufficient historical data, we will have to assume in most cases a reasonable distribution for the variations in the loss rate. Unfortunately, unlike the distribution for PD, the loss rate distribution can take a number of shapes, which result in different equations for the variance of LR. Possible candidates are the binomial, the uniform, or the normal distribution. Whereas the binomial distribution overstates the variance of LR (when a customer defaults, either all of the exposure amount is lost or nothing), the uniform distribution assumes that all defaulted borrowers would have the same probability of losing anywhere between 0% and 100%. Therefore, the most reasonable assumption is the normal distribution, because of the lack of better knowledge in most cases. The shape of this assumed normal distribution should take into account the empirical fact that some customers lose almost nothing, that is, almost fully recover, and it is very unlikely that all of the money is lost during the work-out process."

#### kausthub

##### New Member
Hi @kausthub Really glad to hear the exam went well for you! Of course you are correct that the FRM syllabus assumes PD is a binomial. See below assigned Schroeck for his thoughts on this. Realistically, I think credit spreads are often used to estimate or proxy default probabilities. In regard to this, Hull's Chapter 24 (Credit Risk) discusses the primary decision around estimation based on historical data versus implied from forward-looking credit spreads. Keep in mind that if we have a dataset, we don't need to specify a parametric (aka, analytical) distribution: we can use an empirical distribution. Not unrelated, when I've practiced data science on this topic, it's less about distributions than simulations; or even some/most of the machine-learning techniques don't really assume any distribution. I hope that's a bit helpful!

Here is Schroeck Chapter 5, page 177:
Thanks David.

cheers,
kausthub