Annualized actual probability

Question

I have a dataset of bank loans over different periods. Let's say that most of the loans have a horizon over 5,10,15 years. I obtain the actual default rate over these different type of loans. I would like to annualize these default rates to make them comparable. If I assume that the default rate is constant over time, is it appropriate to do: $dr_a = 1 - (1-dr_y)^{(1/y)}$ where $dr_a$ is the annualized default rate and $dr_y$ is the observed default probability for the considered different class of loans?

Do you have any reference to suggest me?

Answer 1

Yes your formula seems correct under the simplifying assumptions as you can easily verify:

Assume annual default rate is d, and the portfolio size is N. In the first year, we will have Nd defaults. In second year, the number of obligors would have declined by the number of default in the previous year to N-Nd=N(1-d), so the number of defaults in the second year would be N(1-d)d, and so on. So the total number of defaults over n years is:

$D_n=Nd+N(1-d)d+,\dots,+N(1-d)^{n-1}d$

$D_n= Nd \frac{1-(1-d)^n}{1-(1-d)}$

$\frac{D_n}{N}=1-(1-d)^n $

The left hand side is the n-year default rate, and solving this for d, one year default rate, gives your formula.

There are a few technicalities as to how to account for attrition etc, for which the below is a good reference:

https://www.moodys.com/sites/products/DefaultResearch/2006200000425249.pdf