Annualizing Multi-Year Default Rates Under Constant Risk
Summary
The document asks whether a multi-year observed loan default probability can be converted into an annual rate by assuming a constant annual default probability. The accepted explanation models a portfolio in which each year’s defaults are drawn from the borrowers who remain after earlier defaults. Summing those successive cohorts gives a cumulative default probability over the full horizon, and rearranging the relationship yields the annualized rate formula posed in the question.
This derivation makes the annualization assumption explicit: the annual default probability is constant across years, and borrowers who default leave the population at risk. The answer also flags technical complications such as attrition and points to a credit-risk reference, but does not develop those adjustments. The formula is thus a simplifying conversion for comparable horizons, not a complete treatment of loan performance, censoring, or changes in risk over time.
Key ideas
- With a constant annual default probability, cumulative defaults accrue from the surviving borrower population each year.
- The multi-year default probability follows a geometric accumulation of annual default risk.
- Solving that relationship for the annual rate gives the proposed annualization formula.
- Attrition and other technical details can affect the calculation beyond the simplifying assumptions.
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Full text
# Annualized actual probability
# Annualized actual probability
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 by Magic is in the chain (score 5, accepted)
https://quant.stackexchange.com/a/42270
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.pdfShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.