Skip to content
All library documents

Recovering Annual Default Probabilities from a Cumulative PD Curve

Article Quant Q&A · Author: Vinod Khera
Read the original

Summary

The note explains how to convert a cumulative probability of default (PD) term structure into the conditional probability of default during a particular year. It starts from the relationship between default by the end of a year and the two ways that can happen: default occurred earlier, or the borrower survived through the prior year and defaulted during the current year.

For year X, subtract cumulative PD through year X−1 from cumulative PD through year X, then divide by the probability of surviving through year X−1. This produces the one-year conditional PD for year X. The explanation is algebraic and gives no worked numerical example or details about estimating the original curve. The calculation assumes the table entries are cumulative PDs; if they are already annual conditional probabilities, this conversion is unnecessary.

Key ideas

  • A cumulative PD includes defaults from all earlier periods.
  • The probability of default in a given year is conditional on having survived through the previous year.
  • Find the annual conditional PD by taking the increase in cumulative PD and dividing by prior-year survival probability.
  • Confirm that the input term structure is cumulative before applying the conversion.

Tags

Fixed incomeStatisticsRisk management
Full text
# How to apply PD term structure?


# How to apply PD term structure?












I have a table containing PD term structure with their varying PD values over time such as below:

Now, if I know the starting value of a PD (for e.g. 0.0025), how can i canculate its value in a given year, let's say in year 4 using the above table. Any help?

## Answer by Attack68 (score 2)

https://quant.stackexchange.com/a/40241

By the looks of it your table is cumulative PD.

You can use the argument:

$$P(\text{Default by end year X}) = P(\text{Def. by end year X-1}) + P(\text{Not def. by end year X-1})P(\text{Def. in year X}) $$

So that,

$$ P(\text{Def. in year X}) = \frac{P(\text{Def. by end year X})-P(\text{Def. by end year X-1}) }{1-P(\text{Def. by end year X-1})} $$

Shown 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.