Why Annualized Default Probabilities Can Decline with Horizon

Summary

The document explains how an annualized default probability can be lower for a longer horizon even when cumulative default risk does not fall. Its example assumes a firm faces a single default-triggering event during the first year, with a stated chance of default from that event. If the event does not cause default, the firm survives through both the shorter and longer horizons in the example, so the cumulative survival probability is the same for each horizon.

To convert cumulative survival into a constant annualized default rate, the answer takes the root of the survival probability over the horizon and subtracts the result from one. Applying this conversion to the same cumulative survival over four and five years produces a lower annualized figure for five years. This illustrates a mathematical effect of annualizing across different time spans, rather than showing that cumulative risk has decreased. The example is deliberately simplified; it does not establish whether the user’s sourced data is correct or model real-world default timing.

Key ideas

• Cumulative survival can be identical at two horizons when all modeled risk occurs early.
• Annualized default probability converts cumulative survival into a per-year rate over the selected horizon.
• The same survival probability can yield a lower annualized rate over a longer period.
• A simplified timing example explains the effect but cannot validate a particular dataset.

Tags

# Annualized Default Probabilities (short term dp's larger than long term dp's)


# Annualized Default Probabilities (short term dp's larger than long term dp's)












Was looking at some sourced data and noticed that a 4 year annualized default probability was greater than a 5 year annualized default probability. This seems counter intuitive even in the case that most of their debt is in the short term with expectations of large positive cashflows in the mid/long term. Is it possible that this is just an artifact of annualizing raw default probabilities or could the sourced data be incorrect?

## Answer by Rylan (score 2)

https://quant.stackexchange.com/a/76164

I don't think this is necessarily wrong.

For a contrived example, suppose the only thing that could possibly cause a given firm to default within the next five years is an event that occurs in one year (maybe a lawsuit). The firm has a 20% chance of defaulting as a result of this event in year 1.

Then, the (un-annualized) probability of survival for 4 years is $80\%$, and the probability of survival for 5 years is also $80\%$.

We can annualize and convert to probability of default for 4 years by taking $1 - 0.8 ^ \frac{1}{4} \sim 5.43\%$, or for 5 years by taking $1 - 0.8 ^ \frac{1}{5} \sim 4.36\%$

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.