Calculating Default Probability Across a Specific Time Interval
Summary
The document clarifies how to calculate the probability of surviving the first year and then defaulting during the second when the hazard rate is constant. It distinguishes that joint interval event from a conditional probability of default given survival to a specified date. With a constant hazard rate, cumulative default probability by time t is one minus the probability of survival to t. Subtracting cumulative default probability through year one from cumulative default probability through year two gives the probability that default occurs after year one and by year two.
The accepted explanation formalizes the event using the default time and the interval from year one to year two. The reported result follows from the difference between the two cumulative probabilities. The key caveat is terminology: a marginal or conditional default probability may refer to a rate or probability conditioned on survival, whereas the question asks for an unconditional probability over a particular interval. The distinction depends on interpreting the event as survival first, followed by default in year two.
Key ideas
- Cumulative default probability measures default by a given time, while survival probability is its complement.
- The probability of default during an interval is the later cumulative probability minus the earlier one.
- A probability conditional on survival to a date differs from the unconditional probability of an interval event.
- The calculation assumes a constant hazard rate across the periods.
Tags
Full text
# Cumulative vs marginal probability of default
# Cumulative vs marginal probability of default
I understood the cumulative (aka unconditional) probability of default to be the probability of defaulting in a given period eg: between years 1 and 5. Further $\pi_{cumulative} = 1-e^{-\lambda*t}$ where lambda is a hazard rate.
I understood the marginal (aka conditional) probability of default to be the probability of defaulting at time $T$ given survival up to that point. Further $\pi_{marginal} = \lambda e^{-\lambda*t}$ where lambda is a hazard rate.
Attempting to solve the following problem, I came up with a close but off value.
Problem
> 1 year hazard rate = 0.1. What is the probability of surviving in the first year followed by defaulting in the second?
My solution was to calculate the marginal probability of default = $0.1\lambda e^{0.1*2}$ = 8.19%
But the given answer was 8.61% arrived at by:
1 year cumulative (also called unconditional) PD = 1 - e^(- hazard*time) = 9.516%
2 year cumulative (also called unconditional) PD = 1 - e^(- hazard*time) = 18.127%
solution - 18.127% - 9.516% = 8.611%
Is my approach incorrect or merely an approximation?
## Answer by Gordon (score 3, accepted)
https://quant.stackexchange.com/a/21820
The question sounds like a conditional probability problem. However, note that, for conditional probability, people will generally say if survived to or conditional on. Here it says that survived in year one and (i.e., followed by) will default in year two. Then we should not treat this as a conditional or marginal probability.
Based on the above understanding, the probability can be computed as follows: \begin{align*} P(\tau >1 \ and \ \tau \le 2) &= P(1 < \tau \le 2)\\ &=P\big((\tau \le 2) \setminus(\tau \le 1) \big)\\ &=P(\tau \le 2) - P(\tau \le 1)\\ &= \big(1- e^{-2\lambda}\big) - \big(1- e^{-\lambda}\big)\\ &= 18.127\,\% - 9.516\,\% \\ &= 8.611\,\%. \end{align*} Here, $\tau$ is the default time.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.