Credit Risk Management Practice Exam

What is the formula for cumulative default probability when the hazard rate is constant?

1. 1 − e^−λt

2. λ*t

3. e^λt

4. λ*t + e^λt

The correct answer is: 1 − e^−λt

The formula for cumulative default probability in the context of a constant hazard rate relies on the relationship between the hazard rate and the time until default. When the hazard rate is constant, it can be denoted by λ, which represents the instantaneous probability of default at any given moment in time. The cumulative default probability over a period of time t is derived from the exponential function, specifically showing how the probability accumulates as the time progresses. The formula 1 − e^−λt captures this relationship effectively. The term e^−λt represents the probability of survival without defaulting until time t, and thus subtracting this from 1 gives the cumulative probability of default by that time. This fundamental concept is rooted in survival analysis and is widely utilized in credit risk modeling to assess the likelihood of borrower defaults within a specified timeframe when the risk (hazard) remains constant. The other options present alternative formulations but do not correctly reflect the established relationship, where only the probability of default by time t is represented as 1 − e^−λt.