# Incremental Default Probability

## Definition

The term Incremental Default Probability is used in the context of multi-period credit risk analysis to denote the likelihood that a legal entity is observed to have experienced a Credit Event during a defined period of time.

The incremental default probability can be considered as another building block of the Credit Curve, alternative to the Cumulative Default Probability representation.

## Notation

Observing whether an entity is defaulted over a period ${\displaystyle [t_{k-1},t_{k}]}$, the incremental default probability is denoted ${\displaystyle p_{k}}$.

In the context of a Credit Rating System, incremental default probabilities} during period k, given an initial rating of m, are denoted by

${\displaystyle {\mbox{PD}}_{t,k}^{m}=p_{k}^{m}=1_{E}[1_{\{R_{k}=D\}}|F_{t},R_{0}=m]=P(R_{k}=D|F_{t},R_{0}=m)}$

## Relationships with related measures

• In terms of the Cumulative Default Probability we have ${\displaystyle p_{k}=q_{k}-q_{k-1}}$ where we denote with ${\displaystyle q_{k}}$ the cumulative default probability up to time ${\displaystyle t_{k}}$
• In terms of the Marginal Default Probability we have ${\displaystyle p_{k}=h_{k}*(1-q_{k-1})}$ where ${\displaystyle h_{k}}$ is the marginal default probability during period ${\displaystyle [t_{k-1},t_{k}]}$. The marginal default probability is also denoted the Hazard Rate
• In terms of the Survival Probability we have ${\displaystyle p_{k}=S_{k-1}-S_{k}}$ where ${\displaystyle S_{k}}$ is the survival probability up to point ${\displaystyle t_{k}}$

## Issues and Challenges

• It is important to distinguish the incremental default probability from the Marginal Default Probability which is conditional on no default prior to the current period.