# Cumulative Default Probability

## Definition

The term Cumulative 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 defined Credit Event up to a particular timepoint.

## Notation

The cumulative default probability can be considered as the primary representation of the Credit Curve as a set of non-decreasing probabilities $q_k$.

## Relationships with related measures

• In terms of the Incremental Default Probability we have $q_T = \sum_{k=1}^{T}{p_k}$ where we denote with $p_k$ the incremental default probability during time $[t_{k-1}, t_k]$
• In terms of the Marginal Default Probability we have $q_T = 1 - \prod_{k=1}^{T} (1 - h_k)$ where $h_k$ is the marginal default probability during period $[t_{k-1}, t_k]$. The marginal default probability is also denoted the Hazard Rate
• In terms of the Survival Probability we have $q_k = 1 - S_k$ where $S_k$ is the survival probability up to point $t_k$