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 the building block of the Credit Curve. Observing whether an entity is defaulted over a period $[t_{k-1}, t_k]$, the incremental default probability is denoted $p_k$

NB: 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.

Notation

Incremental default probabilities} during period k, given an initial rating of m, are denoted by

$\mbox{PD}^{m}_{t,k} = p^m_k = 1_E [ 1_{\{ R_{k} = D \}} | F_t, R_0=m] = P(R_k=D | F_t, R_0=m)$

Relationships with related measures

• In terms of the Cumulative Default Probability we have $p_k = q_k - q_{k-1}$ where we denote with $q_k$ the cumulative default probability up to time $t_k$
• In terms of the Marginal Default Probability we have $p_k = h_k / (1 - q_{k-1})$ 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 $p_k = S_{k-1} - S_k$ where $S_k$ is the survival probability up to point $t_k$

---