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 $[t_{k-1}, t_k]$, the incremental default probability is denoted $p_k$.

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

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

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 $t_k$

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.

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