Once the implied CFADS volatility  \sigma_{CFADS} is calibrated, it is straightforward to compute the probability that the CFADS become smaller than a certain threshold given all scenarii with this value of \sigma.

We report the following two default probabilities credit metrics:

Technical defaults

PD0: the probability of CFADS going below the default threshold.

\[PD0 = \sum_{j \in D_{1}}^{}{p_{1,j}\left( t = 1 \right)}\]

where D_{1} is the set of the defaulting scenarios during the next year.

Hard defaults

PD1: the probability of CFADS going to the scenarios where the a 'workout' fails and a hard default ensues.

\[PD1 = \sum_{j \in L_{1}}^{}{p_{1,j}\left( t = 1 \right)}\]

where L_{1} is the set of the failed restructuring/workout scenarios during the next year.

Technical note

The computed metrics PD0 and PD1 have acceptable discrimination power between predicted and realised default (PD0's AUC is approximately 0.9 and PD1's AUC 0.7).

Still, as is often the case for other asset classes such as listed equities, implied volatility tends to be higher than realised volatility and this would overestimate the PDs when comparing
against the historical observations. We introduce a scaling factor to adjust the implied CFADS volatility to align PD estimation with historical data.

Given this scaling factor k, the final PD output is computed as

\[\tilde{PD0} = 1 - N(k*N^{- 1}\left( 1 - PD0 \right))\]

\[\tilde{PD1} = 1 - N(k*N^{- 1}\left( 1 - PD1 \right))\]

where \(N(.)\) and \(N^{- 1}(.)\) are the cumulative probability function (CDF) and the inverse CDF of the standard normal distribution respectively.

The constant \(k\) is a hyper-parameter as the VOL_ADJUSTMENT in the appendix.

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