That's a great point
@Dr. Jayanthi Sankaran : Basel's own internal ratings-based (IRB) approach is derived from the Merton model (i.e., it computes the default probability for an obligor based precisely on the above Merton model theory that an obligor's asset return has a normal/Gaussian distribution and will default if it falls below are certain quantile). But I would emphasis the "
derived from" as Basel's IRB is not the same Merton as I outlined above. Above I outlined the classic Merton which estimates the PD for a single asset (and the associated KMV variation). Basel's IRB is certainly based on the same theory but extends (or really just begins from it) using a so-called ASRF framework (which makes huge assumptions about a
portfolio of obligors for basically the sole purpose of achieving the convenience of portfolio invariance). In the way, the way I would put it is,
Basel's IRB extends Merton's by utilizing a Gaussian copula (another convenience purchased at the cost of simplifying assumptions), where the
asset's correlation to the systematic risk factor becomes a key assumption. In this portfolio-based model (i.e., Basel's IRB), under both the Foundation and Advanced IRB, the "average default probability" is estimated by the bank; technically, the IRB maps an average PD to a conditional PD. My point is: in Basel IRB, PD is an
estimated input (just as Dr. Jayanthi says!) into a model that utilizes Merton's theory, compared to my description of the classic single-asset model which generates PD as the
output.
That's one of the fascinating aspects, to me, of the IRB model. If you look at the IRB formula, it begins with N(N^-1(pd) .... if the asset correlation is zero (which is not allowed, there is a floor), then the numerator would start with this sort of dynamic: supply a PD of 2% and retrieve = N[N^-1(pd)] = NORM.S.DIST(NORM.S.INV(2%), TRUE) = 2.0%. Ha! The PD can be estimated any number of ways in order to generate the IRB input. Once it gets to the IRB, it's a Gaussian copula which is "merely using" the multivariate normal distribution to achieve the supreme convenience of relying all obligors on a single-factor (ie, correlation to systematic risk) so it's really about the copula more so at that point, and to an extent the multivariate normal distribution.
I am less qualified to say how many banks use a Merton-type model to derive these initial "average" PDs which become inputs in the PD. My understanding is that a pure Merton (specifically the lognormal asset value assumption) significantly underestimates the actual (and historical) defaults (as mentioned by Moody's above: "However, actual default experience departs significantly from the predictions of normally distributed DDs. For example, when a firm’s DD is greater than 4, a normal distribution predicts that default will occur 6 in 100,000 times. Given that the median DD of the entire sample of firms in the EDF dataset is not far from 4, this would lead to about one half of actual firms being essentially default risk-free. This is highly improbable"). So I would expect where Merton's
theory is applied, it's applied in a way similar to Moody's mapping of DD to PD; or alternatively, a heavier-tail distribution can be assumed. I hope that's additive!
(see "Tap Dancing to Work: Warren Buffett on practically everything" by C. Loomis.)
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