Jun 24

Beta distribution for credit recovery

by David Harper, CFA, FRM, CIPM

FRM |

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  • FRM AIM (loss given default): "Discuss the beta function …"

Generally, credit risk methods in the FRM treat probability of default (PD) and loss given default (LGD = 1 – recovery rate) as uncorrelated. Put another way, the formula for expected loss implies independence:

Expected loss (EL) = PD * LGD

Recall (from Gujarati Property 3.6) that E[PD*LGD] = E[PD] * E[LGD] only if PD and LGD are independent. If, however, they are positively correlated, then EL is understated. In the FRM curriculum, I can think of only one exception to the assumption of independence:

  • Ong's EL = PD* LGD assumes independence
  • De Servigny's application of the beta function to LGD assumes independence from PD
  • Basel II IRB (while assuming PD correlates to the single factor) assumes independence between PD & LGD (more in this forum thread) (also, blog here with more detail including Fitch's empirical finding that no PD/LGD correlation assumption is warranted.)
  • The only exception is the (first-generation) structural Merton Model in de Servigny/Stulz. Under the structural (balance sheet) approach, LGD can be expressed as a function of firm value and default threshold (debt, or function of debt). This is not independence because here PD & LGD are both commonly a function of [firm assets, asset volatility]. This speaks to a key difference between first-generation and second-generation structural form models: first-generation naively treat PD & LGD as correlated (i.e., as an endogenous variable) while second-generation treat LGD like the other approaches, as independent of PD (exogenous)

Beta distribution for recovery

As de Servigny says, almost everybody seems to use the beta distribution to model recovery/LGD. The beta takes two parameters, the mean and variance, just like the normal. In the EditGrid below are a four plots to illustrate its flexibility. You can see why he says "this type of parametric distribution is very appealing, as it offers a lot of flexibility. It only requires the mean and variance for calibration." But also note the two weaknesses:

  • You can't make it bi-model (like a two-humped camel), it "peaks" only once. (This is the point of the kernel estimation, to overcome this limitation)
  • It doesn't do so well if there is high density at 0.0 or 1.0 (at no recovery or full recovery, which are distributionally extreme but entirely plausible)

EditGrid:

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