Overview
Loan defaults refer to the event when the obligor on a debt, such as a loan mortgage, fails to pay according to the terms of the loan. At such a point, the obligor is judged to be in default.
Default does not necessarily mean that the lender has lost all future payments. The obligor may cure the loan by paying the missed payments. However, institutions will still experience financial consequences as a result of the delayed timing of any payments, plus any accounting loss the institution takes as a result of writing down the loan.
As such, the primary concern of institutions is not modeling defaults per se, but the losses that occur due to a default. Most models will incorporate the following concepts.
- PD - probability of default
- LGD - loss given default. The percentage of the remaining value of the loan that the lender recoups.
- EAD - exposure at default. This is the size of the remaining value of the loan at the time the default occurs. That is, the exposure is much less at the end of the loan than at the beginning.
Of the three concepts, typically the probability of default is the one that gets the most attention, and as with any concept, there are multiple ways to model it.
Default Statistics
Given a PD/LGD/EAD model, it is common to compute the moments of the implied loss distribution.
We model the loss on a loan as the variable {% L %}
{% L=EAD \times LGD \times D %}
If the variables are independent
{% \mathbb{E} [L]= \mathbb{E} [EAD] \times \mathbb{E} [LGD] \times PD %}
Unexpected Loss
{% UL = \sqrt{Var(L)} = \sqrt{Var(EAD \times LGD \times L)} %}
Assuming that {% LGD %} and default {% D %} are
independent,
and given that {% EAD %} is deterministic, the follwoing relation holds
{% UL = EAD \times \sqrt{Var(LGD) \times PD + \mathbb{E}[LGD]^2 \times PD(1-PD)} %}
(Bluhm pg. 22)