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Capital Structure in Banks 173
Hence,
EL H
= EA H
– E(EA H
)
= EA H
– [(1 – PD H
) ⋅ EA H
+ PD H
⋅ (EA H
⋅ (1 – LR H
))]
= PD H
⋅ EA H
⋅ LR H
(5.4)
where PD H
= Probability of default up to time H (horizon)
EA H
= Exposure amount at time H
LR H
= Loss rate experienced at time H
E(⋅) = Expected Value of (⋅)
The expected loss experienced at time H (EL H
), that is, at the end of the
predetermined estimation period, is the difference between the promised
exposure amount (EA H
) at that time (including all promised interest payments)
and the amount that the bank can expect to receive at that time—
given that, with a certain probability of default (PD H
) between time 0 and
H, a loss (EA H
⋅ LR H
) will be experienced. 182
Therefore, EL is the product of its three determining components, which
we will briefly describe in turn below:
1. Probability of default (PD): This probability determines whether a
counterparty or client goes into default 183 over a predetermined period
of time. PD is a borrower-specific estimate 184 that is typically linked
to the borrower’s risk rating, that is, estimated independently 185 of
182 This assumes—for the sake of both simplicity and practicability—that all default
events occurring between time 0 and the predetermined period of time ending at H
will be considered in this framework. However, the exposure amount and the loss
experienced after recoveries will be considered/calculated only at time H and not
exactly at the time when the actual default occurs.
183 Default is typically defined as a failure to make a payment of either principal or
interest, or a restructuring of obligations to avoid a payment failure. This is the
definition also used by most external rating agencies, such as Standard & Poor’s and
Moody’s. Independently of what default definition has been chosen, a bank should
ensure an application of this definition of default as consistent as possible across the
credit portfolio.
184 This assumes that either all credit obligations of one borrower are in default or
none of them.
185 This is not true for some facility types such as project finance or commercial real
estate lending where the probability of default (PD) is not necessarily linked to a
specific borrower but rather to the underlying business. Additionally, PD is not
independent from the loss rate (LR – as discussed later), that is, the recovery rates
change with the credit quality of the underlying business. This requires obviously a
different modeling approach (usually a Monte Carlo simulation), whose specifics
will not be discussed in this book.