6.1 The standalone loan
The average loss expected on an individual loan. This is given by:
where EL is the yearly expected loss, exposure is the amount of the loan,
EDF is the expected default frequency and loss severity is the amount
recoverable in the event of a default. The word EDF is quite intimidating
but is simply the probability of default in a 1 year period.
This equation is already far too complicated and needs to be broken
down further. Let us just consider the case of a loss severity of 100
per cent and an exposure of €1 then we can write the expected loss as
Now, we only have one abstract variable which is just the probability
of default. To understand why this is the case, let us apply the bino-
mial model of default (over a 1 year period) to the case of a single loan
of €10 million. It is based on the quite natural assumption that the
loan either defaults or it does not. If it defaults we get nothing back
and if it does not we have our original exposure of €10 million. This
arrangement is displayed in Figure 6.1 for the case where the default
probability is 2 per cent.
EL exposure EDF loss severity,
Then we can determine our expected loss as follows:
that is the expected loss is just the default probability scaled.
Now we introduce the notion of the unexpected loss, which is the
variation of the expected loss. This is very simple to evaluate because
there are only two possible outcomes within a binomial model either a
loss or no loss. This will give a volatility,
called the unexpected loss
This will be formalized into a general expression for the expected loss
and the volatility of the expected loss. Again assume the losses are
binomial and that the default probability is constant. We consider two
distinct cases (the basis of the distinction is the amount lost given a
default). In the ﬁrst case we examine a recovery rate of zero.
The probability that the asset defaults within the horizon period
under consideration is
Expected loss default probability exposure
(1 default probability) 0.
UL 2% (10 000 000 200 000) 98% (0 200 000)
UL 2% (9 800 000) 98% (200 000)
UL 10 000 000 98%
EL 2% 10 000 000 98% 0 200 000,
234 Credit risk: from transaction to portfolio management
Figure 6.1 The outcomes for the loan.
Notice this is a very special case of a constant default rate. There is a volatility
of loss but no volatility in the default rate, more generally the default rate will itself
This simpliﬁes down because one half is zero:
The volatility of this expected loss is given by the standard derivation.
This is just
in the context of asset returns (since this is the application that the
majority of readers will be familiar with). For our topic the loss replaces
the return, and thus average loss replaces the average return. As we
have a binomial process the counter stops at two and so we have:
This simpliﬁes down into
Now we relax the assumptions on constant recovery and countenance
extra volatility because the losses, given a default have a distribution,
in this case the volatility of loss is modiﬁed according to the formula:
We depict this situation graphically as shown in Figure 6.2.
(Volatility of loss) (exposure) [default probability
default probability default probability (volatility)
(Volatility of loss) exposure) default probability
(1 default probability
(Volatility of loss) default probability (exposure expected loss
default probability (0 expected loss
Volatility prob. return (asset return average return)
Expected loss default probability exposure.
The fundamentals of credit 235
Figure 6.2 Including volatility after default.