6

The fundamentals

of credit

Methodologies

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 PD.

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,

1

called the unexpected loss

UL of

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%

UL

222

222

,

,

%,

.

2

1400 000€

EL 2% 10 000 000 98% 0 200 000,

234 Credit risk: from transaction to portfolio management

€10 million

€0

€10 million

Default

No default

Figure 6.1 The outcomes for the loan.

1

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

be stochastic.

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)

22

2

1

() ].

(Volatility of loss) exposure) default probability

(1 default probability

22

(

).

(Volatility of loss) default probability (exposure expected loss

default probability (0 expected loss

2 2

2

1

)

()).

Volatility prob. return (asset return average return)

2

number of

returns

∑

Expected loss default probability exposure.

The fundamentals of credit 235

10 million

10 million

?

No default

Default

Loss volatility

Figure 6.2 Including volatility after default.

Get *Credit Risk: From Transaction to Portfolio Management* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.