of two loans with an exposure of €50 million with the same default

probability of 1 per cent and one of the counterparties defaults the

loss will be €500 000. However the expected loss for the other is the

same so the total expected loss is still €1 million despite the different

structure of the portfolio.

The loss volatility is the sum of the individual loss volatilities for the

very special case that the portfolio is equally weighted and the obligors

have no correlation.

The fact that all the issuers depend on the same default rate enables

some simpliﬁcation because the volatility of the loss for an individual is

just the exposure times the volatility of the default rate. This is very dif-

ferent from the binomial approach where the default rate is constant

and the loss just has two outcomes. Thus substituting in the constant

volatility:

and

For the example above if the default probability volatility was 3 per cent

then the unexpected loss would be €3 million.

Volatility with different rates which vary and are correlated

The next level of complexity is to assume a variable default rate but

further to assume there is some correlation between obligors. The nat-

ural assumption is to break the portfolio down into sub-portfolios

which have a common default rate. So in essence we have a small

number of classes of known default rates. But within each class the

rate is random. Let us ﬁrst consider the case where our portfolio can

be divided into two sub-portfolios.

In this situation the default probability for sector one is d

1

and for

two is d

2

. The expected loss for this portfolio is

and the unexpected loss on the portfolio is

Volatility loss

correlation

2

1

2

12

2

212 1

2

12

( ) () ( ) ()

() (,).

EdEdEEd

d

EL E Ed E Ed() ( ) ( )

112 2

Volatility(loss assets exposure )(). d

Volatility loss exposure asset

asset

assets

222

() ( ) ()id

i

∑

The fundamentals of credit 245

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