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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