Remember that rather than pairwise correlation we are interested in
an average correlation and define this as follows:
As there are only ‘cross’ elements. We can now substitute this to get
Now, finally, the average loss is observable, together with the loss
volatility. Consequently we can imply the default correlation.
6.5 Other approaches to default
A major approach to portfolio default modelling is not unsurprisingly
a type of mean variance approach under which the default rate is
assumed to have some mean and randomness in the same sense that
a stock has a random nature. The basic requirement is that the data
on defaults is available as a time series, that is, an average and volatil-
ity can be calculated. Once this not inconsiderable foundation is in
place all we need is to link the volatility of default to the volatility of
loss at the portfolio level.
We start off with the assumption that once again there is a homo-
geneous default rate, but this time it can be volatile. We want to find
out the expected loss and the unexpected loss for this portfolio.
The expected loss for an individual counterpart is simply the expos-
ure multiplied by the expected probability of default. The portfolio
expected loss is just the sum over the counterparties:
because the default probability is the same, the expected loss is just
the total exposure times the default probability. In equation form
Consequently, the expected loss does not depend on the properties of
the individual loans. For example, if we have 100 loans within the
portfolio with an exposure of €1 million each and a default probability
of 1 per cent then the expected loss is €1 million. If we had a portfolio
EL E Ed() ().
EL E i() ( ) asset
assets
∑
2
(average average (assets (assets assets)
correlation
22
)
).
Correlation
correlation
(assets assets)
firm ,
firms
firm
firms
(, )
.
ij
jj ii
∑∑
2
244 Credit risk: from transaction to portfolio management
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