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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