Advances in Risk Management

134 A COMPARATIVE ANALYSIS OF DEPENDENCE LEVELS

In this model, the correlation between default events is related to the
correlation between assets values. Here, the latter correlation coefficient is
equal to

corrij=corr(Ai,Aj)

Even if there exist many alternative models for setting the dynamic of the
asset value, we will consider in this paper the usual simple one factor model:

Ai=ρV+

√

1 −ρ^2 εi, (7.1)

whereVfollows a standard normally distributed random variable. It may be
seen as an overall macro-economic factor that influences all the firm values.
ρis a constant between−1 and 1. We will consider positiveρonly because it
is the case most of the time in practice.^2 εiis a standard normally distributed
random variable, specific to the obligori. As usual, we assume that all the
εiare mutually independent and independent fromV.
Therefore, the firm’s value is also normally distributed and

corrij=corr(Ai,Aj)=ρ^2. (7.2)
In order to simulate the portfolio loss distribution, we follow these
successive steps:

1 For any firmi, we get its mean historical default probabilitypiat the

horizon T, as given by the rating agencies (here Standard & Poor’s).

2 We calculate the barrierli=−^1 (pi) whereis the cumulated distribution

function of a N(0, 1) (see (7.1)).

3 We generate some random variablesVfor the whole portfolio andεifor

every firm. Both are N(0, 1). Then, we compareρV+

√

1 −ρ^2 εiwithli

and record if ai’s default is triggered or not.

4 We finally cumulate the losses and repeat the same procedure many times

in order to get the loss distribution.

The calibration will be done onρ. Below is an example of what we get
with the following parameters:

ρ=

√

0 .2 (the choice promoted by Basel 2).

A time-horizonT=1 year.

One year default probabilities given by Standard & Poor’s in Table 7.1:

A portfolio of 100 firms:^3

• 10 firms rated AAA


• 20 firms rated AA