Advances in Risk Management

144 A COMPARATIVE ANALYSIS OF DEPENDENCE LEVELS

200

0

4

8

12

% Frequency

16

20

24

600 1000 1400

Losses

alpha 0  0.5 et alpha  0.5

Figure 7.4 Histogram of the losses in the multi-factor intensity-based

model (T=1 year)

first model. We just have to draw the realizations of additional gamma ran-
dom variables (one for each obligor). In practice, there are now two free
parametersα 0 andαi, related toZ 0 andZirespectively. This may cause
some estimation complications, even if the log-likelihood of the observa-
tions can be written in closed form (Parner, 1998). In Figure 7.4, we draw
the histogram of the losses obtained with model (7.10). Since we impose
that the expectation of the global frailty componentZ 0 +Ziequals one,
we drawZ 0 ∼G(α 0 ,α 0 +α) and Zi∼ G(α, α 0 +α). We have chosen the
parameter valuesα 0 = 0 .5 andα= 0 .5 for everyiin Figure 7.4.
In this case

Var(Z 0 +Zi)=α 0 /(α 0 +α)^2 +α/(α 0 +α)^2 = 1

and the correlated frailtyZ 0 +Zihas the same two first moments as in
Figure 7.2. The loss distributions seem to be very similar. At first glance,
the introduction of specific components does not lessen too much the
dependence between defaults.^9