Advances in Risk Management

136 A COMPARATIVE ANALYSIS OF DEPENDENCE LEVELS

Table 7.2One-year default events correlations between firms as a

function of their ratings (%), withρ=

√

0. 2

AAA AA A BBB BB B CCC

AAA 0.27 0.27 0.32 0.58 0.80 1.04 1.09

AA 0.27 0.27 0.32 0.58 0.80 1.04 1.09

A 0.32 0.32 0.38 0.69 0.96 1.27 1.35

BBB 0.58 0.58 0.69 1.33 1.94 2.70 3.06

BB 0.80 0.80 0.96 1.94 2.90 4.20 5.02

B 1.04 1.04 1.27 2.70 4.20 6.42 8.23

CCC 1.09 1.09 1.35 3.06 5.02 8.23 11.65

Forρ=

√
0 .2, we also calculate the linear correlation between the default
events for couples of firms that belong to pre-specified rating classes. The
results are gathered in Table 7.2. In the Appendix we explain how we calcu-
late such correlations. As empirically measured previously, the correlation
levels we get among speculative grade firms are higher than those obtained
with investment firms. They cover a range between 0.7 percent up to 11.6
percent, which is coherent with the empirical literature (de Servigny and
Renault, 2002).

7.3 INTENSITY-BASED MODELS

Such models are based on a direct evaluation of the intensity processes them-
selves. We are reminded that the default intensity is the instantaneous arrival
rate of default:

λ(t)= lim

t→ 0

1

t

P(τ∈[t,t+t]|τ>t)

denoting byτthe default time. Letfbe the probability density function ofτ
andSits survival function. For every timet, we have obviously:

λ(t)=

f(t)

S(t)

Just as the densityf, the functionsλandSdetermine the law ofτ, because

S(t)=exp

(

−

∫t

0

λ(s)ds

)

The model we consider now belongs to the well-known frailty models
family (Clayton and Cuzick, 1985). It has been used extensively in Survival
Analysis (Hougaard, 2000). Frailty models are extensions of the Cox model