Advances in Risk Management

152 A COMPARATIVE ANALYSIS OF DEPENDENCE LEVELS

APPENDIX B: EXTENSIONS OF THE RESULTS TO LARGE TIME

HORIZONS

We use the same method as in sections 7.2 and 7.3. We choose the following default
probabilities:

Average default rates over 1981–2002
Rating CCC B BB BBB A AA AAA
PD (%) (5 years) 61.35 33.02 14.45 3.83 0.75 0.27 0,11
PD (%) (20 years) 73.94 67.68 48.09 19.48 6.78 4.38 1.13
Source: Standard & Poor’s Credit Pro.
We obtain default event correlations for the time horizonsT=5 years andT= 20
years (Tables 7.7, 7.8, 7.10 and 7.11), and the usual dependence indicators withT= 10
years (Tables 7.9 and 7.12). In the latter case, particularly, the scope of values obtained in
both cases is similar.
Note that when the time horizonTis increasing, it is surely questionable to assume the
same valuesρandαas whenT=1 year apply. Indeed, in the Merton-style models there
is some empirical evidence that the asset correlations depend onT(see the discussion in
de Servigny and Renault, 2002, for example).
Moreover, since we assumed the random default intensitiesλiare constant func-
tions between 0 andT, their (random) levels should be less and less variable whenTis
increasing.^13 It should be more relevant to simulate an annual process (Zt) for the frailty,
but this does not belong in our simple framework. Thus, a realistic range ofα-values is

Table 7.7 5-years default events correlations in the Merton model, withρ=

√

0 .2 (%)

AAA AA A BBB BB B CCC

AAA 0.63 0.82 1.10 1.59 1.89 1.85 1.51

AA 0.82 1.10 1.48 2.21 2.68 2.67 2.22

A 1.10 1.48 2.04 3.12 3.90 3.98 3.40

BBB 1.59 2.21 3.12 5.05 6.67 7.11 6.37

BB 1.89 2.68 3.90 6.67 9.32 10.43 9.84

B 1.85 2.67 3.98 7.12 10.43 12.15 12.01

CCC 1.51 2.22 3.40 6.37 9.84 12.01 12.53

Table 7.820-years default events correlations in the Merton model, withρ=

√

0 .2 (%).

AAA AA A BBB BB B CCC

AAA 2.60 3.68 4.02 4.64 4.41 3.78 3.50

AA 3.68 5.41 6.01 7.22 7.20 6.35 5.92

A 4.02 6.01 6.70 8.17 8.30 7.40 6.93

BBB 4.64 7.22 8.17 10.41 11.16 10.27 9.73

BB 4.41 7.20 8.30 11.16 12.81 12.28 11.81

B 3.78 6.35 7.40 10.27 12.28 12.09 11.74

CCC 3.50 5.92 6.93 9.73 11.81 11.74 11.43