Default probability definitions:
di  hazard rate – marginal probability of default in period
(1-di)  marginal probability of surviving period.
cN = 1 – SN  cumulative probability of default through period N.
N
SN =  (1  d i )  cumulative probability of surviving through period N.
i 1
kN = SN-1*dN  conditional marginal default probability, i.e. chance of surviving through
period N-1 and defaulting in period N.
For some applications it is desirable to express average per period default probability
over 0 ----- N, as an average hazard rate
d a  1  (1  C N )1 / N
Default probabilities may be extracted from rating agency (Standard and Poor’s, Moody’s
Duff and Phelps, etc.) cumulative default studies or they can be extracted from
information provided by the rating agencies on transition probabilities.
Transition probabilities are another means to estimate hazard rates (d), cumulative default
probabilities (c), and conditional marginal default probabilities (k) from actuarial data.
Transition matrices are available from credit rating agencies and indicate the probability
of a credit risk of a given rating migrating to a specific credit rating by the end of the
period. If the process of credit migration is a Markov process (i.e. the conditional
likelihood of migrating to a specific credit rating from the beginning of period credit
rating is independent of the path taken to the beginning of the period), hazard rates may
be found from simple summing of the probabilities of all possible paths leading to default
and cumulative default probabilities may be found from summing of hazard rates.
The Markov assumption implies that conditional on beginning of period credit rating
distribution of default is constant over time, i.e. identically and independently distributed.
Credit rating transition probabilities, Table 19-5 page 425 Jorion
A
B
C
Def
total
A
0.97
0.03
0
0
1
B
0.02
0.93
0.02
0.03
1
C
0.01
0.12
0.64
0.23
1
Def
0
0
0
1
1
1
Hazard rates, cumulative default probabilities,
conditional marginal default probabilities for a B
rated credit risk for one, two and three period
horizons
period (0,1)
d1
c1
k1
B_D
3.0000% 3.0000% 3.0000%
period (1,2)
B_A_D
0.0000%
B_B_D
2.7900%
B_C_D
0.4600%
d2
c2
k2
3.2500% 6.2500% 3.1525%
period (2,3)
B_A_A_D
0.0000%
B_A_B_D
0.0018%
B_A_C_D 0.0000%
B_B_A_D
B_B_B_D
B_B_C_D
0.0000%
2.5947%
0.4278%
B_C_A_D
B_C_B_D
B_C_C_D
0.0000%
0.0072%
0.2944%
d3
c3
k3
3.3259% 9.5759% 3.1180%
2