Question 14.5: A firm has Standard and Poor’s credit rating of A. Using the...
A firm has Standard and Poor’s credit rating of A. Using the credit migration rates averaged over 1981 to 2009, what is the probability that the firm will have defaulted in two years time? What is the answer using an N times one-year approximation? How do these results compare to the actual two-year default rates for 1981 to 2009?
The firm has a 0.08% chance of defaulting before the end of the first year. Scaling this up to allow for the 4.83% of issuers losing their rating leaves the result at 0.08%. If the firm survives the first year, then the following probabilities can be calculated, again allowing for the issuers losing their ratings:
- the probability that the firm will be promoted to AAA and then default is (0.04%/(1−4.83%)×(0.00%/(1−3.31%)=0.00%;
- the probability that the firm will be promoted to AA and then default is (1.95%/(1−4.83%)×(0.00%/(1−4.00%)=0.00%;
- the probability that the firm will remain at A and then default is (87.05%/(1−4.83%)×(0.08%/(1−4.83%)=0.08%;
- the probability that the firm will be demoted to BBB and then default is (5.47%/(1−4.83%)×(0.26%/(1−6.68%)=0.02%;
- the probability that the firm will be demoted to BB and then default is (0.40%/(1−4.83%)×(0.97%/(1−9.82%)=0.00%;
- the probability that the firm will be demoted to B and then default is (0.16%/(1−4.83%)×(4.93%/(1−11.83%)=0.01%; and
- the probability that the firm will be demoted to CCC-C and then default is (0.02%/(1−4.83%)×(27.98%/(1−14.37%)=0.01%.
Summing these probabilities gives a total of 0.20% over two years.
If the two-year default probability is simply taken to be 2×0.08%, then the result is instead 0.16%, which is close to the value obtained using the migration approach.
However, both of these results are lower than the two-year default rate calculated directly from the data, which is 0.21%.