In order to test the quality of the output monthly TPM we compare TPM121−month to the annual TPM from Table 19 which was used to construct the generator; Table 33 shows the difference between these two matrices in percent. Table 34 shows the relative difference between these two matrices.
We use the following matrix norms as a measure of the distance between the monthly TPM of
AAA AA A BBB BB B D AAA 0.001248 -0.00126 1.83E-05 -1.90E-06 -1.70E-07 -4.60E-08 -7.70E-07 AA 0.000301 -0.00415 0.003831 9.46E-06 1.20E-06 2.17E-07 1.23E-05 A 6.56E-07 -7.40E-05 0.000177 -0.0001 2.42E-07 2.15E-07 5.97E-07 BBB 2.33E-07 0.000515 0.011891 -0.01732 0.00414 0.000415 0.000363 BB 1.52E-07 4.58E-06 0.000442 0.004854 -0.00864 0.002662 0.000674 B 2.23E-09 3.77E-07 6.90E-05 0.000169 0.002829 -0.00466 0.001589
D 0 0 0 0 0 0 0
Table 26: Monthly TPM for the financial sector consistent with Basel PDs: Table 24 minus Table 25.
AAA AA A BBB BB B D
AAA -12.5696% 13.1447% -0.6051% 0.0186% 0.0021% 0.0005% 0.0088%
AA 0.7138% -9.6363% 9.1061% -0.1838% -0.0223% -0.0063% 0.0289%
A 0.0117% 6.2251% -13.2248% 6.0481% 0.7447% 0.1345% 0.0607%
BBB -0.0056% 0.7855% 19.3005% -28.0677% 6.7580% 0.6536% 0.5757%
BB 0.0005% -0.1247% 1.1121% 15.5310% -27.0168% 8.4560% 2.0420%
B -8.05⋅10−6% -0.0153% 0.4648% 0.9171% 22.7612% -36.4100% 12.2822%
D 0 0 0 0 0 0 0
Table 27: The generator G after taking the natural logarithm of the TPM in Table 19; results have been rounded to nearest displayed accuracy. Note that the elements𝑔13,𝑔24,𝑔25,𝑔26,𝑔41,𝑔52,𝑔61, 𝑔62 will be nullified in the next step.
Table 29 raised to the 12-th power and the annual TPM of Table 19:
∥𝐴∥1 = max that though the difference seems large at first, it is actually an exaggeration of the difference:
∥𝐴∥Frobenius, for e.g., should really be divided by 72 = 49, the number of elements in the matrix, to get an estimate of the typical size of the elements in Table 33.
AAA AA A BBB BB B D AAA -12.8651% 12.8358% 0 0.0181% 0.0021% 0.0005% 0.0086%
AA 0.7060% -9.7414% 9.0068% 0 0 0 0.0286%
A 0.0117% 6.2251% -13.2248% 6.0481% 0.7447% 0.1345% 0.0607%
BBB 0 0.7854% 19.2986% -28.0704% 6.7573% 0.6535% 0.5756%
BB 0.0005% 0 1.1095% 15.4952% -27.0790% 8.4365% 2.0373%
B 0 0 0.4647% 0.9169% 22.7564% -36.4177% 12.2796%
D 0 0 0 0 0 0 0
Table 28: The generator ˆ𝐺after rescaling the generator in Table 27 according to (12) – (14); results have been rounded to nearest displayed accuracy.
AAA AA A BBB BB B D
AAA 98.9339% 1.0596% 0.0040% 0.0015% 0.0002% 4.39⋅10−5% 0.0007%
AA 0.0583% 99.1937% 0.7434% 0.0019% 0.0002% 4.23⋅10−5% 0.0024%
A 0.0011% 0.5140% 98.9099% 0.4958% 0.0625% 0.0113% 0.0053%
BBB 2.76⋅10−5% 0.0685% 1.5814% 97.6955% 0.5513% 0.0550% 0.0482%
BB 3.89⋅10−5% 0.0007% 0.1012% 1.2625% 97.7788% 0.6851% 0.1717%
B 5.77⋅10−7% 0.0001% 0.0395% 0.0864% 1.8471% 97.0173% 1.0096%
D 0 0 0 0 0 0 100%
Table 29: Monthly TPM for the financial sector after exponentiating the generator ˆ𝐺in Table 28 according to (15).
AAA AA A BBB BB B D
AAA 0.989366 0.010634 0 0 0 0 0
AA 0.000546 0.991982 0.007472 0 0 0 0
A 1.12E-05 0.00514 0.9891 0.004958 0.000625 0.000113 5.26E-05 BBB 0 0.000684 0.015814 0.976957 0.005513 0.00055 0.000481
BB 0 0 0.000995 0.012635 0.97782 0.006848 0.001702
B 0 0 0.000392 0.000862 0.018473 0.970177 0.010095
D 0 0 0 0 0 0 1
Table 30: Monthly TPM for the financial sector: using the QOM scheme.
AAA AA A BBB BB B D
AAA -2.60E-05 -3.80E-05 3.99E-05 1.49E-05 1.75E-06 4.40E-07 7.26E-06 AA 3.65E-05 -4.40E-05 -3.80E-05 1.87E-05 2.33E-06 4.23E-07 2.39E-05 A -6.50E-09 -1.90E-07 -2.10E-07 3.77E-07 3.06E-08 8.07E-09 -2.00E-09 BBB 2.76E-07 9.30E-07 -9.10E-07 -1.60E-06 2.22E-08 6.12E-07 6.52E-07 BB 3.89E-07 6.70E-06 1.71E-05 -1.00E-05 -3.20E-05 3.06E-06 1.51E-05 B 5.77E-09 1.29E-06 2.38E-06 1.97E-06 -2.00E-06 -3.90E-06 2.47E-07
D 0 0 0 0 0 0 0
Table 31: Monthly TPM for the financial sector: Table 24 minus Table 30.
QOM Generator approach
Table 32: Absolute row sum of the monthly TPM for the financial sector from Tables 24 and 30.
AAA AA A BBB BB B D
AAA -0.2616% -0.2831% 0.5152% 0.0250% 0.0037% 0.0008% 5.09⋅10−6% AA -0.0082% -0.0988% -0.0760% 0.1520% 0.0240% 0.0063% 0.0008%
A -0.0001% -0.0025% -0.0025% 0.0046% 0.0005% 0.0001% -1.48⋅10−5% BBB 0.0045% 0.0031% -0.0023% -0.0023% -0.0022% -0.0006% -0.0002%
BB 0.0007% 0.1048% -0.0008% -0.0314% -0.0507% -0.0166% -0.0059%
B 0.0001% 0.0234% 0.0003% -0.0038% -0.0096% -0.0072% -0.0031%
D 0 0 0 0 0 0 0
AA -0.0129 -0.0011 -0.0093 1.9000 2.3982 – 0.0251 A -0.0038 -0.0005 -2.84⋅10−5 0.0009 0.0006 0.0007 -0.0002 BBB – 0.0027 -0.0001 -3.06⋅10−5 -0.0004 -0.0009 -0.0004
BB – – -0.0004 -0.0026 -0.0007 -0.0027 -0.0026
B – – 0.0005 -0.0019 -0.0006 -0.0001 -0.0003
D 0 0 0 0 0 0 0
Table 34: The relative error(
TPM121−month−TPM of Table 19)
/(TPM of Table 19); dash entries represent cases where the TPM of Table 19 had a zero entry.
Norm Govt. Corp. Fin.
∥ ⋅ ∥1 0.5632% 0.0339% 0.5971%
∥ ⋅ ∥2 0.4742% 0.0419% 0.6460%
∥ ⋅ ∥∞ 0.8375% 0.0709% 1.0894%
∥ ⋅ ∥Frobenius 0.5752% 0.0485% 0.6853%
Table 35: Norms of the difference(
TPM121−month−TPM of Table 19) .
6 Discussion
This paper summarizes most of our exercise to compute TPMs for IRC. There are large uncertain-ties in the computed TPMs due to the lack of sufficiently accurate input data and the multiple ways in which the matrices can be manipulated. Due to varying portfolio composition among different institutions we refrain from making specific recommendations on which method performs best. Therefore, given the importance of TPMs and their PDs in the IRC, financial institutions will need to make discretionary choices regarding their preferred methodology while ensuring that uncertainties are well understood, managed and communicated properly to local regulators.
We also performed other tests and there are still several issues that we need to investigate further.
For example, one of the spurious effects of the manipulations done on the annual generator is to introduce non-zero probabilities in places that had zero probabilities in the original TPM; compare, for e.g., the elements in the first row of Table 29 to those in Table 19. These non-zero probabilities in the first row of Table 29 will introduce non-zero probabilities in the annual TPM in TPM121−month. This poses two problems that at the moment are not addressed. The obvious question is how to correct these non-zero probabilities that defy the zero probabilities in the original TPM. A more subtle issue is the fact that since the original TPM is constructed from historic data it extracts probabilities from a finite number of events; therefore, if we are to address the previous question, perhaps we should first bear in mind that from a formal perspective, a zero probability entry in the original TPM has, in fact, zero probability... The right thing to do is perhaps to replace all zero entries in the original TPM with some error of the data, something like standard error
∼ 𝜎/√
𝑁measurements. That way, the original input data will be given some error bars correcting the misleading perception of its accuracy, and allowing for some leeway in manipulating it in a self-consistent manner.
There are several other future developments that we will continue to investigate. For example, we may continue to look for any other available information that we can integrate to the estimation of TPM; better rating mapping methodologies; other statistic measures to understand the rating migration behavior, etc.
Finally, it may be worthwhile mentioning that the exercise reported in this report can be applied to other projects such as counterparty economic capital calculation. Currently, most such com-putations are based on default only approach and a similar TPM is needed if we want to include migration risk. Contrary to what we have done here, we may adjust sector and rating based TPMs to include idiosyncratic credit risk on the obligor level.
Appendix: TPMs
This appendix brings the relevant TPMs mentioned in the calculations in this document.
Moody’s TPMs
Tables A1 – A3 list the monthly TPMs generated from MCRC. These can be compared against the monthly TPMs resulting from our calculations.
AAA AA A BBB BB B CCC D
AAA 99.69% 0.30% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00%
AA 0.30% 99.56% 0.13% 0.00% 0.00% 0.00% 0.00% 0.00%
A 0.04% 0.43% 99.34% 0.14% 0.05% 0.00% 0.00% 0.00%
BBB 0.00% 0.00% 0.45% 99.13% 0.40% 0.03% 0.00% 0.00%
BB 0.00% 0.00% 0.00% 0.64% 98.95% 0.36% 0.06% 0.00%
B 0.00% 0.00% 0.00% 0.00% 0.73% 97.95% 1.06% 0.26%
CCC 0.00% 0.00% 0.00% 0.00% 0.00% 0.64% 98.95% 0.41%
Table A1: Moody’s monthly TPM for the government sector.
AAA AA A BBB BB B CCC D
AAA 98.77% 1.23% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
AA 0.10% 99.19% 0.70% 0.01% 0.00% 0.00% 0.00% 0.00%
A 0.01% 0.48% 99.03% 0.46% 0.02% 0.00% 0.00% 0.00%
BBB 0.00% 0.07% 1.41% 97.83% 0.62% 0.03% 0.01% 0.02%
BB 0.00% 0.04% 0.15% 1.16% 97.84% 0.76% 0.01% 0.04%
B 0.00% 0.00% 0.03% 0.13% 1.55% 97.13% 0.87% 0.28%
CCC 0.00% 0.00% 0.00% 0.00% 0.00% 2.14% 93.26% 4.61%
Table A2: Moody’s monthly TPM for the financial sector.
AAA AA A BBB BB B CCC D
AAA 99.06% 0.84% 0.09% 0.00% 0.00% 0.00% 0.00% 0.00%
AA 0.03% 99.03% 0.92% 0.02% 0.00% 0.00% 0.00% 0.00%
A 0.00% 0.09% 99.33% 0.56% 0.01% 0.00% 0.01% 0.00%
BBB 0.00% 0.00% 0.31% 99.25% 0.40% 0.03% 0.01% 0.00%
BB 0.00% 0.00% 0.03% 0.53% 98.34% 1.03% 0.03% 0.03%
B 0.00% 0.01% 0.01% 0.04% 0.45% 98.37% 0.77% 0.35%
CCC 0.00% 0.00% 0.00% 0.04% 0.06% 0.67% 96.30% 2.93%
Table A3: Moody’s monthly TPM for the corporate sector.
Tables A4 – A6 list the annual TPMs generated from MCRC. These are used as input for our calculations.
AAA AA A BBB BB B CCC D
AAA 96.50% 3.32% 0.00% 0.05% 0.12% 0.00% 0.00% 0.00%
AA 3.52% 94.97% 1.51% 0.00% 0.00% 0.00% 0.00% 0.00%
A 0.43% 5.22% 93.00% 1.13% 0.21% 0.00% 0.00% 0.00%
BBB 0.00% 0.00% 5.23% 89.98% 4.44% 0.36% 0.00% 0.00%
BB 0.00% 0.00% 0.00% 8.22% 86.64% 2.36% 2.56% 0.23%
B 0.00% 0.00% 0.00% 0.00% 8.33% 82.54% 6.62% 2.50%
CCC 0.00% 0.00% 0.00% 0.00% 0.00% 7.40% 89.57% 3.03%
Table A4: Moody’s annual TPM for the government sector.
AAA AA A BBB BB B CCC D
AAA 88.24% 11.76% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
AA 0.64% 91.11% 8.13% 0.08% 0.01% 0.00% 0.00% 0.03%
A 0.03% 5.59% 88.36% 4.99% 0.79% 0.15% 0.02% 0.07%
BBB 0.00% 1.16% 15.85% 76.40% 5.28% 0.70% 0.00% 0.61%
BB 0.00% 0.00% 2.13% 11.93% 77.46% 6.23% 0.99% 1.27%
B 0.00% 0.00% 0.62% 1.99% 16.69% 70.17% 7.30% 3.22%
CCC 0.00% 0.00% 0.00% 0.00% 4.17% 20.83% 29.56% 45.44%
Table A5: Moody’s annual TPM for the financial sector.
AAA AA A BBB BB B CCC D
AAA 89.23% 9.82% 0.95% 0.00% 0.00% 0.00% 0.00% 0.00%
AA 0.35% 88.97% 10.20% 0.48% 0.00% 0.00% 0.00% 0.00%
A 0.03% 1.04% 92.16% 6.28% 0.35% 0.04% 0.05% 0.04%
BBB 0.01% 0.03% 3.76% 91.37% 3.90% 0.55% 0.25% 0.12%
BB 0.00% 0.02% 0.44% 6.17% 81.87% 9.33% 0.99% 1.18%
B 0.02% 0.04% 0.14% 0.51% 5.14% 81.92% 6.83% 5.39%
CCC 0.00% 0.00% 0.03% 0.00% 1.14% 8.22% 68.57% 22.03%
Table A6: Moody’s annual TPM for the corporate sector.
IRC TPMs: Basel PD Floored Results
Tables A7 – A9 list the monthly TPMs generated by the process described in the text, including flooring PDs at the Basel PDs.
AAA AA A BBB BB B D
AAA 99.6974% 0.2869% 2.01⋅10−4% 4.00⋅10−3% 0.0108% 1.32⋅10−5% 6.49⋅10−4% AA 0.3048% 99.5611% 0.1332% 7.41⋅10−5% 2.79⋅10−5% 8.81⋅10−6% 8.27⋅10−4% A 0.0299% 0.4602% 99.3890% 0.1016% 0.0171% 3.47⋅10−5% 2.26⋅10−3% BBB 7.02⋅10−5% 1.09⋅10−3% 0.4701% 99.0862% 0.4125% 0.0293% 8.29⋅10−4% BB 4.57⋅10−5% 8.24⋅10−4% 1.82⋅10−3% 0.7675% 98.6777% 0.2310% 0.3211%
B 4.09⋅10−7% 6.86⋅10−6% 1.49⋅10−3% 3.19⋅10−3% 0.8193% 98.1120% 1.0640%
Table A7: Monthly TPM for the government sector.
AAA AA A BBB BB B D
AAA 98.9339% 1.0596% 0.0040% 0.0015% 0.0002% 0.0000% 0.0007%
AA 0.0583% 99.1937% 0.7434% 0.0019% 0.0002% 0.0000% 0.0024%
A 0.0011% 0.5140% 98.9099% 0.4958% 0.0625% 0.0113% 0.0053%
BBB 0.0000% 0.0685% 1.5814% 97.6955% 0.5513% 0.0550% 0.0482%
BB 0.0000% 0.0007% 0.1012% 1.2625% 97.7788% 0.6851% 0.1717%
B 0.0000% 0.0001% 0.0395% 0.0864% 1.8471% 97.0173% 1.0096%
Table A8: Monthly TPM for the financial sector.
AAA AA A BBB BB B D
AAA 99.0508% 0.9083% 0.0399% 1.65⋅10−4% 5.18⋅10−5% 2.95⋅10−6% 7.89⋅10−4% AA 0.0323% 99.0214% 0.9303% 0.0146% 1.21⋅10−4% 7.64⋅10−6% 1.23⋅10−3% A 2.54⋅10−3% 0.0948% 99.3054% 0.5665% 0.0213% 1.27⋅10−3% 8.15⋅10−3% BBB 8.63⋅10−4% 8.72⋅10−4% 0.3389% 99.2186% 0.3706% 0.0334% 0.0367%
BB 1.23⋅10−5% 1.52⋅10−3% 0.0299% 0.5859% 98.3042% 0.9349% 0.1436%
B 1.92⋅10−3% 3.64⋅10−3% 0.0114% 0.0317% 0.5150% 98.3259% 1.1105%
Table A9: Monthly TPM for the corporate sector.
Tables A10 – A12 list the annual TPM generated by raising to the power of 12 the monthly TPMs of Tables A7 – A9. Results have been rounded to the nearest displayed accuracy.
AAA AA A BBB BB B D
AAA 96.4847% 3.3062% 0.0277% 0.0501% 0.1196% 0.0017% 0.0101%
AA 3.5147% 94.9526% 1.5086% 0.0101% 0.0039% 0.0001% 0.0101%
A 0.4297% 5.2187% 92.9775% 1.1292% 0.2099% 0.0046% 0.0304%
BBB 0.0111% 0.1462% 5.1903% 89.7869% 4.3971% 0.3560% 0.1123%
BB 0.0010% 0.0134% 0.2366% 8.1485% 85.5301% 2.3356% 3.7348%
B 0.0001% 0.0010% 0.0232% 0.3945% 8.2381% 79.6607% 11.6824%
Table A10: Annual TPM for the government sector generated by raising Table A7 to the power of 12.
AAA AA A BBB BB B D
AAA 87.9684% 11.4769% 0.5152% 0.0250% 0.0037% 0.0008% 0.0100%
AA 0.6318% 91.0112% 8.0540% 0.2320% 0.0340% 0.0063% 0.0308%
A 0.0299% 5.5875% 88.3575% 4.9946% 0.7905% 0.1501% 0.0900%
BBB 0.0045% 1.1631% 15.8477% 76.3977% 5.2778% 0.6994% 0.6098%
BB 0.0007% 0.1048% 2.1292% 11.8986% 77.3993% 6.2134% 2.2541%
B 0.0001% 0.0234% 0.6203% 1.9862% 16.6804% 70.1728% 10.5169%
Table A11: Annual TPM for the financial sector generated by raising Table A8 to the power of 12.
AAA AA A BBB BB B D
AAA 89.2035% 9.8026% 0.9485% 0.0334% 0.0019% 0.0002% 0.0101%
AA 0.3496% 88.9396% 10.1888% 0.4803% 0.0201% 0.0020% 0.0196%
A 0.0300% 1.0399% 92.1483% 6.2800% 0.3501% 0.0400% 0.1117%
BBB 0.0100% 0.0300% 3.7600% 91.2633% 3.9000% 0.5500% 0.4867%
BB 0.0015% 0.0201% 0.4400% 6.1697% 81.8392% 9.3295% 2.1999%
B 0.0200% 0.0400% 0.1400% 0.5100% 5.1400% 81.9300% 12.2200%
Table A12: Annual TPM for the corporate sector generated by raising Table A9 to the power of 12.
IRC TPMs: Non-Floored Results
Tables A13 – A15 list the monthly TPMs generated by the process described in the text, excluding flooring PDs at the Basel PDs (i.e., using Moody’s original PDs for each annual TPM). Results have been rounded to the nearest displayed accuracy.
AAA AA A BBB BB B D
AAA 99.6982% 0.2868% 0.0002% 0.0040% 0.0107% 0.0000% 0.0000%
AA 0.3048% 99.5620% 0.1331% 0.0001% 0.0000% 0.0000% 0.0000%
A 0.0299% 0.4600% 99.3916% 0.1015% 0.0170% 0.0000% 0.0000%
BBB 0.0001% 0.0011% 0.4682% 99.0924% 0.4088% 0.0288% 0.0006%
BB 0.0000% 0.0008% 0.0018% 0.7627% 98.7720% 0.2262% 0.2364%
B 0.0000% 0.0000% 0.0014% 0.0031% 0.8016% 98.3855% 0.8084%
Table A13: Monthly TPM for the government sector.
AAA AA A BBB BB B D
AAA 98.9348% 1.0595% 0.0040% 0.0015% 0.0002% 0.0000% 0.0000%
AA 0.0583% 99.1937% 0.7434% 0.0019% 0.0002% 0.0000% 0.0024%
A 0.0011% 0.5140% 98.9099% 0.4958% 0.0625% 0.0113% 0.0053%
BBB 0.0000% 0.0685% 1.5814% 97.6955% 0.5513% 0.0550% 0.0482%
BB 0.0000% 0.0007% 0.1012% 1.2625% 97.7788% 0.6851% 0.1717%
B 0.0000% 0.0001% 0.0395% 0.0864% 1.8471% 97.0173% 1.0096%
Table A14: Monthly TPM for the financial sector.
AAA AA A BBB BB B D
AAA 99.0517% 0.9082% 0.0399% 0.0002% 0.0001% 0.0000% 0.0000%
AA 0.0323% 99.0230% 0.9300% 0.0146% 0.0001% 0.0000% 0.0000%
A 0.0025% 0.0948% 99.3073% 0.5661% 0.0213% 0.0013% 0.0066%
BBB 0.0009% 0.0009% 0.3387% 99.2292% 0.3704% 0.0334% 0.0266%
BB 0.0000% 0.0015% 0.0299% 0.5854% 98.3073% 0.9347% 0.1412%
B 0.0019% 0.0036% 0.0114% 0.0317% 0.5149% 98.3259% 1.1106%
Table A15: Monthly TPM for the corporate sector.
Tables A16 – A18 list the annual TPM generated by raising to the power of 12 the monthly TPMs of Tables A13 – A15. Results have been rounded to the nearest displayed accuracy.
AAA AA A BBB BB B D
AAA 96.4939% 3.3054% 0.0276% 0.0500% 0.1195% 0.0017% 0.0018%
AA 3.5147% 94.9626% 1.5086% 0.0100% 0.0039% 0.0001% 0.0001%
A 0.4297% 5.2178% 93.0063% 1.1289% 0.2097% 0.0046% 0.0031%
BBB 0.0111% 0.1456% 5.1724% 89.8520% 4.3820% 0.3548% 0.0821%
BB 0.0010% 0.0133% 0.2350% 8.1422% 86.5089% 2.3343% 2.7653%
B 0.0001% 0.0010% 0.0227% 0.3884% 8.2264% 82.3602% 9.0011%
Table A16: Annual TPM for the government sector generated by raising Table A13 to the power of 12.
AAA AA A BBB BB B D
AAA 87.9776% 11.4758% 0.5152% 0.0250% 0.0037% 0.0008% 0.0019%
AA 0.6318% 91.0112% 8.0540% 0.2320% 0.0340% 0.0063% 0.0308%
A 0.0299% 5.5875% 88.3575% 4.9946% 0.7905% 0.1501% 0.0900%
BBB 0.0045% 1.1631% 15.8477% 76.3977% 5.2778% 0.6994% 0.6098%
BB 0.0007% 0.1048% 2.1292% 11.8986% 77.3993% 6.2134% 2.2541%
B 0.0001% 0.0234% 0.6203% 1.9862% 16.6804% 70.1728% 10.5169%
Table A17: Annual TPM for the financial sector generated by raising Table A14 to the power of 12.
AAA AA A BBB BB B D
AAA 89.2134% 9.8025% 0.9484% 0.0333% 0.0018% 0.0002% 0.0004%
AA 0.3495% 88.9568% 10.1865% 0.4802% 0.0201% 0.0020% 0.0049%
A 0.0300% 1.0399% 92.1699% 6.2800% 0.3501% 0.0400% 0.0900%
BBB 0.0100% 0.0300% 3.7600% 91.3800% 3.9000% 0.5500% 0.3700%
BB 0.0015% 0.0201% 0.4400% 6.1697% 81.8692% 9.3295% 2.1699%
B 0.0200% 0.0400% 0.1400% 0.5100% 5.1400% 81.9300% 12.2200%
Table A18: Annual TPM for the corporate sector generated by raising Table A15 to the power of 12.
IRC TPMs: Using Basel PD Results
Tables A19 – A21 list the monthly TPMs generated by the process described in the text, except that we replace all of Moody’s original PDs for each annual TPM with Basel PDs. Results have been rounded to the nearest displayed accuracy.
AAA AA A BBB BB B D
AAA 99.6974% 0.2869% 0.0002% 0.0040% 0.0108% 0.0000% 0.0006%
AA 0.3048% 99.5611% 0.1332% 0.0001% 0.0000% 0.0000% 0.0008%
A 0.0299% 0.4602% 99.3890% 0.1016% 0.0171% 0.0000% 0.0023%
BBB 0.0001% 0.0011% 0.4701% 99.0862% 0.4125% 0.0293% 0.0008%
BB 0.0000% 0.0008% 0.0018% 0.7675% 98.6777% 0.2310% 0.3211%
B 0.0000% 0.0000% 0.0015% 0.0032% 0.8193% 98.1120% 1.0640%
Table A19: Monthly TPM for the government sector.
AAA AA A BBB BB B D
AAA 98.9340% 1.0596% 0.0040% 0.0015% 0.0002% 0.0000% 0.0008%
AA 0.0583% 99.1952% 0.7433% 0.0019% 0.0002% 0.0000% 0.0011%
A 0.0011% 0.5139% 98.9117% 0.4945% 0.0623% 0.0113% 0.0052%
BBB 0.0000% 0.0684% 1.5770% 97.7446% 0.5470% 0.0546% 0.0083%
BB 0.0000% 0.0007% 0.1014% 1.2523% 97.8740% 0.6748% 0.0969%
B 0.0000% 0.0001% 0.0392% 0.0870% 1.8195% 97.1919% 0.8624%
Table A20: Monthly TPM for the financial sector.
AAA AA A BBB BB B D
AAA 99.0508% 0.9083% 0.0399% 0.0002% 0.0001% 0.0000% 0.0008%
AA 0.0323% 99.0214% 0.9303% 0.0146% 0.0001% 0.0000% 0.0012%
A 0.0025% 0.0948% 99.3054% 0.5665% 0.0213% 0.0013% 0.0082%
BBB 0.0009% 0.0009% 0.3389% 99.2186% 0.3706% 0.0334% 0.0367%
BB 0.0000% 0.0015% 0.0299% 0.5859% 98.3043% 0.9339% 0.1446%
B 0.0019% 0.0036% 0.0114% 0.0317% 0.5144% 98.3449% 1.0920%
Table A21: Monthly TPM for the corporate sector.
Tables A22 – A24 list the annual TPM generated by raising to the power of 12 the monthly TPMs of Tables A19 – A21. Results have been rounded to the nearest displayed accuracy.
AAA AA A BBB BB B D
AAA 96.4847% 3.3062% 0.0277% 0.0501% 0.1196% 0.0017% 0.0101%
AA 3.5147% 94.9526% 1.5086% 0.0101% 0.0039% 0.0001% 0.0101%
A 0.4297% 5.2187% 92.9775% 1.1292% 0.2099% 0.0046% 0.0304%
BBB 0.0111% 0.1462% 5.1903% 89.7869% 4.3971% 0.3560% 0.1123%
BB 0.0010% 0.0134% 0.2366% 8.1485% 85.5301% 2.3356% 3.7348%
B 0.0001% 0.0010% 0.0232% 0.3945% 8.2381% 79.6607% 11.6824%
Table A22: Annual TPM for the government sector generated by raising Table A19 to the power of 12.
AAA AA A BBB BB B D
AAA 87.9685% 11.4770% 0.5152% 0.0249% 0.0036% 0.0008% 0.0100%
AA 0.6318% 91.0266% 8.0542% 0.2318% 0.0339% 0.0063% 0.0154%
A 0.0299% 5.5875% 88.3745% 4.9946% 0.7905% 0.1501% 0.0730%
BBB 0.0045% 1.1631% 15.8477% 76.8506% 5.2778% 0.6994% 0.1569%
BB 0.0007% 0.1045% 2.1290% 11.8976% 78.2821% 6.2129% 1.3732%
B 0.0001% 0.0233% 0.6203% 1.9861% 16.6802% 71.6796% 9.0103%
Table A23: Annual TPM for the financial sector generated by raising Table A20 to the power of 12.
AAA AA A BBB BB B D
AAA 89.2035% 9.8026% 0.9485% 0.0334% 0.0019% 0.0002% 0.0101%
AA 0.3496% 88.9397% 10.1888% 0.4803% 0.0201% 0.0020% 0.0196%
A 0.0300% 1.0399% 92.1482% 6.2800% 0.3501% 0.0400% 0.1117%
BBB 0.0100% 0.0300% 3.7600% 91.2633% 3.9000% 0.5500% 0.4867%
BB 0.0015% 0.0201% 0.4400% 6.1697% 81.8392% 9.3295% 2.1999%
B 0.0200% 0.0400% 0.1400% 0.5100% 5.1400% 82.1200% 12.0300%
Table A24: Annual TPM for the corporate sector generated by raising Table A21 to the power of 12.
IRC TPMs: Using Maximum Basel PD Results
Tables A25 – A27 list the monthly TPMs generated by the process described in the text, except that we replace all of Moody’s original PDs for each annual TPM with Basel maximum PDs. Results have been rounded to the nearest displayed accuracy.
AAA AA A BBB BB B D
AAA 99.6974% 0.2869% 0.0002% 0.0040% 0.0108% 0.0000% 0.0006%
AA 0.3048% 99.5611% 0.1332% 0.0001% 0.0000% 0.0000% 0.0008%
A 0.0299% 0.4602% 99.3881% 0.1016% 0.0171% 0.0000% 0.0030%
BBB 0.0001% 0.0011% 0.4706% 99.0820% 0.4147% 0.0296% 0.0020%
BB 0.0000% 0.0008% 0.0018% 0.7713% 98.6009% 0.2348% 0.3903%
B 0.0000% 0.0000% 0.0015% 0.0033% 0.8330% 97.9039% 1.2582%
Table A25: Monthly TPM for the government sector.
AAA AA A BBB BB B D
AAA 98.9339% 1.0596% 0.0040% 0.0015% 0.0002% 0.0000% 0.0008%
AA 0.0583% 99.1947% 0.7434% 0.0019% 0.0002% 0.0000% 0.0014%
A 0.0011% 0.5140% 98.9091% 0.4947% 0.0624% 0.0113% 0.0074%
BBB 0.0000% 0.0684% 1.5776% 97.7399% 0.5477% 0.0548% 0.0115%
BB 0.0000% 0.0007% 0.1014% 1.2539% 97.8551% 0.6789% 0.1099%
B 0.0000% 0.0001% 0.0393% 0.0870% 1.8306% 97.1013% 0.9417%
Table A26: Monthly TPM for the financial sector.
AAA AA A BBB BB B D
AAA 99.6974% 0.2869% 0.0002% 0.0040% 0.0108% 0.0000% 0.0006%
AA 0.3048% 99.5598% 0.1332% 0.0001% 0.0000% 0.0000% 0.0021%
A 0.0299% 0.4603% 99.3837% 0.1017% 0.0171% 0.0000% 0.0072%
BBB 0.0001% 0.0011% 0.4713% 99.0625% 0.4152% 0.0296% 0.0202%
BB 0.0000% 0.0008% 0.0018% 0.7722% 98.6008% 0.2348% 0.3895%
B 0.0000% 0.0000% 0.0015% 0.0033% 0.8330% 97.9039% 1.2583%
Table A27: Monthly TPM for the corporate sector.
Tables A28 – A30 list the annual TPM generated by raising to the power of 12 the monthly TPMs of Tables A25 – A27. Results have been rounded to the nearest displayed accuracy.
AAA AA A BBB BB B D
AAA 96.4847% 3.3062% 0.0277% 0.0501% 0.1196% 0.0017% 0.0101%
AA 3.5147% 94.9526% 1.5086% 0.0101% 0.0039% 0.0001% 0.0101%
A 0.4297% 5.2187% 92.9674% 1.1293% 0.2099% 0.0046% 0.0403%
BBB 0.0111% 0.1464% 5.1952% 89.7421% 4.4012% 0.3563% 0.1476%
BB 0.0010% 0.0135% 0.2374% 8.1524% 84.7408% 2.3366% 4.5185%
B 0.0001% 0.0010% 0.0235% 0.3992% 8.2449% 77.6611% 13.6702%
Table A28: Annual TPM for the government sector generated by raising Table A25 to the power of 12.
AAA AA A BBB BB B D
AAA 87.9684% 11.4769% 0.5152% 0.0250% 0.0036% 0.0008% 0.0100%
AA 0.6318% 91.0215% 8.0542% 0.2318% 0.0339% 0.0063% 0.0204%
A 0.0299% 5.5875% 88.3475% 4.9946% 0.7905% 0.1501% 0.1000%
BBB 0.0045% 1.1631% 15.8477% 76.8076% 5.2778% 0.6994% 0.1999%
BB 0.0007% 0.1045% 2.1291% 11.8978% 78.1091% 6.2130% 1.5458%
B 0.0001% 0.0234% 0.6203% 1.9861% 16.6803% 70.8927% 9.7971%
Table A29: Annual TPM for the financial sector generated by raising Table A26 to the power of 12.
AAA AA A BBB BB B D
AAA 96.4847% 3.3062% 0.0277% 0.0501% 0.1196% 0.0017% 0.0101%
AA 3.5147% 94.9376% 1.5086% 0.0101% 0.0039% 0.0001% 0.0251%
A 0.4297% 5.2187% 92.9174% 1.1293% 0.2099% 0.0046% 0.0903%
BBB 0.0111% 0.1465% 5.1960% 89.5322% 4.4018% 0.3564% 0.3560%
BB 0.0010% 0.0135% 0.2378% 8.1523% 84.7406% 2.3365% 4.5184%
B 0.0001% 0.0010% 0.0236% 0.3993% 8.2447% 77.6608% 13.6705%
Table A30: Annual TPM for the corporate sector generated by raising Table A27 to the power of 12.
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