Portfolio Credit Risk -Random Correlation Matrix
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Portfolio.Distributi Exposure Weights PD AAA 1,154,172.76 1.34% 0.00% AA+ 7,465,389.06 8.69% 0.03% AA 3,230,347.74 3.76% 0.03% AA- 2,454,446.60 2.86% 0.03% A+ 6,865,322.46 7.99% 0.70% A 1,534,429.39 1.79% 0.70% A- 663,563.63 0.77% 0.70% BBB+ 5,761,189.94 6.70% 1.48% BBB 1,064,356.08 1.24% 1.48% BBB- 1,807,724.13 2.10% 1.48% BB+ 1,557,220.48 1.81% 4.98% BB 6,511,702.91 7.58% 4.98% BB- 6,752,740.90 7.86% 4.98% B+ 9,425,868.83 10.97% 8.86% B 8,028,625.15 9.34% 8.86% B- 622,731.35 0.72% 8.86% CCC 9,037,938.96 10.52% 18.89% CC 37,663.26 0.04% 18.89% C 9,456,525.20 11.00% 18.89% D 2,516,245.06 2.93% 100% Total exposure ### Krishna To note 1) In th MATLAB f platform 2. As we level is negative 3. In re 4. Howev Portfoli
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Transcript of Portfolio Credit Risk -Random Correlation Matrix
Portfolio.Distribution Exposure Weights PD LGDAAA 1,154,172.76 1.34% 0.00% 10%AA+ 7,465,389.06 8.69% 0.03% 10%AA 3,230,347.74 3.76% 0.03% 10%AA- 2,454,446.60 2.86% 0.03% 20%A+ 6,865,322.46 7.99% 0.70% 45%A 1,534,429.39 1.79% 0.70% 60%A- 663,563.63 0.77% 0.70% 60%BBB+ 5,761,189.94 6.70% 1.48% 60%BBB 1,064,356.08 1.24% 1.48% 60%BBB- 1,807,724.13 2.10% 1.48% 75%BB+ 1,557,220.48 1.81% 4.98% 75%BB 6,511,702.91 7.58% 4.98% 75%BB- 6,752,740.90 7.86% 4.98% 75%B+ 9,425,868.83 10.97% 8.86% 75%B 8,028,625.15 9.34% 8.86% 75%B- 622,731.35 0.72% 8.86% 75%CCC 9,037,938.96 10.52% 18.89% 90%CC 37,663.26 0.04% 18.89% 90%C 9,456,525.20 11.00% 18.89% 95%D 2,516,245.06 2.93% 100% 100%Total exposure 85,948,203.90
Krishnan Chari:To note1) In the absence of any real market Default correlation data , a definite positive correlation matrix has been simulated with the help of MATLAB functions using Random Matrix Theory. The algorithm for generating Random Matrices using Eigen values are available in opensoure R platforms also.
2. As we generated a synthetic Random Matrice with sufficient negative correlations between rating categories, the Credit VaR at portfolio level is lesser than the expected losses at individual asset exposure level. This situation arises as the synthetic correlation matrix has negative correlation numbers in equal symmetry thereby creating an ideal diversification.
3. In reality, Banks may never acheive this level of diversification.
4. However this template enables the Bank to calibrate an internal credit correlations matrix and use the same in the mdoel to generate Portfolio Credit Risk using default or asset credit corrrelations
Variance.of.LGD Variance.of.PD Expected.Loss Unexpected.Loss(assuming.Corre=0)0% 0 - - 0% 0.00029991 223.96 12,928.49 0% 0.00029991 96.91 5,594.29 0% 0.00029991 147.27 8,501.18 7% 0.006951 21,625.77 299,061.63 8% 0.006951 6,444.60 84,913.17
10% 0.006951 2,786.97 37,550.67 10% 0.01458096 51,159.37 472,598.30 10% 0.01458096 9,451.48 87,310.59 10% 0.01458096 20,065.74 177,872.98 10% 0.04731996 58,162.19 276,806.52 10% 0.04731996 243,212.10 1,157,499.43 10% 0.04731996 252,214.87 1,200,345.57 10% 0.08075004 626,348.98 2,196,083.77 10% 0.08075004 533,502.14 1,870,547.29 10% 0.08075004 41,380.50 145,086.91 10% 0.15321679 1,536,540.00 3,417,674.84 10% 0.15321679 6,403.13 14,242.27 10% 0.15321679 1,697,020.73 3,748,986.35 10% 0 2,516,245.06 795,706.55
7,623,031.77 16,009,310.79 Undiversified VaR 16,009,310.79
4,946,279.50 Expected Losses 7,623,031.77 Economic Capital -2,676,752.27
Diversified VaR ( Factoring default Asset correlations)
Krishnan Chari:To note1) In the absence of any real market Default correlation data , a definite positive correlation matrix has been simulated with the help of MATLAB functions using Random Matrix Theory. The algorithm for generating Random Matrices using Eigen values are available in opensoure R platforms also.
2. As we generated a synthetic Random Matrice with sufficient negative correlations between rating categories, the Credit VaR at portfolio level is lesser than the expected losses at individual asset exposure level. This situation arises as the synthetic correlation matrix has negative correlation numbers in equal symmetry thereby creating an ideal diversification.
3. In reality, Banks may never acheive this level of diversification.
4. However this template enables the Bank to calibrate an internal credit correlations matrix and use the same in the mdoel to generate Portfolio Credit Risk using default or asset credit corrrelations
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Krishnan Chari: To note 1) In the absence of any real market Default correlation data , a definite positive correlation matrix has been simulated with the help of MATLAB functions using Random Matrix Theory. The algorithm for generating Random Matrices using Eigen values are available in opensoure R platforms also. 2. As we generated a synthetic Random Matrice with sufficient negative correlations between rating categories, the Credit VaR at portfolio level is lesser than the expected losses at individual asset exposure level. This situation arises as the synthetic correlation matrix has negative correlation numbers in equal symmetry thereby creating an ideal diversification. 3. In reality, Banks may never acheive this level of diversification. 4. However this template enables the Bank to calibrate an internal credit correlations matrix and use the same in the mdoel to generate Portfolio Credit Risk using default or asset credit corrrelations
Krishnan Chari:To note1) In the absence of any real market Default correlation data , a definite positive correlation matrix has been simulated with the help of MATLAB functions using Random Matrix Theory. The algorithm for generating Random Matrices using Eigen values are available in opensoure R platforms also.
2. As we generated a synthetic Random Matrice with sufficient negative correlations between rating categories, the Credit VaR at portfolio level is lesser than the expected losses at individual asset exposure level. This situation arises as the synthetic correlation matrix has negative correlation numbers in equal symmetry thereby creating an ideal diversification.
3. In reality, Banks may never acheive this level of diversification.
4. However this template enables the Bank to calibrate an internal credit correlations matrix and use the same in the mdoel to generate Portfolio Credit Risk using default or asset credit corrrelations
AAA AA+ AA AA- A+ A A- BBB+AAA 1.00 -0.22 -0.03 0.16 -0.44 0.42 0.10 0.15AA+ -0.22 1.00 -0.40 0.13 0.50 -0.44 -0.15 0.03AA -0.03 -0.40 1.00 -0.20 -0.42 0.03 -0.11 -0.08AA- 0.16 0.13 -0.20 1.00 -0.15 0.18 0.19 -0.17A+ -0.44 0.50 -0.42 -0.15 1.00 -0.43 -0.08 0.05A 0.42 -0.44 0.03 0.18 -0.43 1.00 -0.05 0.00A- 0.10 -0.15 -0.11 0.19 -0.08 -0.05 1.00 0.11BBB+ 0.15 0.03 -0.08 -0.17 0.05 0.00 0.11 1.00BBB -0.32 -0.04 -0.08 -0.29 -0.13 -0.09 0.02 -0.06BBB- -0.09 0.05 0.33 -0.46 -0.18 -0.11 -0.20 0.19BB+ 0.05 0.44 0.03 0.30 0.25 -0.03 -0.17 -0.09BB -0.16 0.39 -0.02 -0.29 0.29 -0.57 0.35 -0.12BB- 0.33 -0.28 0.00 -0.10 -0.27 0.42 -0.06 0.01B+ 0.10 -0.11 0.15 0.08 -0.16 0.11 0.13 0.24B -0.34 0.10 -0.39 0.24 0.28 -0.35 0.04 -0.06B- 0.31 -0.10 -0.20 -0.07 -0.03 0.02 0.35 -0.05CCC -0.18 0.21 -0.56 0.02 0.45 0.00 0.12 -0.03CC -0.21 0.26 0.40 -0.40 -0.07 -0.37 -0.22 0.05C -0.09 -0.34 0.17 0.02 -0.12 0.42 -0.55 -0.24D -0.15 -0.20 0.19 -0.07 0.24 -0.56 0.01 -0.03
BBB BBB- BB+ BB BB- B+ B B- CCC CC-0.32 -0.09 0.05 -0.16 0.33 0.10 -0.34 0.31 -0.18 -0.21-0.04 0.05 0.44 0.39 -0.28 -0.11 0.10 -0.10 0.21 0.26-0.08 0.33 0.03 -0.02 0.00 0.15 -0.39 -0.20 -0.56 0.40-0.29 -0.46 0.30 -0.29 -0.10 0.08 0.24 -0.07 0.02 -0.40-0.13 -0.18 0.25 0.29 -0.27 -0.16 0.28 -0.03 0.45 -0.07-0.09 -0.11 -0.03 -0.57 0.42 0.11 -0.35 0.02 0.00 -0.370.02 -0.20 -0.17 0.35 -0.06 0.13 0.04 0.35 0.12 -0.22
-0.06 0.19 -0.09 -0.12 0.01 0.24 -0.06 -0.05 -0.03 0.051.00 0.28 -0.48 0.10 -0.16 0.00 0.13 -0.16 0.06 -0.090.28 1.00 -0.13 0.11 0.23 0.23 -0.12 0.11 0.00 0.59
-0.48 -0.13 1.00 0.10 -0.06 -0.22 0.12 -0.19 0.19 0.170.10 0.11 0.10 1.00 -0.20 -0.17 0.13 0.10 0.13 0.28
-0.16 0.23 -0.06 -0.20 1.00 -0.23 -0.19 0.54 0.03 -0.130.00 0.23 -0.22 -0.17 -0.23 1.00 -0.05 -0.18 -0.23 -0.060.13 -0.12 0.12 0.13 -0.19 -0.05 1.00 0.07 0.54 -0.09
-0.16 0.11 -0.19 0.10 0.54 -0.18 0.07 1.00 0.16 0.110.06 0.00 0.19 0.13 0.03 -0.23 0.54 0.16 1.00 -0.21
-0.09 0.59 0.17 0.28 -0.13 -0.06 -0.09 0.11 -0.21 1.000.06 0.03 -0.03 -0.64 0.26 -0.07 -0.22 -0.26 -0.24 -0.14
-0.16 -0.20 -0.01 0.15 -0.34 -0.22 0.21 -0.14 -0.09 0.08
C D-0.09 -0.15-0.34 -0.200.17 0.190.02 -0.07
-0.12 0.240.42 -0.56
-0.55 0.01-0.24 -0.030.06 -0.160.03 -0.20
-0.03 -0.01-0.64 0.150.26 -0.34
-0.07 -0.22-0.22 0.21-0.26 -0.14-0.24 -0.09-0.14 0.081.00 -0.12
-0.12 1.00
Vt 0.00 12928.49 5594.29 8501.18
C 1.00 -0.22 -0.03 0.16-0.22 1.00 -0.40 0.13-0.03 -0.40 1.00 -0.200.16 0.13 -0.20 1.00
-0.44 0.50 -0.42 -0.150.42 -0.44 0.03 0.180.10 -0.15 -0.11 0.190.15 0.03 -0.08 -0.17
-0.32 -0.04 -0.08 -0.29-0.09 0.05 0.33 -0.460.05 0.44 0.03 0.30
-0.16 0.39 -0.02 -0.290.33 -0.28 0.00 -0.100.10 -0.11 0.15 0.08
-0.34 0.10 -0.39 0.240.31 -0.10 -0.20 -0.07
-0.18 0.21 -0.56 0.02-0.21 0.26 0.40 -0.40-0.09 -0.34 0.17 0.02-0.15 -0.20 0.19 -0.07
1x20 Result 1 -1267148.1 -387952.7 -1685416 146756.739512549
Resultant 1x120X1 0.00
12928.495594.298501.18
299061.63 24465680907263.9084913.17 SQRT 4,946,279.5015 37550.67
472598.3087310.59 Comment
177872.98276806.52
1157499.43
Portfolio diversified VAR=Unexpected Losses after taking into account Default Correlations/Asset Correlations
Due to non availability of true default correlations across rating grades or asset classes, Correlation Matrice was generated under suitable constraints using the Random Matrix Theory and simulation using MATLAB function
1200345.572196083.771870547.29
145086.913417674.84
14242.273748986.35
795706.55
299061.63 84913.17 37550.67 472598.30 87310.59 177872.98 276806.52 1157499.43
-0.44 0.42 0.10 0.15 -0.32 -0.09 0.05 -0.160.50 -0.44 -0.15 0.03 -0.04 0.05 0.44 0.39
-0.42 0.03 -0.11 -0.08 -0.08 0.33 0.03 -0.02-0.15 0.18 0.19 -0.17 -0.29 -0.46 0.30 -0.291.00 -0.43 -0.08 0.05 -0.13 -0.18 0.25 0.29
-0.43 1.00 -0.05 0.00 -0.09 -0.11 -0.03 -0.57-0.08 -0.05 1.00 0.11 0.02 -0.20 -0.17 0.350.05 0.00 0.11 1.00 -0.06 0.19 -0.09 -0.12
-0.13 -0.09 0.02 -0.06 1.00 0.28 -0.48 0.10-0.18 -0.11 -0.20 0.19 0.28 1.00 -0.13 0.110.25 -0.03 -0.17 -0.09 -0.48 -0.13 1.00 0.100.29 -0.57 0.35 -0.12 0.10 0.11 0.10 1.00
-0.27 0.42 -0.06 0.01 -0.16 0.23 -0.06 -0.20-0.16 0.11 0.13 0.24 0.00 0.23 -0.22 -0.170.28 -0.35 0.04 -0.06 0.13 -0.12 0.12 0.13
-0.03 0.02 0.35 -0.05 -0.16 0.11 -0.19 0.100.45 0.00 0.12 -0.03 0.06 0.00 0.19 0.13
-0.07 -0.37 -0.22 0.05 -0.09 0.59 0.17 0.28-0.12 0.42 -0.55 -0.24 0.06 0.03 -0.03 -0.640.24 -0.56 0.01 -0.03 -0.16 -0.20 -0.01 0.15
1752750.6 478456.66 -941789.8 -259760.2 371044.29 876962.71 553343.65 -952177.81786
1200345.57 2196083.77 1870547.29 145086.91 3417674.84 14242.27 3748986.35 795706.55
0.33 0.10 -0.34 0.31 -0.18 -0.21 -0.09 -0.15-0.28 -0.11 0.10 -0.10 0.21 0.26 -0.34 -0.200.00 0.15 -0.39 -0.20 -0.56 0.40 0.17 0.19
-0.10 0.08 0.24 -0.07 0.02 -0.40 0.02 -0.07-0.27 -0.16 0.28 -0.03 0.45 -0.07 -0.12 0.240.42 0.11 -0.35 0.02 0.00 -0.37 0.42 -0.56
-0.06 0.13 0.04 0.35 0.12 -0.22 -0.55 0.010.01 0.24 -0.06 -0.05 -0.03 0.05 -0.24 -0.03
-0.16 0.00 0.13 -0.16 0.06 -0.09 0.06 -0.160.23 0.23 -0.12 0.11 0.00 0.59 0.03 -0.20
-0.06 -0.22 0.12 -0.19 0.19 0.17 -0.03 -0.01-0.20 -0.17 0.13 0.10 0.13 0.28 -0.64 0.151.00 -0.23 -0.19 0.54 0.03 -0.13 0.26 -0.34
-0.23 1.00 -0.05 -0.18 -0.23 -0.06 -0.07 -0.22-0.19 -0.05 1.00 0.07 0.54 -0.09 -0.22 0.210.54 -0.18 0.07 1.00 0.16 0.11 -0.26 -0.140.03 -0.23 0.54 0.16 1.00 -0.21 -0.24 -0.09
-0.13 -0.06 -0.09 0.11 -0.21 1.00 -0.14 0.080.26 -0.07 -0.22 -0.26 -0.24 -0.14 1.00 -0.12
-0.34 -0.22 0.21 -0.14 -0.09 0.08 -0.12 1.00
958749.02918 420045.385 2910630.42 53932.4136 3333401.81 -1186731.9 1658906.03 -361427.77
