1 Introduction
2 Data and LGD estimation approach
2.1 Data description
Quantiles | Mean | Obs. | |||||
---|---|---|---|---|---|---|---|
0.05 | 0.25 | 0.50 | 0.75 | 0.95 | |||
\(LGD_{overall}\) | \(-7.92\) | 0.08 | 5.45 | 61.36 | 100.00 | 28.86 | 32851 |
\(\log (EAD)\) | 8.11 | 10.00 | 11.36 | 12.64 | 14.39 | 11.33 | 32851 |
Number of collaterals | 0.00 | 0.00 | 1.00 | 1.00 | 3.00 | 1.18 | 32851 |
Number of guarantors | 0.00 | 0.00 | 0.00 | 0.00 | 2.00 | 0.31 | 32851 |
LGD conditional to guarantee availability: | |||||||
No guarantee | \(-7.91\) | 0.08 | 5.55 | 65.73 | 100.00 | 29.89 | 28774 |
Guarantee | \(-7.93\) | 0.04 | 5.04 | 33.94 | 100.00 | 21.57 | 4077 |
LGD conditional to collateral type: | |||||||
No collateral | \(-2.62\) | 1.88 | 24.09 | 98.22 | 100.00 | 43.37 | 9008 |
Real estate | \(-8.47\) | \(-0.55\) | 2.12 | 26.37 | 99.45 | 18.13 | 7188 |
Other | \(-9.71\) | \(-0.09\) | 4.61 | 51.09 | 100.00 | 25.64 | 16655 |
LGD conditional to facility type: | |||||||
Medium term | \(-3.99\) | 0.25 | 4.69 | 48.95 | 100.00 | 26.26 | 19463 |
Short term | \(-15.54\) | \(-0.39\) | 8.30 | 82.77 | 100.00 | 32.72 | 12658 |
Other | \(-6.80\) | 0.08 | 4.43 | 79.98 | 100.00 | 31.35 | 730 |
LGD conditional to seniority type: | |||||||
Pari-passu | \(-8.47\) | 0.01 | 5.64 | 60.03 | 100.00 | 28.63 | 27013 |
Super senior | \(-4.67\) | 0.99 | 4.10 | 62.91 | 100.00 | 29.04 | 5301 |
Non-senior | \(-9.81\) | 0.73 | 20.65 | 84.76 | 100.00 | 38.40 | 537 |
LGD conditional to industry type: | |||||||
Finance, insurance, real estate | \(-8.74\) | \(-0.48\) | 1.95 | 45.30 | 100.00 | 23.18 | 4726 |
Agriculture, forestry, fishing, hunting | \(-6.17\) | \(-0.66\) | 2.43 | 66.13 | 100.61 | 27.20 | 1486 |
Mining | \(-5.52\) | 0.65 | 1.13 | 53.88 | 100.00 | 26.82 | 120 |
Construction | \(-10.42\) | \(-0.02\) | 3.54 | 62.49 | 100.00 | 27.74 | 3720 |
Manufacturing | \(-10.29\) | 0.01 | 5.37 | 64.12 | 100.00 | 29.35 | 4707 |
Transp., commu.,elec., gas, sani. serv. | \(-6.76\) | 0.61 | 2.47 | 51.53 | 100.00 | 25.08 | 2587 |
Wholesale and retail trade | \(-7.82\) | 0.08 | 6.28 | 74.09 | 100.00 | 32.42 | 5483 |
Services | \(-6.85\) | 0.82 | 14.41 | 76.85 | 100.00 | 34.12 | 5877 |
Other | \(-5.18\) | 0.84 | 10.27 | 45.26 | 100.00 | 26.64 | 4145 |
2.2 LGD estimation methods
Method | Exemplary literature |
---|---|
Ordinary Least Squares (OLS) | |
Variable selection methods: | |
\(\begin{array}{l} {{(1)}\,\text{OLS}}\hbox { with Backward Elimination }(bOLS)\\ \hbox {(2) Least Angle Regression (LAR)}\end{array}\) | |
Penalized Regressions: | |
\(\begin{array}{l} \hbox {(1) Ridge Regression }(RR)\\ \hbox {(2) Lasso Regression }(LR)\\ \hbox {(3) Elastic Regression }(ER) \end{array}\) |
Loterman et al. (2012) |
Fractional Logit Regression (FLR) | |
Regression Tree (RT) | |
Conditional Inference Tree (CIT) | |
Random Forest (RF) | |
Boosting Methods: | |
(1) Adaptive Boosting (ADA), (2) Gradient Boosting (GB) |
Tanoue and Yamashita (2019) |
Cubist Regression Model (CUB) | |
Artificial Neural Network (ANN) | |
Support Vector Regression (SVR) | |
Relevance Vector Regression (RVR) | |
Gaussian Process Regression (GAPR) |
Bellotti et al. (2021) |
K-nearest Neighbors (KNN) | |
Multivariate Adaptive Regression Splines (MARS) | |
Finite Mixture Model (FMM) |
2.3 Clustering and LGD distribution analysis
3 Comparative analysis
3.1 Model comparison procedure
3.2 Hyperparameter tuning
Method | Hyperparameter | Description of Hyperparameter | Hyperparameter Set | Set inspired by | Hyperparameter Choice | ||
---|---|---|---|---|---|---|---|
Cluster 1 | Cluster 2 | Cluster 3 | |||||
RR LR ER | \(\lambda \) | Lagrangian parameter that control the amount of shrinkage of the model parameters: The larger the value of \(\lambda \), the greater is the amount of shrinkage. | \(\{10^{-6},10^{-5},...,10^{6}\}\) |
Zou and Hastie (2005) | 100 0.001 0.001 | 100 0.0001 0.0001 | 100 0.0001 0.0001 |
RT | Tree size | Tuning parameter that controls the tree’s complexity and indicates how deep the tree is allowed to be. | \(\{3,4,...,10\}\) |
Qi and Zhao (2011) | 6 | 7 | 6 |
Min node size | The minimum number of observations required to split an internal node. By increasing the node size, the tree becomes more constrained because it has to consider more samples at each node. | \(\{5,6,...,20\}\) |
Qi and Zhao (2011) | 14 | 9 | 17 | |
CIT | Tree size | Same description as for Regression Tree. | \(\{3,4,...,10\}\) |
Talaba (2019) | 8 | 8 | 8 |
Mincriterion | 1 - p-value that must be exceeded in order to implement a split. | \(\{0.01,0.02,...,1\}\) |
Talaba (2019) | 0.65 | 0.6 | 0.15 | |
RF | Tree size | Same description as for Regression Tree. | \(\{3,4,...,10\}\) |
Qi and Zhao (2011) | 9 | 8 | 9 |
Min node size | Same description as for Regression Tree. | \(\{5,6,...,20\}\) |
Qi and Zhao (2011) | 6 | 5 | 7 | |
# splitting variables | The number of variables to consider when looking for the best split. It handles the so-called bias-variance trade-off. | \(\{1,6,...,10\}\) |
Breiman (2001) | 8 | 9 | 9 | |
# trees | This parameter specifies the number of trees in the forest of the model and thus controls the model’s complexity. | \(\{100,101...,1000\}\) |
Hastie et al. (2017) | 833 | 873 | 777 | |
ADA | Tree size | Same description as for Regression Tree. | \(\{3,4,...,10\}\) |
Qi and Zhao (2011) | 4 | 4 | 6 |
Min node size | Same description as for Regression Tree. | \(\{5,6,...,20\}\) |
Qi and Zhao (2011) | 13 | 12 | 11 | |
# splitting variables | Same description as for Random Forest. | \(\{1,6,...,10\}\) |
Breiman (2001) | 6 | 6 | 8 | |
# trees | Same description as for Random Forest. | \(\{100,101,...,1000\}\) |
Hastie et al. (2017) | 112 | 134 | 152 | |
GB | Tree size | Same description as for Regression Tree. | \(\{3,4,...,10\}\) |
Qi and Zhao (2011) | 5 | 5 | 4 |
Min node size | Same description as for Regression Tree. | \(\{5,6,...,20\}\) |
Qi and Zhao (2011) | 8 | 9 | 8 | |
# splitting variables | Same description as for Random Forest. | \(\{1,6,...,10\}\) |
Breiman (2001) | 6 | 5 | 7 | |
# trees | Same description as for Random Forest. | \(\{100,101,...,1000\}\) |
Hastie et al. (2017) | 123 | 137 | 168 | |
CUB | # committees | Tuning parameter that controls the number of boosting iterations. | \(\{1,2,...,100\}\) |
Kuhn and Johnson (2016) | 23 | 62 | 64 |
k | The Number of nearest training set instances that are used to adjust the model prediction. | \(\{1,2,...,9\}\) |
Kuhn and Johnson (2016) | 8 | 7 | 7 | |
ANN | # hidden neurons | The number of neurons in the first and second hidden layer. The parameter affects the network’s complexity. | (\(\{1,2,...,10\}\) ; \(\{0,1,...,10\}\)) |
Heaton (2008) | (6,0) | (5,0) | (5,0) |
Activation | The function is attached to each neuron in the network, and, determines whether a neuron should be activated, based on whether its input is relevant to the model’s prediction. | \(\{\)logistic, tanh, relu\(\}\) |
Hastie et al. (2017) | Logistic | Logistic | Logistic | |
Error function | The function that optimizes the weights of the network. | \(\{\)stochastic gradient -based optimizer (sgo)\(\}\) |
Kingma and Ba (2014) | sgo | sgo | sgo | |
SVR | C | The so-called cost parameter controls the penalty imposed on observations that lie outside a defined margin of tolerance. Larger values of C focus attention on (correctly classified) points near the decision boundary, while smaller values involve data further away. | \(\{0.001, 0.01, 0.1, 1, 5, 10,100\}\) |
van Gestel et al. (2004) | 1 | 1 | 1 |
\(\epsilon \) | The parameter controls the width of the so-called \(\epsilon \)-insensitive zone, used to fit the training data. The value of \(\epsilon \) can affect the number of support vectors used to construct the regression function. A bigger value of \(\epsilon \) indicates that, fewer support vectors are selected. | \(\{0,0.1,...,1\}\) |
van Gestel et al. (2004) | 0.2 | 0.3 | 0.4 | |
RVR | Kernel | The kernel function used in training and predicting. | \(\{\)radial basis function (rbf)\(\}\) |
Karatzoglou et al. (2004) | rbf | rbf | rbf |
\(\sigma \) | Tuning parameter that determines the inverse kernel width for the radial basis function. | \(\{0.01,0.02,...,1\}\) |
Karatzoglou et al. (2004) | 0.06 | 0.09 | 0.10 | |
GAPR | Kernel | Same description as for Relevance Vector Regression. | \(\{\)radial basis function (rbf)\(\}\) |
Karatzoglou et al. (2004) | rbf | rbf | rbf |
\(\sigma \) | Same description as for Relevance Vector Regression. | \(\{0.01,0.02,...,1\}\) |
Karatzoglou et al. (2004) | 0.03 | 0.05 | 0.06 | |
KNN | k | Number of nearest neighbors. | \(\{1,2,...,100\}\) |
Hastie et al. (2017) | 22 | 20 | 11 |
MARS | Degree | Complexity parameter that controls the maximum degree of input terms in the regression function. | \(\{1,2,...,5\}\) |
Boehmke and Greenwell (2020) | 2 | 4 | 3 |
nprune | Number of terms to retain in the final regression function. | \(\{2,3,...,100\}\) |
Boehmke and Greenwell (2020) | 18 | 32 | 15 | |
FMM | \(k_{c}\) | Number of mixture components. | \(\{1,2,...,10\}\) |
Min et al. (2020) | 5 | 5 | 5 |
3.3 Out-of-sample results
Split | OLS | bOLS | LAR | RR | LR | ER | FLR | RT | CIT | RF | ADA | GB | CUB | ANN | SVR | RVR | GAPR | KNN | MARS | FMM |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cluster 1: (nearly) symmetric bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.1338 | 0.1344 | 0.1336 | 0.1349 | 0.1330 | 0.1316 | 0.1330 | 0.1350 | 0.1379 | 0.1268 | 0.1361 | 0.1321 | 0.1345 | 0.1374 | 0.1354 | 0.1372 | 0.1357 | 0.1463 | 0.1360 | 0.1350 |
70/30 | 0.1359 | 0.1355 | 0.1362 | 0.1332 | 0.1347 | 0.1348 | 0.1343 | 0.1353 | 0.1320 | 0.1235 | 0.1347 | 0.1320 | 0.1314 | 0.1346 | 0.1321 | 0.1363 | 0.1323 | 0.1442 | 0.1337 | 0.1327 |
80/20 | 0.1316 | 0.1332 | 0.1317 | 0.1329 | 0.1329 | 0.1346 | 0.1341 | 0.1333 | 0.1294 | 0.1238 | 0.1315 | 0.1308 | 0.1322 | 0.1314 | 0.1326 | 0.1311 | 0.1317 | 0.1419 | 0.1342 | 0.1320 |
90/10 | 0.1311 | 0.1299 | 0.1309 | 0.1324 | 0.1326 | 0.1326 | 0.1324 | 0.1302 | 0.1309 | 0.1221 | 0.1303 | 0.1270 | 0.1317 | 0.1311 | 0.1285 | 0.1282 | 0.1309 | 0.1399 | 0.1282 | 0.1314 |
Mean | 0.1331 | 0.1332 | 0.1331 | 0.1334 | 0.1333 | 0.1334 | 0.1335 | 0.1335 | 0.1326 | 0.1241 | 0.1332 | 0.1305 | 0.1324 | 0.1336 | 0.1322 | 0.1332 | 0.1327 | 0.1431 | 0.1330 | 0.1328 |
Cluster 2: asymmetric (positively skewed) bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.1388 | 0.1383 | 0.1388 | 0.1382 | 0.1383 | 0.1380 | 0.1369 | 0.1357 | 0.1352 | 0.1330 | 0.1474 | 0.1287 | 0.1351 | 0.1356 | 0.1348 | 0.1358 | 0.1340 | 0.1358 | 0.1351 | 0.1354 |
70/30 | 0.1376 | 0.1376 | 0.1376 | 0.1376 | 0.1376 | 0.1378 | 0.1357 | 0.1344 | 0.1340 | 0.1324 | 0.1462 | 0.1282 | 0.1341 | 0.1346 | 0.1343 | 0.1345 | 0.1335 | 0.1345 | 0.1344 | 0.1349 |
80/20 | 0.1366 | 0.1366 | 0.1381 | 0.1373 | 0.1376 | 0.1378 | 0.1359 | 0.1339 | 0.1334 | 0.1325 | 0.1443 | 0.1273 | 0.1335 | 0.1340 | 0.1332 | 0.1343 | 0.1331 | 0.1344 | 0.1354 | 0.1334 |
90/10 | 0.1353 | 0.1361 | 0.1364 | 0.1360 | 0.1361 | 0.1366 | 0.1360 | 0.1331 | 0.1317 | 0.1296 | 0.1438 | 0.1260 | 0.1305 | 0.1335 | 0.1303 | 0.1337 | 0.1309 | 0.1335 | 0.1318 | 0.1316 |
Mean | 0.1371 | 0.1371 | 0.1377 | 0.1373 | 0.1374 | 0.1375 | 0.1361 | 0.1343 | 0.1336 | 0.1319 | 0.1454 | 0.1276 | 0.1333 | 0.1344 | 0.1331 | 0.1346 | 0.1329 | 0.1345 | 0.1342 | 0.1338 |
Cluster 3: (positively skewed) unimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.0828 | 0.0815 | 0.0857 | 0.0810 | 0.0817 | 0.0819 | 0.0847 | 0.0682 | 0.0658 | 0.0603 | 0.0777 | 0.0607 | 0.0642 | 0.0805 | 0.0642 | 0.0660 | 0.0644 | 0.0646 | 0.0689 | 0.0496 |
70/30 | 0.0813 | 0.0812 | 0.0825 | 0.0809 | 0.0807 | 0.0799 | 0.0825 | 0.0675 | 0.0649 | 0.0551 | 0.0773 | 0.0576 | 0.0599 | 0.0791 | 0.0603 | 0.0636 | 0.0627 | 0.0627 | 0.0689 | 0.0412 |
80/20 | 0.0795 | 0.0817 | 0.0837 | 0.0804 | 0.0801 | 0.0802 | 0.0829 | 0.0677 | 0.0646 | 0.0549 | 0.0761 | 0.0568 | 0.0591 | 0.0787 | 0.0598 | 0.0656 | 0.0636 | 0.0630 | 0.0682 | 0.0455 |
90/10 | 0.0790 | 0.0808 | 0.0843 | 0.0809 | 0.0803 | 0.0807 | 0.0805 | 0.0674 | 0.0647 | 0.0540 | 0.0720 | 0.0536 | 0.0570 | 0.0768 | 0.0565 | 0.0636 | 0.0624 | 0.0642 | 0.0674 | 0.0456 |
Mean | 0.0807 | 0.0813 | 0.0840 | 0.0808 | 0.0807 | 0.0807 | 0.0826 | 0.0677 | 0.0650 | 0.0561 | 0.0758 | 0.0572 | 0.0600 | 0.0788 | 0.0602 | 0.0647 | 0.0633 | 0.0636 | 0.0683 | 0.0455 |
Split | OLS | bOLS | LAR | RR | LR | ER | FLR | RT | CIT | RF | ADA | GB | CUB | ANN | SVR | RVR | GAPR | KNN | MARS | FMM |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cluster 1: (nearly) symmetric bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.3225 | 0.3234 | 0.3244 | 0.3217 | 0.3278 | 0.3244 | 0.3107 | 0.3178 | 0.3296 | 0.3128 | 0.3354 | 0.3134 | 0.3099 | 0.2969 | 0.3279 | 0.3209 | 0.3221 | 0.3391 | 0.3271 | 0.3206 |
70/30 | 0.3250 | 0.3253 | 0.3254 | 0.3237 | 0.3296 | 0.3284 | 0.3311 | 0.3247 | 0.3147 | 0.3075 | 0.3339 | 0.3133 | 0.3057 | 0.3140 | 0.3236 | 0.3106 | 0.3182 | 0.3354 | 0.3210 | 0.3419 |
80/20 | 0.3188 | 0.3221 | 0.3201 | 0.3208 | 0.3276 | 0.3286 | 0.3253 | 0.3156 | 0.3097 | 0.3077 | 0.3327 | 0.3115 | 0.3065 | 0.3090 | 0.3242 | 0.3052 | 0.3178 | 0.3352 | 0.3199 | 0.3205 |
90/10 | 0.3176 | 0.3169 | 0.3172 | 0.3196 | 0.3261 | 0.3245 | 0.3298 | 0.3115 | 0.3099 | 0.2989 | 0.3323 | 0.3129 | 0.3053 | 0.3229 | 0.3170 | 0.3014 | 0.3186 | 0.3278 | 0.3118 | 0.3197 |
Mean | 0.3210 | 0.3219 | 0.3218 | 0.3214 | 0.3278 | 0.3265 | 0.3242 | 0.3174 | 0.3160 | 0.3067 | 0.3336 | 0.3128 | 0.3068 | 0.3107 | 0.3232 | 0.3095 | 0.3192 | 0.3343 | 0.3200 | 0.3257 |
Cluster 2: asymmetric (positively skewed) bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.2963 | 0.2953 | 0.2958 | 0.2953 | 0.2975 | 0.2959 | 0.3017 | 0.2844 | 0.2843 | 0.2887 | 0.3430 | 0.2731 | 0.2736 | 0.2940 | 0.2780 | 0.2826 | 0.2861 | 0.2807 | 0.2867 | 0.3470 |
70/30 | 0.2950 | 0.2950 | 0.2950 | 0.2950 | 0.2970 | 0.2966 | 0.2929 | 0.2817 | 0.2815 | 0.2884 | 0.3434 | 0.2745 | 0.2732 | 0.2659 | 0.2779 | 0.2811 | 0.2869 | 0.2795 | 0.2868 | 0.3467 |
80/20 | 0.2924 | 0.2924 | 0.2951 | 0.2940 | 0.2971 | 0.2962 | 0.2954 | 0.2806 | 0.2789 | 0.2881 | 0.3388 | 0.2697 | 0.2705 | 0.2604 | 0.2751 | 0.2796 | 0.2859 | 0.2776 | 0.2869 | 0.3450 |
90/10 | 0.2923 | 0.2939 | 0.2945 | 0.2939 | 0.2954 | 0.2956 | 0.2570 | 0.2785 | 0.2764 | 0.2853 | 0.3394 | 0.2676 | 0.2678 | 0.2847 | 0.2712 | 0.2794 | 0.2829 | 0.2780 | 0.2845 | 0.3411 |
Mean | 0.2940 | 0.2942 | 0.2951 | 0.2946 | 0.2968 | 0.2961 | 0.2867 | 0.2813 | 0.2803 | 0.2876 | 0.3411 | 0.2712 | 0.2713 | 0.2763 | 0.2756 | 0.2807 | 0.2854 | 0.2790 | 0.2862 | 0.3450 |
Cluster 3: (positively skewed) unimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.2130 | 0.2077 | 0.2238 | 0.2067 | 0.2091 | 0.2108 | 0.2217 | 0.1641 | 0.1679 | 0.1518 | 0.2116 | 0.1599 | 0.1603 | 0.2228 | 0.2143 | 0.1691 | 0.1679 | 0.1633 | 0.1704 | 0.1499 |
70/30 | 0.2123 | 0.2112 | 0.2162 | 0.2111 | 0.2091 | 0.2067 | 0.2162 | 0.1621 | 0.1661 | 0.1498 | 0.1961 | 0.1569 | 0.1521 | 0.1944 | 0.2097 | 0.1648 | 0.1669 | 0.1498 | 0.1687 | 0.1493 |
80/20 | 0.2047 | 0.2121 | 0.2202 | 0.2079 | 0.2045 | 0.2051 | 0.2163 | 0.1636 | 0.1675 | 0.1440 | 0.2288 | 0.1527 | 0.1528 | 0.1940 | 0.2095 | 0.1699 | 0.1706 | 0.1563 | 0.1737 | 0.1470 |
90/10 | 0.1997 | 0.2046 | 0.2183 | 0.2068 | 0.2022 | 0.2036 | 0.2408 | 0.1666 | 0.1763 | 0.1496 | 0.2318 | 0.1574 | 0.1557 | 0.2305 | 0.2071 | 0.1684 | 0.1684 | 0.1787 | 0.1725 | 0.1460 |
Mean | 0.2074 | 0.2089 | 0.2196 | 0.2081 | 0.2062 | 0.2065 | 0.2238 | 0.1641 | 0.1694 | 0.1488 | 0.2171 | 0.1568 | 0.1552 | 0.2104 | 0.2101 | 0.1680 | 0.1685 | 0.1620 | 0.1713 | 0.1480 |
4 Robustness checks
4.1 Inclusion of enterprise-specific variables
Split | OLS | bOLS | LAR | RR | LR | ER | FLR | RT | CIT | RF | ADA | GB | CUB | ANN | SVR | RVR | GAPR | KNN | MARS | FMM |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cluster 1: (nearly) symmetric bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.1168 | 0.1182 | 0.1168 | 0.1173 | 0.1172 | 0.1166 | 0.1150 | 0.1181 | 0.1149 | 0.1085 | 0.1153 | 0.1121 | 0.1189 | 0.1163 | 0.1129 | 0.1196 | 0.1142 | 0.1249 | 0.1152 | 0.1168 |
70/30 | 0.1140 | 0.1168 | 0.1158 | 0.1157 | 0.1146 | 0.1155 | 0.1148 | 0.1182 | 0.1136 | 0.1062 | 0.1149 | 0.1134 | 0.1158 | 0.1161 | 0.1121 | 0.1177 | 0.1145 | 0.1233 | 0.1144 | 0.1158 |
80/20 | 0.1111 | 0.1135 | 0.1143 | 0.1153 | 0.1147 | 0.1156 | 0.1146 | 0.1163 | 0.1160 | 0.1052 | 0.1136 | 0.1173 | 0.1121 | 0.1141 | 0.1136 | 0.1143 | 0.1132 | 0.1204 | 0.1144 | 0.1121 |
90/10 | 0.1150 | 0.1089 | 0.1103 | 0.1108 | 0.1124 | 0.1116 | 0.1138 | 0.1121 | 0.1107 | 0.1045 | 0.1131 | 0.1063 | 0.1094 | 0.1127 | 0.1117 | 0.1062 | 0.1085 | 0.1185 | 0.1127 | 0.1107 |
Mean | 0.1142 | 0.1143 | 0.1143 | 0.1148 | 0.1147 | 0.1148 | 0.1145 | 0.1162 | 0.1138 | 0.1061 | 0.1142 | 0.1123 | 0.1140 | 0.1148 | 0.1126 | 0.1144 | 0.1126 | 0.1218 | 0.1142 | 0.1139 |
Cluster 2: asymmetric (positively skewed) bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.1279 | 0.1260 | 0.1258 | 0.1268 | 0.1250 | 0.1257 | 0.1242 | 0.1260 | 0.1250 | 0.1238 | 0.1246 | 0.1233 | 0.1270 | 0.1270 | 0.1251 | 0.1277 | 0.1268 | 0.1247 | 0.1254 | 0.1252 |
70/30 | 0.1237 | 0.1244 | 0.1242 | 0.1255 | 0.1257 | 0.1252 | 0.1233 | 0.1172 | 0.1204 | 0.1216 | 0.1235 | 0.1210 | 0.1228 | 0.1232 | 0.1257 | 0.1237 | 0.1225 | 0.1227 | 0.1169 | 0.1218 |
80/20 | 0.1219 | 0.1217 | 0.1243 | 0.1235 | 0.1246 | 0.1242 | 0.1234 | 0.1173 | 0.1163 | 0.1099 | 0.1236 | 0.1074 | 0.1149 | 0.1184 | 0.1115 | 0.1188 | 0.1090 | 0.1219 | 0.1175 | 0.1182 |
90/10 | 0.1192 | 0.1197 | 0.1216 | 0.1191 | 0.1198 | 0.1198 | 0.1177 | 0.1176 | 0.1149 | 0.1088 | 0.1235 | 0.1058 | 0.1062 | 0.1171 | 0.1077 | 0.1179 | 0.1052 | 0.1205 | 0.1148 | 0.1061 |
Mean | 0.1232 | 0.1230 | 0.1240 | 0.1237 | 0.1238 | 0.1237 | 0.1221 | 0.1195 | 0.1192 | 0.1160 | 0.1238 | 0.1144 | 0.1177 | 0.1214 | 0.1175 | 0.1221 | 0.1159 | 0.1225 | 0.1186 | 0.1178 |
Cluster 3: (positively skewed) unimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.0789 | 0.0817 | 0.0835 | 0.0809 | 0.0805 | 0.0805 | 0.0835 | 0.0694 | 0.0664 | 0.0585 | 0.0780 | 0.0589 | 0.0624 | 0.0780 | 0.0592 | 0.0671 | 0.0639 | 0.0681 | 0.0705 | 0.0485 |
70/30 | 0.0811 | 0.0815 | 0.0838 | 0.0832 | 0.0791 | 0.0796 | 0.0829 | 0.0679 | 0.0646 | 0.0553 | 0.0764 | 0.0565 | 0.0592 | 0.0778 | 0.0582 | 0.0652 | 0.0649 | 0.0675 | 0.0688 | 0.0472 |
80/20 | 0.0769 | 0.0799 | 0.0820 | 0.0769 | 0.0790 | 0.0790 | 0.0818 | 0.0653 | 0.0628 | 0.0532 | 0.0741 | 0.0546 | 0.0575 | 0.0759 | 0.0567 | 0.0646 | 0.0617 | 0.0658 | 0.0663 | 0.0437 |
90/10 | 0.0788 | 0.0785 | 0.0812 | 0.0788 | 0.0797 | 0.0791 | 0.0762 | 0.0630 | 0.0621 | 0.0536 | 0.0725 | 0.0525 | 0.0566 | 0.0725 | 0.0563 | 0.0634 | 0.0621 | 0.0661 | 0.0647 | 0.0437 |
Mean | 0.0789 | 0.0804 | 0.0826 | 0.0800 | 0.0796 | 0.0795 | 0.0811 | 0.0664 | 0.0640 | 0.0551 | 0.0753 | 0.0556 | 0.0589 | 0.0760 | 0.0576 | 0.0651 | 0.0631 | 0.0669 | 0.0676 | 0.0458 |
4.2 Clustering based on loan-specific variable
Split | OLS | bOLS | LAR | RR | LR | ER | FLR | RT | CIT | RF | ADA | GB | CUB | ANN | SVR | RVR | GAPR | KNN | MARS | FMM |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cluster 1: (nearly) symmetric bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.1366 | 0.1365 | 0.1366 | 0.1370 | 0.1372 | 0.1374 | 0.1369 | 0.1370 | 0.1357 | 0.1300 | 0.1367 | 0.1348 | 0.1353 | 0.1370 | 0.1353 | 0.1372 | 0.1362 | 0.1448 | 0.1360 | 0.1348 |
70/30 | 0.1362 | 0.1357 | 0.1362 | 0.1370 | 0.1369 | 0.1367 | 0.1366 | 0.1366 | 0.1356 | 0.1305 | 0.1357 | 0.1328 | 0.1357 | 0.1368 | 0.1343 | 0.1361 | 0.1359 | 0.1437 | 0.1353 | 0.1345 |
80/20 | 0.1355 | 0.1359 | 0.1360 | 0.1363 | 0.1362 | 0.1361 | 0.1360 | 0.1361 | 0.1347 | 0.1262 | 0.1353 | 0.1325 | 0.1340 | 0.1366 | 0.1346 | 0.1364 | 0.1359 | 0.1435 | 0.1344 | 0.1344 |
90/10 | 0.1357 | 0.1372 | 0.1361 | 0.1362 | 0.1363 | 0.1363 | 0.1361 | 0.1361 | 0.1354 | 0.1244 | 0.1364 | 0.1313 | 0.1328 | 0.1359 | 0.1331 | 0.1366 | 0.1357 | 0.1438 | 0.1345 | 0.1344 |
Mean | 0.1360 | 0.1363 | 0.1362 | 0.1366 | 0.1367 | 0.1366 | 0.1364 | 0.1364 | 0.1353 | 0.1277 | 0.1360 | 0.1328 | 0.1345 | 0.1366 | 0.1343 | 0.1366 | 0.1359 | 0.1439 | 0.1350 | 0.1345 |
Cluster 2: asymmetric (positively skewed) bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.1254 | 0.1257 | 0.1258 | 0.1256 | 0.1257 | 0.1256 | 0.1255 | 0.1252 | 0.1235 | 0.1207 | 0.1293 | 0.1194 | 0.1241 | 0.1249 | 0.1216 | 0.1252 | 0.1222 | 0.1253 | 0.1249 | 0.1234 |
70/30 | 0.1253 | 0.1254 | 0.1257 | 0.1253 | 0.1253 | 0.1254 | 0.1257 | 0.1247 | 0.1232 | 0.1199 | 0.1302 | 0.1183 | 0.1242 | 0.1248 | 0.1211 | 0.1248 | 0.1225 | 0.1251 | 0.1245 | 0.1229 |
80/20 | 0.1252 | 0.1252 | 0.1253 | 0.1253 | 0.1255 | 0.1253 | 0.1249 | 0.1245 | 0.1220 | 0.1193 | 0.1280 | 0.1177 | 0.1239 | 0.1248 | 0.1207 | 0.1247 | 0.1210 | 0.1250 | 0.1239 | 0.1222 |
90/10 | 0.1251 | 0.1251 | 0.1254 | 0.1252 | 0.1253 | 0.1253 | 0.1243 | 0.1242 | 0.1213 | 0.1193 | 0.1273 | 0.1161 | 0.1234 | 0.1238 | 0.1196 | 0.1247 | 0.1193 | 0.1248 | 0.1235 | 0.1216 |
Mean | 0.1252 | 0.1253 | 0.1255 | 0.1254 | 0.1254 | 0.1254 | 0.1251 | 0.1246 | 0.1225 | 0.1198 | 0.1287 | 0.1179 | 0.1239 | 0.1246 | 0.1207 | 0.1249 | 0.1213 | 0.1251 | 0.1242 | 0.1225 |
Cluster 3: (positively skewed) unimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.0806 | 0.0802 | 0.0824 | 0.0797 | 0.0815 | 0.0815 | 0.0796 | 0.0764 | 0.0758 | 0.0708 | 0.0794 | 0.0735 | 0.0729 | 0.0785 | 0.0754 | 0.0742 | 0.0758 | 0.0754 | 0.0777 | 0.0695 |
70/30 | 0.0797 | 0.0790 | 0.0816 | 0.0796 | 0.0797 | 0.0798 | 0.0799 | 0.0757 | 0.0744 | 0.0702 | 0.0777 | 0.0729 | 0.0725 | 0.0787 | 0.0735 | 0.0749 | 0.0760 | 0.0751 | 0.0756 | 0.0674 |
80/20 | 0.0778 | 0.0780 | 0.0815 | 0.0802 | 0.0796 | 0.0795 | 0.0779 | 0.0760 | 0.0749 | 0.0695 | 0.0786 | 0.0733 | 0.0723 | 0.0787 | 0.0730 | 0.0738 | 0.0744 | 0.0751 | 0.0759 | 0.0669 |
90/10 | 0.0781 | 0.0783 | 0.0816 | 0.0813 | 0.0791 | 0.0800 | 0.0782 | 0.0758 | 0.0748 | 0.0677 | 0.0761 | 0.0726 | 0.0735 | 0.0765 | 0.0727 | 0.0729 | 0.0726 | 0.0756 | 0.0750 | 0.0637 |
Mean | 0.0790 | 0.0788 | 0.0818 | 0.0802 | 0.0800 | 0.0802 | 0.0789 | 0.0760 | 0.0750 | 0.0695 | 0.0780 | 0.0731 | 0.0728 | 0.0781 | 0.0737 | 0.0739 | 0.0747 | 0.0753 | 0.0760 | 0.0669 |
4.3 Logarithmic transformation of the positively skewed unimodally distributed LGDs
Split | OLS | bOLS | LAR | RR | LR | ER | FLR | RT | CIT | RF | ADA | GB | CUB | ANN | SVR | RVR | GAPR | KNN | MARS | FMM |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
60/40 | 0.0804 | 0.0809 | 0.0848 | 0.0811 | 0.0806 | 0.0817 | 0.0817 | 0.0659 | 0.0653 | 0.0556 | 0.0737 | 0.0618 | 0.0629 | 0.0806 | 0.0581 | 0.0694 | 0.0639 | 0.0761 | 0.0690 | 0.0444 |
70/30 | 0.0787 | 0.0776 | 0.0823 | 0.0782 | 0.0780 | 0.0776 | 0.0810 | 0.0652 | 0.0652 | 0.0553 | 0.0742 | 0.0617 | 0.0609 | 0.0768 | 0.0564 | 0.0655 | 0.0628 | 0.0751 | 0.0675 | 0.0434 |
80/20 | 0.0775 | 0.0789 | 0.0826 | 0.0779 | 0.0776 | 0.0776 | 0.0813 | 0.0651 | 0.0639 | 0.0549 | 0.0722 | 0.0597 | 0.0627 | 0.0760 | 0.0555 | 0.0655 | 0.0613 | 0.0746 | 0.0675 | 0.0441 |
90/10 | 0.0767 | 0.0800 | 0.0824 | 0.0772 | 0.0780 | 0.0775 | 0.0817 | 0.0652 | 0.0622 | 0.0531 | 0.0703 | 0.0618 | 0.0607 | 0.0757 | 0.0542 | 0.0656 | 0.0614 | 0.0698 | 0.0640 | 0.0444 |
Mean | 0.0783 | 0.0793 | 0.0830 | 0.0786 | 0.0786 | 0.0786 | 0.0814 | 0.0653 | 0.0641 | 0.0547 | 0.0726 | 0.0612 | 0.0618 | 0.0773 | 0.0561 | 0.0665 | 0.0623 | 0.0739 | 0.0670 | 0.0441 |
4.4 Non-European credit portfolios
Split | OLS | bOLS | LAR | RR | LR | ER | FLR | RT | CIT | RF | ADA | GB | CUB | ANN | SVR | RVR | GAPR | KNN | MARS | FMM |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Latin American credit portfolio: (nearly) symmetric bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.1403 | 0.1465 | 0.1391 | 0.1390 | 0.1476 | 0.1478 | 0.1383 | 0.1458 | 0.1414 | 0.1256 | 0.1429 | 0.1491 | 0.1472 | 0.1448 | 0.1429 | 0.1421 | 0.1258 | 0.1444 | 0.1461 | 0.1338 |
70/30 | 0.1388 | 0.1467 | 0.1388 | 0.1371 | 0.1468 | 0.1458 | 0.1367 | 0.1421 | 0.1395 | 0.1110 | 0.1395 | 0.1126 | 0.1112 | 0.1434 | 0.1415 | 0.1384 | 0.1035 | 0.1433 | 0.1459 | 0.1298 |
80/20 | 0.1344 | 0.1449 | 0.1366 | 0.1374 | 0.1444 | 0.1450 | 0.1361 | 0.1440 | 0.1374 | 0.1072 | 0.1375 | 0.1196 | 0.1183 | 0.1409 | 0.1413 | 0.1368 | 0.1331 | 0.1443 | 0.1449 | 0.1267 |
90/10 | 0.1353 | 0.1438 | 0.1378 | 0.1303 | 0.1434 | 0.1443 | 0.1351 | 0.1420 | 0.1260 | 0.1081 | 0.1378 | 0.1123 | 0.1106 | 0.1398 | 0.1362 | 0.1263 | 0.1082 | 0.1373 | 0.1406 | 0.1098 |
Mean | 0.1372 | 0.1455 | 0.1381 | 0.1359 | 0.1455 | 0.1457 | 0.1366 | 0.1435 | 0.1361 | 0.1130 | 0.1394 | 0.1234 | 0.1218 | 0.1422 | 0.1405 | 0.1359 | 0.1176 | 0.1423 | 0.1444 | 0.1250 |
North American credit portfolio: asymmetric (positively skewed) bimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.1363 | 0.1367 | 0.1363 | 0.1363 | 0.1402 | 0.1381 | 0.1334 | 0.1391 | 0.1390 | 0.1265 | 0.1438 | 0.1272 | 0.1334 | 0.1381 | 0.1346 | 0.1384 | 0.1313 | 0.1387 | 0.1371 | 0.1342 |
70/30 | 0.1361 | 0.1362 | 0.1361 | 0.1361 | 0.1395 | 0.1377 | 0.1331 | 0.1387 | 0.1358 | 0.1251 | 0.1429 | 0.1249 | 0.1287 | 0.1378 | 0.1339 | 0.1379 | 0.1286 | 0.1382 | 0.1305 | 0.1337 |
80/20 | 0.1344 | 0.1336 | 0.1345 | 0.1344 | 0.1388 | 0.1371 | 0.1313 | 0.1351 | 0.1362 | 0.1228 | 0.1411 | 0.1179 | 0.1241 | 0.1348 | 0.1334 | 0.1359 | 0.1264 | 0.1380 | 0.1284 | 0.1268 |
90/10 | 0.1289 | 0.1284 | 0.1289 | 0.1289 | 0.1348 | 0.1365 | 0.1258 | 0.1332 | 0.1290 | 0.1198 | 0.1393 | 0.1185 | 0.1262 | 0.1351 | 0.1289 | 0.1351 | 0.1253 | 0.1342 | 0.1250 | 0.1248 |
Mean | 0.1339 | 0.1337 | 0.1340 | 0.1339 | 0.1383 | 0.1373 | 0.1309 | 0.1365 | 0.1350 | 0.1236 | 0.1418 | 0.1221 | 0.1281 | 0.1364 | 0.1327 | 0.1368 | 0.1279 | 0.1373 | 0.1303 | 0.1299 |
Oceanian credit portfolio: (positively skewed) unimodal LGD distribution | ||||||||||||||||||||
60/40 | 0.0992 | 0.1019 | 0.1244 | 0.0994 | 0.0943 | 0.0927 | 0.1010 | 0.0951 | 0.1074 | 0.0764 | 0.0974 | 0.0893 | 0.0958 | 0.1062 | 0.0936 | 0.1094 | 0.0847 | 0.1263 | 0.1080 | 0.0651 |
70/30 | 0.1065 | 0.0951 | 0.1236 | 0.0951 | 0.0922 | 0.0933 | 0.0971 | 0.0918 | 0.1065 | 0.0801 | 0.0943 | 0.0888 | 0.0922 | 0.1058 | 0.0923 | 0.1077 | 0.0828 | 0.1254 | 0.1083 | 0.0560 |
80/20 | 0.1187 | 0.1007 | 0.1216 | 0.0977 | 0.0916 | 0.0922 | 0.1040 | 0.0919 | 0.1075 | 0.0748 | 0.0941 | 0.0885 | 0.0937 | 0.1054 | 0.0914 | 0.1016 | 0.0808 | 0.1262 | 0.1075 | 0.0514 |
90/10 | 0.1117 | 0.0964 | 0.1182 | 0.0959 | 0.0908 | 0.0914 | 0.0988 | 0.0904 | 0.1071 | 0.0764 | 0.0972 | 0.0882 | 0.0922 | 0.1014 | 0.0915 | 0.1023 | 0.0727 | 0.1241 | 0.1030 | 0.0410 |
Mean | 0.1090 | 0.0985 | 0.1220 | 0.0970 | 0.0922 | 0.0924 | 0.1002 | 0.0923 | 0.1071 | 0.0769 | 0.0957 | 0.0887 | 0.0935 | 0.1047 | 0.0922 | 0.1052 | 0.0803 | 0.1255 | 0.1067 | 0.0534 |