# Predicting Multi-Period Corporate Default Based on Bayesian Estimation of Forward Intensity—Evidence from China

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Research Questions and Main Work

#### 1.3. Contributions and Novelties

## 2. Literature Review

## 3. The Forward Intensity Model with Bayesian Estimation

## 4. Data and Preliminary Analysis

## 5. Empirical Results

#### 5.1. Parameter Estimates

#### 5.2. Prediction Accuracy Ratio

#### 5.3. Revised PDs of Firms with Default out of Sample

## 6. Conclusions

- (1)
- We found that the PDs or POEs of the firms with default records were significantly higher than the average level. The prediction performance of the forward intensity model can be improved by introducing firm’s default heterogeneity with information concerning a firm’s past default situation;
- (2)
- Bayesian estimation can help measure a firm’s default heterogeneity. By applying Bayesian estimation to a reduced-form credit risk model such as the forward intensity model, the original model can be optimized and the information of firm default heterogeneity can be taken into account to improve the prediction accuracy;
- (3)
- The empirical results show that the PDs revised by Bayesian estimation are higher than the original PDs of the original forward intensity model. For all prediction horizons, the accuracy ratio of the revised PDs out of sample increased by more than 10% as compared to the original PDs. Moreover, the accuracy ratio increased by almost 15% for prediction horizons of less than 6 months.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Monthly occurrence rate of default and other exits of Chinese-listed firms. Source: NUS-CRI database.

**Figure 2.**Monthly occurrence rate of default and other exits of Chinese-listed firms for the 7 to 240 months following default. Source: NUS-CRI database.

**Figure 3.**This figure shows the in-sample cumulative accuracy profiles of the revised PDs based on the experimental group (3513 firms) in the period 2000.01–2019.12 for different prediction horizons.

**Figure 4.**This figure shows the out-of-sample cumulative accuracy profiles of the revised PDs based on the evaluation group (703 firms) in the period 2000.01–2019.12 for different prediction horizons.

**Figure 5.**This figure shows the out-of-sample revised average cumulative PDs of the defaulted 42 firms in the period 2000.01–2019.12 as compared with the overall mean level for different prediction horizons.

**Figure 6.**This figure shows the ranking of out-of-sample revised average cumulative PDs of the defaulted 42 firms in the period 2000.01–2019.12 for all 703 firms out of sample for different prediction horizons.

Common Variables | Interpretation |
---|---|

1 Stock index return 2 Interest rate 3 Financial aggregate DTD 4 Non-financial aggregate DTD | Shanghai SE composite index China time deposit rate, 3 months median DTD of financial firms in China median DTD of non–financial firms in China |

Firm-Specific Variables | Interpretation |
---|---|

1. DTD (level) | the 1-year average of distance-to-default (DTD) |

2. DTD (trend) | the current value of DTD minus level of DTD |

3. CASH/TA (level) | level of {ln[(cash + short–term investments)/total assets]} (only applies to financial firms) |

4. CA/CL (level) | level of {ln[(cash + short–term investments)/total assets]} (only applies to financial firms) |

5. CA/CL (trend) | trend of [ln(current assets/current liabilities)] (only applies to non–financial firms) |

6. NI/TA (level) | level of (net income/total assets) |

7. NI/TA (trend) | trend of (net income/total assets) |

8. SIZE (level) | level of [ln(firm market capitalization/China’s median market capitalization over the past 1 year)] |

9. SIZE (trend) | trend of [ln(firm market capitalization/China’s median market capitalization over the past one year)] |

10. M/B | firm’s M/B (current value)/China’s M/B median (current value) |

11. SIGMA | current value of SIGMA is defined to be the standard deviation of the residuals of this regression |

12. Default record | If the firm has defaulted before, the value is 1, otherwise the value is 0. Only for the other exit intensity function. |

$\mathit{\gamma}\left(1\right)$ | $\mathit{\gamma}\left(2\right)$ | $\mathit{\gamma}\left(3\right)$ | $\mathit{\gamma}\left(6\right)$ | $\mathit{\gamma}\left(12\right)$ | $\mathit{\gamma}\left(24\right)$ | $\mathit{\gamma}\left(36\right)$ | |
---|---|---|---|---|---|---|---|

Intercept | −3.19567 | −3.22324 | −3.14231 | −2.75779 | −3.35508 | −3.48794 | −5.61305 |

Stock Index | 0.364704 | 0.345455 | 0.37122 | 0.495571 | 0.295483 | 0.036342 | 0.401967 |

Interest R | −0.09484 | −0.14648 | −0.20179 | −0.26951 | −0.27651 | −0.48271 | −0.59288 |

DTD (L) | −0.44485 | −0.43777 | −0.41021 | −0.37806 | −0.30378 | −0.23925 | −0.159 |

DTD (T) | −0.24401 | −0.25766 | −0.24311 | −0.17357 | −0.15487 | −0.16768 | −0.11836 |

Liquidity (L) | −0.35206 | −0.37554 | −0.3775 | −0.36836 | −0.36478 | −0.32862 | −0.35424 |

Liquidity (T) | −0.21799 | −0.13562 | −0.20819 | −0.36271 | −0.2559 | −0.67751 | 0.370419 |

NI/TA (L) | −27.1892 | −27.5629 | −29.3298 | −34.5291 | −49.0559 | −17.5311 | −11.7268 |

NI/TA(T) | −3.86283 | −5.04577 | −3.95147 | 1.605639 | 1.843335 | −6.48836 | −0.6641 |

Size (L) | −0.58644 | −0.58365 | −0.55998 | −0.53765 | −0.30256 | −0.12317 | 0.034523 |

Size (T) | −0.80865 | −0.78113 | −0.71216 | −1.01524 | −0.64005 | −0.41534 | −0.43872 |

M/B | 0.000531 | 0.002197 | 0.005594 | −0.01068 | −0.02653 | 0.040047 | 0.029517 |

SIGMA | 0.907969 | 0.815252 | −0.09427 | −2.19432 | −0.943 | 0.995337 | 0.990411 |

DTD median | 0.153046 | 0.173195 | 0.17692 | 0.111355 | 0.149462 | 0.12051 | 0.460361 |

$\mathit{\delta}\left(1\right)$ | $\mathit{\delta}\left(2\right)$ | $\mathit{\delta}\left(3\right)$ | $\mathit{\delta}\left(6\right)$ | $\mathit{\delta}\left(12\right)$ | $\mathit{\delta}\left(24\right)$ | $\mathit{\delta}\left(36\right)$ | |
---|---|---|---|---|---|---|---|

Intercept | −5.15746 | −4.60163 | −4.52391 | −3.51009 | −3.91152 | −5.01547 | −5.34153 |

Stock Index | 0.434484 | 0.451039 | 0.381008 | 0.410661 | 0.068886 | 0.432342 | −0.02859 |

Interest R | −0.07809 | −0.10849 | −0.13178 | −0.10656 | −0.03138 | 0.182722 | 0.265114 |

DTD (L) | −0.14544 | −0.16673 | −0.1723 | −0.13182 | −0.04259 | 0.012357 | −0.03725 |

DTD (T) | 0.164378 | 0.085248 | 0.0327 | 0.016497 | −0.1359 | 0.135816 | −0.09274 |

Liquidity (L) | −0.18405 | −0.17164 | −0.16201 | −0.08269 | −0.10729 | −0.03911 | −0.0625 |

Liquidity (T) | −0.47692 | −0.27531 | −0.35687 | −0.64659 | −0.80462 | −0.68153 | 0.217479 |

NI/TA (L) | −36.8954 | −35.3574 | −36.5993 | −33.0425 | −44.9344 | −42.5486 | −35.9121 |

NI/TA(T) | −1.82402 | 0.497658 | 2.619956 | 8.891169 | −1.08149 | −3.4806 | −2.80377 |

Size (L) | −0.1468 | −0.15416 | −0.15002 | −0.11128 | 0.012586 | 0.033836 | −0.01243 |

Size (T) | −0.02591 | −0.05804 | −0.39226 | −0.19978 | −0.39479 | 0.519196 | 1.02308 |

M/B | 0.001692 | 0.030278 | 0.023947 | 0.022703 | −0.05535 | −0.06474 | −0.0949 |

SIGMA | 4.698693 | 2.55448 | 2.967165 | 0.175135 | −0.74486 | −0.80469 | 0.271515 |

DTD median | −0.00836 | −0.04844 | −0.05661 | −0.27929 | −0.24043 | −0.14105 | −0.06458 |

Default or not | 1.179376 | 1.172637 | 1.186272 | 1.169366 | 1.060942 | 0.864298 | 0.766159 |

$\mathbf{Maximum}\mathbf{Pseudo}-\mathbf{Likelihood}\mathbf{Estimates}\mathbf{for}\mathit{\beta}\left(\mathit{l}\right):1\u201336\mathbf{Months}$ | ||||||||
---|---|---|---|---|---|---|---|---|

$\beta \left(1\right)$ | $\beta \left(2\right)$ | $\beta \left(3\right)$ | $\beta \left(4\right)$ | $\beta \left(5\right)$ | $\beta \left(6\right)$ | $\beta \left(7\right)$ | $\beta \left(8\right)$ | $\beta \left(9\right)$ |

226.581 | 238.277 | 269.5841 | 279.075 | 322.4939 | 357.731 | 390.9475 | 471.209 | 498.9649 |

$\beta \left(10\right)$ | $\beta \left(11\right)$ | $\beta \left(12\right)$ | $\beta \left(13\right)$ | $\beta \left(14\right)$ | $\beta \left(15\right)$ | $\beta \left(16\right)$ | $\beta \left(17\right)$ | $\beta \left(18\right)$ |

517.6742 | 577.8831 | 654.2477 | 621.2871 | 593.5402 | 693.7856 | 659.347 | 727.2746 | 698.5131 |

$\beta \left(19\right)$ | $\beta \left(20\right)$ | $\beta \left(21\right)$ | $\beta \left(22\right)$ | $\beta \left(23\right)$ | $\beta \left(24\right)$ | $\beta \left(25\right)$ | $\beta \left(26\right)$ | $\beta \left(27\right)$ |

763.7637 | 922.9653 | 1200.17 | 1240.038 | 1179.648 | 1167.229 | 1218.584 | 1107.113 | 976.984 |

$\beta \left(28\right)$ | $\beta \left(29\right)$ | $\beta \left(30\right)$ | $\beta \left(31\right)$ | $\beta \left(32\right)$ | $\beta \left(33\right)$ | $\beta \left(34\right)$ | $\beta \left(35\right)$ | $\beta \left(36\right)$ |

1076.028 | 1198.735 | 1166.115 | 1075.318 | 988.6297 | 1115.703 | 1175.41 | 1270.019 | 1331.149 |

Accuracy Ratio | 1 Month | 2 Months | 3 Months | 6 Months | 12 Months | 24 Months | 36 Months |
---|---|---|---|---|---|---|---|

Original PDs | 0.5882 | 0.5876 | 0.6032 | 0.5902 | 0.5696 | 0.5269 | 0.5161 |

Revised PDs | 0.6760 | 0.6766 | 0.6931 | 0.6905 | 0.6378 | 0.5996 | 0.5775 |

Increased (%) | 14.9 | 15.2 | 14.9 | 17.0 | 12.0 | 13.8 | 11.9 |

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## Share and Cite

**MDPI and ACS Style**

Ni, Z.; Jiang, M.; Zhan, W.
Predicting Multi-Period Corporate Default Based on Bayesian Estimation of Forward Intensity—Evidence from China. *Systems* **2023**, *11*, 18.
https://doi.org/10.3390/systems11010018

**AMA Style**

Ni Z, Jiang M, Zhan W.
Predicting Multi-Period Corporate Default Based on Bayesian Estimation of Forward Intensity—Evidence from China. *Systems*. 2023; 11(1):18.
https://doi.org/10.3390/systems11010018

**Chicago/Turabian Style**

Ni, Zhengfang, Minghui Jiang, and Wentao Zhan.
2023. "Predicting Multi-Period Corporate Default Based on Bayesian Estimation of Forward Intensity—Evidence from China" *Systems* 11, no. 1: 18.
https://doi.org/10.3390/systems11010018