# Firm’s Credit Risk in the Presence of Market Structural Breaks

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## Abstract

**:**

## 1. Introduction

## 2. Our Modeling Approach and Related Literature

## 3. A Modulated Semi-Markov Model

#### 3.1. Information Filtration

#### 3.2. Conventional Models for Firms’ Rating Transition Intensities

#### 3.3. Our Specification for Firms’ Rating Transition Intensities

#### 3.4. Dynamics of Market Structural Breaks

- (A1)
- The number of jumps in ${\mathit{\beta}}^{(i,j)}(t)$ follows a Poisson process $\{{J}^{(i,j)}(t);t\ge 0\}$ with rate $\eta $ and are independent of ${\mathbf{X}}_{l}(t)$.
- (A2)
- If a jump occurs at time t, the post-change value of ${\mathit{\theta}}^{(i,j)}(t)$ is independent of its pre-change value, in particular, denote ${\mathit{\theta}}^{(i,j)}(t)={\mathit{\omega}}_{{J}^{(i,j)}(t)}^{(i,j)}$, where ${\mathit{\omega}}_{0}^{(i,j)},{\mathit{\omega}}_{1}^{(i,j)},{\mathit{\omega}}_{2}^{(i,j)},\dots $ are independent and identically distributed (i.i.d.) normal random vectors with mean ${\mathit{\mu}}^{(i,j)}$ and covariance ${\mathbf{V}}^{(i,j)}$.

## 4. Inference Procedure

#### 4.1. Inference When No Structural Breaks Exist

#### 4.2. Mixed Estimating Equations

#### 4.3. Estimation of Informative Prior

## 5. An Empirical Study

#### 5.1. Data Description

#### 5.2. Effect of Firm-Specific Variables on Firms’ Credit Risk and Baseline Cumulative Intensities

#### 5.3. Firms’ Rating Transition Intensities and Probabilities

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. A Quasi-EM Approach to Estimate Hyperparameters

- (a)
- $P\left({\mathit{\theta}}^{(i,j)}({t}_{h})\ne {\mathit{\theta}}^{(i,j)}({t}_{h-1})\right)|{\mathcal{F}}_{(0,{t}_{H})})$.
- (b)
- $E\left(logP(d{N}_{l}^{(i,j)}({t}_{h})=1|d{N}_{\xb7}^{(i,j)}({t}_{h})=1,{\mathcal{G}}_{{t}_{h}-})|{\mathcal{F}}_{(0,{t}_{H})}\right)$.
- (c)
- $E\left({({\mathit{\theta}}^{(i,j)}({t}_{h})-{\mathit{\mu}}^{(i,j)})}^{T}{\left[{\mathbf{V}}^{(i,j)}\right]}^{-1}({\mathit{\theta}}^{(i,j)}({t}_{h})-{\mathit{\mu}}^{(i,j)})|{\mathcal{F}}_{(0,{t}_{H})}\right)$.

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**Figure 1.**${\widehat{\theta}}_{1}^{(i,j)}(t)$ (solid) and their 95% confidence bands (dotted) for firms’ distance to default. Note that the values of ${\widehat{\theta}}_{1}^{(i,j)}$ for the corresponding transitions (from left to right and top to down) are 0.069, 0.039, 0.075, −0.004, −0.102, −0.204, −0.426, and −0.378, respectively.

**Figure 2.**${\widehat{\theta}}_{2}^{(i,j)}(t)$ (solid) and their 95% confidence bands (dotted) for firms’ trailing returns. Note that the values of ${\widehat{\theta}}_{2}^{(i,j)}$ for the corresponding transitions (from left to right and top to down) are −1.695, −0.135, −1.277, −0.001, −1.272, −1.088, −1.164, and −4.052, respectively.

**Figure 3.**Estimated baseline cumulative intensities with (solid) and without (dashed) structural breaks.

$\mathcal{AAA}$ | $\mathcal{AA}$ | $\mathcal{A}$ | $\mathcal{BBB}$ | $\mathcal{BB}$ | $\mathcal{B}$ | $\mathcal{CCC}$ | $\mathcal{D}$ | |

January 1986–September 2012 (without structural break assumption) | ||||||||

$\mathcal{AAA}$ | 0.9993 | 7 × 10${}^{-4}$ | 3 × 10${}^{-6}$ | 2 × 10${}^{-8}$ | 2 × 10${}^{-10}$ | 2 × 10${}^{-11}$ | 1 × 10${}^{-12}$ | 2 × 10${}^{-14}$ |

$\mathcal{AA}$ | 1 × 10${}^{-4}$ | 0.9927 | 7 × 10${}^{-3}$ | 9 × 10${}^{-5}$ | 8 × 10${}^{-7}$ | 1 × 10${}^{-7}$ | 9 × 10${}^{-9}$ | 1 × 10${}^{-10}$ |

$\mathcal{A}$ | 9 × 10${}^{-8}$ | 0.0012 | 0.9742 | 0.0241 | 3 × 10${}^{-4}$ | 5 × 10${}^{-5}$ | 5 × 10${}^{-6}$ | 1 × 10${}^{-7}$ |

$\mathcal{BBB}$ | 3 × 10${}^{-10}$ | 6 × 10${}^{-6}$ | 0.0098 | 0.9577 | 0.0274 | 0.0043 | 6 × 10${}^{-4}$ | 2 × 10${}^{-5}$ |

$\mathcal{BB}$ | 1 × 10${}^{-12}$ | 4 × 10${}^{-8}$ | 9 × 10${}^{-5}$ | 0.0183 | 0.9173 | 0.0552 | 0.0087 | 2 × 10${}^{-4}$ |

$\mathcal{B}$ | 7 × 10${}^{-15}$ | 3 × 10${}^{-10}$ | 8 × 10${}^{-7}$ | 2 × 10${}^{-4}$ | 0.0219 | 0.7215 | 0.2463 | 0.0100 |

$\mathcal{CCC}$ | 3 × 10${}^{-17}$ | 1 × 10${}^{-12}$ | 5 × 10${}^{-9}$ | 2 × 10${}^{-6}$ | 3 × 10${}^{-4}$ | 0.0208 | 0.9066 | 0.0722 |

October 1994–March 2001 (with structural break assumption) | ||||||||

$\mathcal{AAA}$ | 0.9998 | 2 × 10${}^{-4}$ | 3 × 10${}^{-7}$ | 6 × 10${}^{-10}$ | 2 × 10${}^{-13}$ | 1 × 10${}^{-14}$ | 2 × 10${}^{-20}$ | 5 × 10${}^{-24}$ |

$\mathcal{AA}$ | 2 × 10${}^{-5}$ | 0.9968 | 0.0031 | 1 × 10${}^{-5}$ | 5 × 10${}^{-9}$ | 3 × 10${}^{-10}$ | 5 × 10${}^{-16}$ | 2 × 10${}^{-19}$ |

$\mathcal{A}$ | 3 × 10${}^{-9}$ | 3 × 10${}^{-4}$ | 0.9933 | 0.0063 | 4 × 10${}^{-6}$ | 2 × 10${}^{-7}$ | 6 × 10${}^{-13}$ | 2 × 10${}^{-16}$ |

$\mathcal{BBB}$ | 4 × 10${}^{-12}$ | 7 × 10${}^{-7}$ | 0.0041 | 0.9944 | 0.0013 | 8 × 10${}^{-5}$ | 3 × 10${}^{-10}$ | 9 × 10${}^{-14}$ |

$\mathcal{BB}$ | 5 × 10${}^{-15}$ | 1 × 10${}^{-9}$ | 1 × 10${}^{-5}$ | 0.0055 | 0.9935 | 0.0010 | 4 × 10${}^{-9}$ | 1 × 10${}^{-12}$ |

$\mathcal{B}$ | 3 × 10${}^{-18}$ | 9 × 10${}^{-12}$ | 1 × 10${}^{-8}$ | 8 × 10${}^{-6}$ | 0.0028 | 0.9971 | 8 × 10${}^{-6}$ | 2 × 10${}^{-9}$ |

$\mathcal{CCC}$ | 2 × 10${}^{-23}$ | 7 × 10${}^{-18}$ | 1 × 10${}^{-13}$ | 1 × 10${}^{-10}$ | 6 × 10${}^{-8}$ | 4 × 10${}^{-5}$ | 0.9999 | 4 × 10${}^{-9}$ |

April 2007–January 2010 (with structural break assumption) | ||||||||

$\mathcal{AAA}$ | 0.9999 | 1 × 10${}^{-4}$ | 1 × 10${}^{-8}$ | 7 × 10${}^{-12}$ | 3 × 10${}^{-15}$ | 3 × 10${}^{-17}$ | 5 × 10${}^{-23}$ | 9 × 10${}^{-27}$ |

$\mathcal{AA}$ | 7 × 10${}^{-6}$ | 0.9998 | 0.0002 | 2 × 10${}^{-7}$ | 1 × 10${}^{-10}$ | 1 × 10${}^{-12}$ | 2 × 10${}^{-18}$ | 4 × 10${}^{-22}$ |

$\mathcal{A}$ | 9 × 10${}^{-11}$ | 3 × 10${}^{-5}$ | 0.9981 | 0.0018 | 2 × 10${}^{-6}$ | 1 × 10${}^{-8}$ | 4 × 10${}^{-14}$ | 8 × 10${}^{-18}$ |

$\mathcal{BBB}$ | 4 × 10${}^{-14}$ | 1 × 10${}^{-8}$ | 0.0011 | 0.9972 | 0.0017 | 2 × 10${}^{-5}$ | 6 × 10${}^{-11}$ | 1 × 10${}^{-14}$ |

$\mathcal{BB}$ | 2 × 10${}^{-17}$ | 1 × 10${}^{-11}$ | 1 × 10${}^{-6}$ | 0.0023 | 0.9967 | 9 × 10${}^{-4}$ | 4 × 10${}^{-9}$ | 8 × 10${}^{-13}$ |

$\mathcal{B}$ | 7 × 10${}^{-21}$ | 5 × 10${}^{-15}$ | 7 × 10${}^{-10}$ | 2 × 10${}^{-6}$ | 0.0016 | 0.9984 | 8 × 10${}^{-6}$ | 2 × 10${}^{-9}$ |

$\mathcal{CCC}$ | 1 × 10${}^{-25}$ | 8 × 10${}^{-20}$ | 2 × 10${}^{-14}$ | 6 × 10${}^{-11}$ | 7 × 10${}^{-8}$ | 9 × 10${}^{-5}$ | 0.9999 | 1 × 10${}^{-5}$ |

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**MDPI and ACS Style**

Xing, H.; Yu, Y. Firm’s Credit Risk in the Presence of Market Structural Breaks. *Risks* **2018**, *6*, 136.
https://doi.org/10.3390/risks6040136

**AMA Style**

Xing H, Yu Y. Firm’s Credit Risk in the Presence of Market Structural Breaks. *Risks*. 2018; 6(4):136.
https://doi.org/10.3390/risks6040136

**Chicago/Turabian Style**

Xing, Haipeng, and Yang Yu. 2018. "Firm’s Credit Risk in the Presence of Market Structural Breaks" *Risks* 6, no. 4: 136.
https://doi.org/10.3390/risks6040136