# The Liquidity Premium in China’s Corporate Bond Market: A Stochastic Liquidity Discount Approach

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## Abstract

**:**

## 1. Introduction

## 2. The Bond Price Model with Liquidity Effect

## 3. Stochastic Liquidity Discount Process

## 4. Monte Carlo Simulation

- (1)
- Simulate a liquidity shock time ${\tau}_{l}$ using ${\tau}_{l}\sim Exp\left({\lambda}_{l}\right)$.
- (2)
- Simulate a random variable ${\epsilon}_{1}$ using ${\epsilon}_{1}\sim N(0,1)$, and calculate the corporate asset value at the time of liquidity shock ${\tau}_{l}$ as $V}_{{\tau}_{l}}={V}_{t}{e}^{(r-0.5{\sigma}_{v}^{2})({\tau}_{l}-t)+{\sigma}_{v}{\epsilon}_{1}\sqrt{{\tau}_{l}}$. Then $P({\tau}_{l},T)$ is calculated by Equation (2).
- (3)
- Simulate the path of ${\alpha}_{u}$ for $t\le u\le {\tau}_{l}$, using the Milstein Scheme$${\alpha}_{u+\Delta u}={\alpha}_{u}+\mu \left({\alpha}_{u}\right)\Delta u+\sigma \left({\alpha}_{u}\right){\epsilon}_{2}\sqrt{\Delta u}+\frac{1}{4}{\sigma}_{\alpha}(\overline{\alpha}+\underline{\alpha}-2{\alpha}_{u})({\epsilon}_{2}^{2}-1)\Delta u,$$
- (4)
- Compute X using Equation (13).
- (5)
- Repeat steps $1-4$ for n times and obtain a sample with size n of X, denoted by $\{{x}_{1},{x}_{2},\cdots ,{x}_{n}\}$. ${E}_{\mathbb{Q}}[X\left|{\mathcal{G}}_{t}\right]$ is estimated by the sample mean $\overline{x}\left(n\right)$, which is$$\begin{array}{c}\overline{x}\left(n\right)=\frac{1}{n}\sum _{i=1}^{n}{x}_{i},\end{array}$$$$\left[\overline{x}\left(n\right)-{t}_{1-\frac{\pi}{2},n-1}\frac{S\left(n\right)}{\sqrt{n}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\overline{x}\left(n\right)+{t}_{1-\frac{\pi}{2},n-1}\frac{S\left(n\right)}{\sqrt{n}}\right],$$$$\begin{array}{c}S\left(n\right)=\sqrt{\frac{1}{n-1}\sum _{i=1}^{n}{[{x}_{i}-\overline{x}\left(n\right)]}^{2}}.\end{array}$$Given an allowed error in the Monte Carlo simulation $\u03f5$, the number of samples in the Monte Carlo simulation, n, should satisfy the condition:$${t}_{1-\frac{\pi}{2},n-1}\frac{S\left(n\right)}{\sqrt{n}}<\u03f5.$$

## 5. Parameter Specification

#### 5.1. The Data

#### 5.2. Risk-Free Interest Rate

#### 5.3. Corporate Asset Value

#### 5.4. Volatility of Corporate Assets

#### 5.5. Liquidity Shock Intensity

#### 5.6. Parameter Calibration

## 6. Numerical Analysis

#### 6.1. Existence of a Liquidity Premium

#### 6.2. The Effect of Liquidity Level, Risk, and Elasticity

#### 6.3. Term Structure of the Liquidity Premium

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | Retrieved from http://www.gtarsc.com/ (accessed on 8 May 2022). |

2 | Retrieved from http://www.resset.com/ (accessed on 8 May 2022). |

3 | Retrieved from http://www.wind.com.cn/ (accessed on 19 May 2022). |

4 | The tables in the paper use three important descriptive statistics: Mean, Standard Deviation (SD), and Percentiles. |

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Mean | SD ^{1} | Percentile | |||||||
---|---|---|---|---|---|---|---|---|---|

0% | 10% | 30% | 50% | 70% | 90% | 100% | |||

IORR ^{2} | 2.41 | 1.22 | 0.81 | 1.10 | 1.77 | 2.27 | 2.82 | 3.76 | 11.74 |

^{1}SD represents standard deviation;

^{2}Interest rates are expressed in percentages.

Mean | SD | Percentile | |||||||
---|---|---|---|---|---|---|---|---|---|

0% | 10% | 30% | 50% | 70% | 90% | 100% | |||

${\eta}_{t}$ | 0.35 | 0.17 | 0.02 | 0.14 | 0.23 | 0.33 | 0.46 | 0.60 | 0.81 |

Term * | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 10 | 15 |
---|---|---|---|---|---|---|---|---|---|

Issue number | 2 | 49 | 3 | 262 | 16 | 76 | 18 | 46 | 3 |

Issue amount | 5.20 | 80.94 | 0.85 | 387.06 | 15.34 | 90.11 | 19.87 | 153.31 | 9.50 |

Mean | SD | Percentile | |||||||
---|---|---|---|---|---|---|---|---|---|

0% | 10% | 30% | 50% | 70% | 90% | 100% | |||

${\sigma}_{e}$ | 0.54 | 0.09 | 0.26 | 0.43 | 0.50 | 0.55 | 0.57 | 0.62 | 1.07 |

${\sigma}_{v}$ | 0.36 | 0.12 | 0.08 | 0.21 | 0.30 | 0.36 | 0.43 | 0.49 | 0.91 |

Mean | SD | Percentile | |||||||
---|---|---|---|---|---|---|---|---|---|

0% | 10% | 30% | 50% | 70% | 90% | 100% | |||

${\lambda}_{l}$ | 0.61 | 0.99 | 0.00 | 0.00 | 0.09 | 0.30 | 0.65 | 1.35 | 7.92 |

Mean | SD | Percentile | |||||||
---|---|---|---|---|---|---|---|---|---|

0% | 10% | 30% | 50% | 70% | 90% | 100% | |||

${\varphi}_{\alpha}$^{1} | 99.55 | 0.86 | 90.49 | 99.08 | 99.61 | 99.77 | 99.88 | 99.95 | 99.99 |

$\overline{\alpha}\phantom{\rule{3.33333pt}{0ex}}$^{2} | 99.99 | 0.08 | 98.87 | 99.99 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |

$\underline{\alpha}\phantom{\rule{3.33333pt}{0ex}}$^{3} | 97.38 | 2.48 | 81.55 | 94.11 | 96.84 | 98.12 | 98.85 | 99.65 | 99.95 |

${\sigma}_{\alpha}$ | 31.88 | 103.60 | 0.43 | 3.50 | 8.14 | 12.14 | 18.05 | 39.88 | 1029.347 |

${\kappa}_{\alpha}$ | 27.53 | 75.54 | 0.12 | 3.91 | 10.43 | 17.30 | 25.37 | 36.79 | 976.75 |

^{1,2,3}${\varphi}_{\alpha}$, $\overline{\alpha}\phantom{\rule{3.33333pt}{0ex}}$, and $\underline{\alpha}\phantom{\rule{3.33333pt}{0ex}}$ are expressed in percentages.

Level Percentile | 0% | 10% | 30% | 50% | 70% | 90% | 100% |
---|---|---|---|---|---|---|---|

LPS | 7.50 | 0.96 | 0.48 | 0.27 | 0.16 | 0.05 | 0.01 |

LYS ^{1} | 1.57 | 0.19 | 0.10 | 0.05 | 0.03 | 0.01 | 0.00 |

LYSR ^{2} | 59.47 | 15.08 | 8.55 | 4.46 | 2.73 | 0.93 | 0.00 |

^{1, 2}LYS and LYSR are expressed in percentages.

Risk | Percentile of ${\mathit{\lambda}}_{\mathit{l}}$ ^{1} | Percentile of ${\mathit{\sigma}}_{\mathit{\alpha}}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Level | 30% | 50% | 90% | 100% | 0% | 10% | 50% | 90% | 100% | |

0% | 3.29 ^{2} | 6.49 | 7.67 | 7.67 | 7.50 | 7.48 | 7.49 | 7.51 | 7.70 | |

(0.67) ^{3} | (1.35) | (1.61) | (1.61) | (1.57) | (1.57) | (1.57) | (1.57) | (1.61) | ||

10% | 0.43 | 0.84 | 0.98 | 0.96 | 0.72 | 0.72 | 0.76 | 1.05 | 2.24 | |

(0.09) | (0.17) | (0.20) | (0.19) | (0.14) | (0.14) | (0.15) | (0.21) | (0.45) | ||

30% | 0.21 | 0.41 | 0.48 | 0.46 | 0.31 | 0.30 | 0.34 | 0.52 | 1.20 | |

(0.04) | (0.08) | (0.10) | (0.09) | (0.06) | (0.06) | (0.07) | (0.10) | (0.24) | ||

50% | 0.12 | 0.24 | 0.28 | 0.27 | 0.18 | 0.19 | 0.20 | 0.31 | 0.72 | |

(0.02) | (0.05) | (0.06) | (0.05) | (0.04) | (0.04) | (0.04) | (0.06) | (0.14) | ||

70% | 0.08 | 0.14 | 0.16 | 0.16 | 0.10 | 0.10 | 0.11 | 0.18 | 0.44 | |

(0.02) | (0.03) | (0.03) | (0.03) | (0.02) | (0.02) | (0.02) | (0.04) | (0.09) | ||

90% | 0.02 | 0.05 | 0.06 | 0.05 | 0.04 | 0.04 | 0.05 | 0.06 | 0.13 | |

(0.00) | (0.01) | (0.01) | (0.01) | (0.01) | (0.01) | (0.01) | (0.01) | (0.03) | ||

100% | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | |

(0.00) | (0.00) | (0.00) | (0.00) | (0.00) | (0.00) | (0.00) | (0.00) | (0.01) |

^{1}The zero and tenth percentiles of ${\lambda}_{l}$ are equal to zero and not considered.

^{2}The numbers without parentheses are the liquidity price spread.

^{3}The numbers in parentheses are the liquidity yield spread as a percentage.

Level | 0% | 10% | 30% | 50% | 70% | 90% | 100% | |
---|---|---|---|---|---|---|---|---|

Elasticity | ||||||||

0% | 7.69 ^{1} | 2.19 | 1.17 | 0.71 | 0.43 | 0.12 | 0.02 | |

(1.61) ^{2} | (0.44) | (0.24) | (0.14) | (0.09) | (0.02) | (0.00) | ||

10% | 7.65 | 1.89 | 0.98 | 0.60 | 0.35 | 0.11 | 0.01 | |

(1.60) | (0.38) | (0.20) | (0.12) | (0.07) | (0.02) | (0.00) | ||

50% | 7.51 | 1.14 | 0.57 | 0.33 | 0.20 | 0.07 | 0.01 | |

(1.57) | (0.23) | (0.11) | (0.07) | (0.04) | (0.01) | (0.00) | ||

90% | 7.51 | 0.89 | 0.42 | 0.25 | 0.15 | 0.05 | 0.01 | |

(1.57) | (0.18) | (0.08) | (0.05) | (0.03) | (0.01) | (0.00) | ||

100% | 7.70 | 2.32 | 1.24 | 0.73 | 0.45 | 0.14 | 0.02 | |

(1.61) | (0.47) | (0.25) | (0.15) | (0.09) | (0.03) | (0.00) |

^{1}The numbers without parentheses are the liquidity price spread.

^{2}The numbers in parentheses are the liquidity yield spread as a percentage.

${\mathit{\lambda}}_{\mathit{l}}$ | ${\mathit{\varphi}}_{\mathit{\alpha}}$ | $\overline{\mathit{\alpha}}\phantom{\rule{3.33333pt}{0ex}}$ | $\underline{\mathit{\alpha}}\phantom{\rule{3.33333pt}{0ex}}$ | ${\mathit{\sigma}}_{\mathit{\alpha}}$ | ${\mathit{\kappa}}_{\mathit{\alpha}}$ | |
---|---|---|---|---|---|---|

c | $1.06$ *** ^{1} | $99.92$ *** | $100.00$ *** | $99.82$ *** | $-35.20$ * | 4.38 |

(0.00) ^{2} | (0.00) | (0.00) | (0.00) | (0.09) | (0.77) | |

$\beta $ | $-0.08$ *** | $-0.06$ ** | −0.00 | $-0.43$ *** | $11.70$ *** | 4.04 |

(0.00) | (0.03) | (0.48) | (0.00) | (0.00) | (0.11) |

^{1}“***, **, *” denote significance of 1%, 5% and 10% respectively.

^{2}The numbers in parentheses are p-values.

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## Share and Cite

**MDPI and ACS Style**

Min, X.; Ji, M.
The Liquidity Premium in China’s Corporate Bond Market: A Stochastic Liquidity Discount Approach. *Risks* **2022**, *10*, 130.
https://doi.org/10.3390/risks10070130

**AMA Style**

Min X, Ji M.
The Liquidity Premium in China’s Corporate Bond Market: A Stochastic Liquidity Discount Approach. *Risks*. 2022; 10(7):130.
https://doi.org/10.3390/risks10070130

**Chicago/Turabian Style**

Min, Xiaoping, and Min Ji.
2022. "The Liquidity Premium in China’s Corporate Bond Market: A Stochastic Liquidity Discount Approach" *Risks* 10, no. 7: 130.
https://doi.org/10.3390/risks10070130