# An Urn-Based Nonparametric Modeling of the Dependence between PD and LGD with an Application to Mortgages

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## Abstract

**:**

## 1. Introduction

- An intuitive bivariate model is proposed for the joint modeling of PD and LGD. The construction exploits the power of Polya urns to generate a Bayesian nonparametric approach to wrong-way risk. The model can be interpreted as a mixture model, following the typical credit risk management classification (McNeil et al. 2015).
- The proposed model is able to combine prior beliefs with empirical evidence and, exploiting the reinforcement mechanism embedded in Polya urns, it learns, thus improving its performances over time.
- The ability of learning and improving gives the model a machine/deep learning flavour. However, differently from the common machine/deep learning approaches, the behavior of the new model can be controlled and studied in a rigorous way from a probabilistic point of view. In other words, the common “black box” argument (Knight 2017) of machine/deep learning does not apply.
- The possibility of eliciting an a priori allows for the exploitation of experts’ judgements, which can be extremely useful when dealing with rare events, historical bias and data problems in general (Cheng and Cirillo 2018; Derbyshire 2017; Shackle 1955).
- The model we propose can only deal with positive dependence. Given the empirical literature, this is not a problem in WWR modeling; however, it is important to be aware of this feature, if other applications are considered.

## 2. Model

#### 2.1. The Two-Color RUP

**Definition**

**1**

**.**A random distribution function F is a beta-Stacy process with jumps at $j\in {\mathbb{N}}_{0}$ and parameters ${\{{\alpha}_{j},{\beta}_{j}\}}_{j\in {\mathbb{N}}_{0}}$, if there exist mutually independent random variables ${\{{V}_{j}\}}_{j\in {\mathbb{N}}_{0}}$, each beta distributed with parameters $({\alpha}_{j},{\beta}_{j})$, such that the random mass assigned by F to $\{j\}$, written $F(\{j\})$, is given by ${V}_{j}{\prod}_{i<j}(1-{V}_{i})$.

#### 2.2. Modeling Dependence

- (1)
- Given the observations ${\mathbf{X}}_{m}={\mathbf{x}}_{m}$ and ${\mathbf{Y}}_{m}={\mathbf{y}}_{m}$, the sequence ${\mathbf{A}}_{m}=({A}_{1},\dots ,{A}_{m})$ is generated via a Gibbs sampling. The full conditional of ${A}_{m}$, $P[{A}_{m}={a}_{m}\mid {\mathbf{A}}_{m-1}={\mathbf{a}}_{m-1},{\mathbf{X}}_{m}={\mathbf{x}}_{m},{\mathbf{Y}}_{m}={\mathbf{y}}_{m}]$, is such that$$\begin{array}{ccc}\hfill P[{A}_{m}={a}_{m}\mid {\mathbf{A}}_{m-1}={\mathbf{a}}_{m-1},{\mathbf{X}}_{m}={\mathbf{x}}_{m},{\mathbf{Y}}_{m}={\mathbf{y}}_{m}]& \propto & P[{A}_{m}={a}_{m}\mid {\mathbf{A}}_{m-1}={\mathbf{a}}_{m-1}]\hfill \\ & & \times P[{X}_{m}-{A}_{m}={x}_{m}-{a}_{m}\mid {\mathbf{B}}_{m-1}={\mathbf{b}}_{m-1}]\hfill \\ & & \times P[{Y}_{m}-{A}_{m}={y}_{m}-{a}_{m}\mid {\mathbf{C}}_{m-1}={\mathbf{c}}_{m-1}].\hfill \end{array}$$Since ${\{{A}_{j}\}}_{j=1}^{m}$ is exchangeable, all the other full conditionals, $P[{A}_{j}={a}_{j}\mid {\mathbf{A}}_{-j}={\mathbf{a}}_{-j},{\mathbf{X}}_{m}={\mathbf{x}}_{m},{\mathbf{Y}}_{m}={\mathbf{y}}_{m}]$, where ${\mathbf{A}}_{-j}=({A}_{1},\cdots ,{A}_{j-1},{A}_{j+1},\cdots ,{A}_{m})$, have an analogous form.
- (2)
- Once ${\mathbf{A}}_{m}$ is obtained, compute ${\mathbf{B}}_{m}={\mathbf{X}}_{m}-{\mathbf{A}}_{m}$ and ${\mathbf{C}}_{m}={\mathbf{Y}}_{m}-{\mathbf{A}}_{m}$.
- (3)
- The quantities ${A}_{m+1}$, ${B}_{m+1}$, and ${C}_{m+1}$ are then sampled according to their beta-Stacy predictive distributions $P({A}_{m+1}\mid {\mathbf{A}}_{m})$, $P({B}_{m+1}\mid {\mathbf{B}}_{m})$, and $P({C}_{m+1}\mid {\mathbf{C}}_{m})$ as per Equation (3).
- (4)
- Finally, set ${X}_{m+1}={A}_{m+1}+{B}_{m+1}$ and ${Y}_{m+1}={A}_{m+1}+{C}_{m+1}$.

## 3. Data

## 4. Results

#### 4.1. Discretisation

#### 4.2. Prior Elicitation

- Independent discrete uniforms for ${\{{A}_{i}\}}_{i=1}^{m}$, ${\{{B}_{i}\}}_{i=1}^{m}$ and ${\{{C}_{i}\}}_{i=1}^{m}$, where the range of variation for B and C is simply inherited from X and Y (but extra conditions can be applied, if needed), while for A the range is chosen to guarantee ${\sigma}_{A}^{2}=Cov(X,Y)$. For instance, if the covariance between X and Y is approximately 3, the interval $[0,5]$ guarantees that ${\sigma}_{A}^{2}\approx 3$ as well. We can simply use the formula for the variance of a discrete uniform, i.e.,$${\sigma}^{2}=\frac{{(b-a+1)}^{2}-1}{12}.$$
- Independent Poisson distributions, such that $A\sim Poi({\lambda}_{A}=Cov(X,Y))$, while for B and C one sets $Poi(\overline{X}-{\lambda}_{A})$ and $Poi(\overline{Y}-{\lambda}_{A})$, where $\overline{X}$ is the empirical mean of X. This guarantees, for example, that $X\sim Poi\left(\overline{X}\right)$. Recall in fact that, in a Poisson random variable, the mean and the variance are both equal to the intensity parameter, and independent Poissons are closed under convolution. Given our data, where the empirical variances of X and Y are not at all equal to the empirical means, but definitely larger, the Poisson prior can be seen as an example of a wrong prior.

#### 4.3. Fitting

#### 4.4. What about the Crisis?

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | In reality, as observed in Zhang and Thomas (2012), the LGD can be slightly negative or slightly above 100% because of fees and interests; however, we exclude that situation here. In terms of applications, all negative values can be set to 0, and all values above 100 can be rounded to 100. |

2 | Please observe that exchangeability only applies among the couples ${\{({X}_{i},{Y}_{i})\}}_{i=1}^{m}$, while within each couple there is a clear dependence, so that ${X}_{i}$ and ${Y}_{i}$ are not exchangeable. |

3 | To avoid any copyright problem with Freddie Mac, which already freely shares its data online, from Maio’s dataset (here attached), we only provide the PD and the LGD estimates, together with the unique alphanumeric identifier. In this way, merging the data sources is straightforward. |

**Figure 1.**Plots of PD vs. LGD for mortgages in the “Very poor” (FICO score below 579) class—on the left, a simple scatter plot; on the right, a hexagonal heatmap with counts. (

**a**) scatter plot; (

**b**) hexagonal heatmap.

**Figure 2.**Histograms of the marginal distributions of PD and LGD in the “Very poor” rating class. (

**a**) PD; (

**b**) LGD.

**Figure 3.**Prior, posterior and ECDF for the discretised PD and LGD in the “Very poor” rating class, when priors are uniform and levels are like those in Equation (12). The strength of belief is always set to 1. (

**a**) PD; (

**b**) LGD.

**Figure 4.**Prior, posterior and ECDF for the discretised PD and LGD in the “Very poor” rating class, when priors are uniform and levels are like those in Equation (12). The strength of belief is always set to 100. (

**a**) PD; (

**b**) LGD.

**Figure 5.**Prior, posteriors (${c}_{j}=1$ and ${c}_{j}$ = 100) and ECDF for the discretised PD in the “Exceptional” rating class, when priors are uniform.

**Figure 6.**Bivariate density distribution of PD and LGD in the “Very poor” rating class, with Poisson priors and strength of belief equal to 1.

**Figure 7.**Bivariate density distribution of PD and LGD in the “Very poor” rating class, with Poisson priors and strength of belief equal to 100.

**Figure 8.**Bivariate density distribution of PD and LGD in the “Exceptional” rating class, with Poisson priors and strength of belief equal to 1.

**Figure 9.**Predictive distribution generated by the bivariate urn model for the “Very poor” class when trained on the training sample (1590 loans), against the empirical ECDF from the validation set (37 loans).

**Table 1.**Some descriptive information about the data used in the analysis. Loans are collected in terms of FICO score.

Class | Number of Loans | Avg. PD | Avg. LGD | $\mathit{\rho}$ |
---|---|---|---|---|

Very Poor | 1627 | 0.0378 | 0.1013 | 0.3370 |

Fair | 46,720 | 0.0238 | 0.1237 | 0.4346 |

Good | 124,824 | 0.0138 | 0.1409 | 0.3159 |

Very good | 177,891 | 0.0083 | 0.1574 | 0.3599 |

Exceptional | 32,403 | 0.0080 | 0.1777 | 0.1858 |

**Table 2.**PD values (%) and correlation. Some descriptive descriptive statistics (mean, median, standard deviation, correlation for the joint) for the predictive distribution (P) and the validation set (V) for the different FICO classes, under uniform priors.

Class | ${\mathit{mean}}_{\mathit{P}}$ | ${\mathit{mean}}_{\mathit{V}}$ | ${\mathit{median}}_{\mathit{P}}$ | ${\mathit{median}}_{\mathit{V}}$ | ${\mathit{SD}}_{\mathit{P}}$ | ${\mathit{SD}}_{\mathit{V}}$ | ${\mathit{\rho}}_{\mathit{P}}$ | ${\mathit{\rho}}_{\mathit{V}}$ |
---|---|---|---|---|---|---|---|---|

Very Poor | 3.85 | 3.81 | 3.02 | 3.03 | 3.24 | 2.53 | 0.34 | 0.20 |

Fair | 2.41 | 2.32 | 1.50 | 1.44 | 2.36 | 2.33 | 0.45 | 0.51 |

Good | 1.43 | 0.49 | 0.83 | 0.25 | 0.71 | 0.66 | 0.34 | 0.38 |

Very good | 0.36 | 0.32 | 0.62 | 0.64 | 0.14 | 0.16 | 0.38 | 0.42 |

Exceptional | 0.16 | 0.18 | 0.60 | 0.54 | 0.82 | 0.83 | 0.19 | 0.18 |

**Table 3.**LGD values (%). Some descriptive descriptive statistics (mean, median, standard deviation) for the predictive distribution (P) and the validation set (V) for the different FICO classes, under uniform priors.

Class | ${\mathit{mean}}_{\mathit{P}}$ | ${\mathit{mean}}_{\mathit{V}}$ | ${\mathit{median}}_{\mathit{P}}$ | ${\mathit{median}}_{\mathit{V}}$ | ${\mathit{SD}}_{\mathit{P}}$ | ${\mathit{SD}}_{\mathit{V}}$ |
---|---|---|---|---|---|---|

Very Poor | 9.82 | 10.2 | 6.63 | 6.14 | 9.51 | 8.32 |

Fair | 12.1 | 11.5 | 9.43 | 10.6 | 9.69 | 5.53 |

Good | 13.8 | 19.8 | 14.5 | 19.2 | 5.70 | 5.26 |

Very good | 15.6 | 18.5 | 18.7 | 18.0 | 5.27 | 5.13 |

Exceptional | 17.8 | 17.2 | 20.4 | 19.7 | 8.00 | 4.99 |

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Cheng, D.; Cirillo, P. An Urn-Based Nonparametric Modeling of the Dependence between PD and LGD with an Application to Mortgages. *Risks* **2019**, *7*, 76.
https://doi.org/10.3390/risks7030076

**AMA Style**

Cheng D, Cirillo P. An Urn-Based Nonparametric Modeling of the Dependence between PD and LGD with an Application to Mortgages. *Risks*. 2019; 7(3):76.
https://doi.org/10.3390/risks7030076

**Chicago/Turabian Style**

Cheng, Dan, and Pasquale Cirillo. 2019. "An Urn-Based Nonparametric Modeling of the Dependence between PD and LGD with an Application to Mortgages" *Risks* 7, no. 3: 76.
https://doi.org/10.3390/risks7030076