Additional Default Probability in Consideration of Firm’s Network

Abstract

In this paper, we propose a methodology to effectively capture credit risk from firms’ network. In short, our target is to numerically obtain additional credit risk from connected firms on network. Recently, commercial networks are available for investing and managing risk on professional information terminals like Bloomberg and Reuters. They enable us to check commercial connection of firms. We utilize them to assess positive and negative effect on observing firms from neighbor firms, especially, when the neighbor firms have any credit events. We propose a methodology to analyze/measure credit impact, which observing firms potentially receive from their neighbors. We applied Merton model (Merton and Robert in J Financ 29(2):449–470, 1974) which is generally utilized for credit risk management to calculate additional risk and simplified the formula for practicability/usability. Also, it enables us to escape from having any difficulties in computation time. We introduce our approach with over-viewing simple model guidance and explaining a few samples of numerical experiments.

We would like to thank Dr. Toshinao Yoshiba (Bank of Japan) for useful discussion. We thank anonymous reviewers for their helpful comments and suggestions. The first author was supported by a grant-in-aid from Zengin Foundation for Studies on Economics and Finance.

5 Appendix

We explain how to calculate default threshold for two neighbors’ case at first. Namely, we calculate conditional default probability $\mathbf {P}(\xi _1<d_1|\xi _2=d_2, \xi _3=d_3)$ and obtain the threshold from $\mathbf {N}^{-1}[\mathbf {P}(\xi _1<d_1|\xi _2=d_2, \xi _3=d_3)]$. Three dimensional standardized probability density function for asset is as follows.

$$\begin{aligned} \phi (\xi _1, \xi _2, \xi _3)=\frac{1}{ (2\pi )^{\frac{3}{2}} \sqrt{ |\Sigma | } } \mathbf {exp} \left\{ -\frac{1}{2} (\xi _1 \xi _2 \xi _3)\Sigma ^{-1}(\xi _1 \xi _2 \xi _3)^\top \right\} \end{aligned}$$

(11)

where $\xi _1$, $\xi _2$, and $\xi _3$ are standardized asset random variables for firm 1, 2, and 3 respectively. $\Sigma$ is 3 $\times$ 3 correlation matrix.

$$\begin{aligned} \Sigma = \left( \begin{array}{rrrrr} 1 \quad \rho _{12} \quad \rho _{13} \\ \rho _{21} \quad 1 \quad \rho _{23} \\ \rho _{31} \quad \rho _{32} \quad 1 \\ \end{array} \right) \end{aligned}$$

(12)

For later calculation, we prepare inverse matrix of $\Sigma$.

$$\begin{aligned} \Sigma ^{-1}= \frac{1}{ |\Sigma | } \left( \begin{array}{rrrrrrr} 1-{\rho _{23}}^2 \quad \quad \rho _{13}\rho _{23}-\rho _{12} \quad \quad \rho _{12}\rho _{23}-\rho _{13} \\ \rho _{13}\rho _{23}-\rho _{12} \quad \quad 1-{\rho _{13}}^2 \quad \quad \rho _{12}\rho _{13}-\rho _{23} \\ \rho _{12}\rho _{23}-\rho _{23} \quad \quad \rho _{12}\rho _{13}-\rho _{23} \quad \quad 1-{\rho _{12}}^2 \\ \end{array} \right) \end{aligned}$$

(13)

where $|\Sigma |=1+2\rho _{12}\rho _{13}\rho _{23}-{\rho _{12}}^2-{\rho _{13}}^2-{\rho _{23}}^2$

$$\begin{aligned} (\xi _1 \xi _2 \xi _3){\Sigma }^{-1}(\xi _1 \xi _2 \xi _3)^\top= & {} \frac{1}{|\Sigma |} (\xi _1 \xi _2 \xi _3) \left( \begin{array}{c} (1-{\rho _{23}}^2)\xi _1+(\rho _{13}\rho _{23}-\rho _{12})\xi _2+(\rho _{12}\rho _{23}-\rho _{13})\xi _3 \\ (\rho _{13}\rho _{23}-\rho _{12})\xi _1+(1-{\rho _{13}}^2)\xi _2+(\rho _{12}\rho _{13}-\rho _{23})\xi _3 \\ (\rho _{12}\rho _{23}-\rho _{23})\xi _1+(\rho _{12}\rho _{13}-\rho _{23})\xi _2+(1-{\rho _{12}}^2)\xi _3 \\ \end{array} \right) \nonumber \\= & {} \frac{1}{|\Sigma |} \{ (1-{\rho _{23}}^2){\xi _1}^2+(1-{\rho _{13}}^2){\xi _2}^2+(1-{\rho _{12}}^2){\xi _3}^2 \nonumber \\&\quad +2(\rho _{13}\rho _{23}-\rho _{12})\xi _1\xi _2 +2(\rho _{12}\rho _{23}-\rho _{13})\xi _1 \xi _3 \nonumber \\&\quad \quad +2(\rho _{12}\rho _{23}-\rho _{13})\xi _1\xi _3+2(\rho _{12}\rho _{13}-\rho _{23})\xi _2\xi _3 \} \end{aligned}$$

(14)

This result is part of coefficient of exponential in Eq. (11). We subtract coefficient of two dimensional normal distribution $\frac{1}{1-{\rho _{23}}^2} ( {\xi _2}^2-2\rho _{23}\xi _2\xi _3+{\xi _3}^2 )$ from Eq. (14). And we obtain below. We utilize this to calculate conditional default probability for two neighbors’ case.

$$\begin{aligned} => \frac{1}{|\Sigma |(1-{\rho _{23}}^2)} \left[ \left\{ (1-{\rho _{23}}^2)\xi _1+(\rho _{13}\rho _{23}-\rho _{12})\xi _2+(\rho _{12}\rho _{23}-\rho _{13})\xi _3 \right\} ^2 \right] \end{aligned}$$

(15)

$$\begin{aligned}&\frac{\int ^{d_1}_{-\infty }\phi (\xi , d_2, d_3)\mathbf {d}\xi _1}{\phi (d_2, d_3)}\nonumber \\&\quad = \int ^{d_1}_{-\infty }\frac{1}{\sqrt{2\pi }\sqrt{ \frac{|\Sigma |}{1-{\rho _{23}}^2} }} \mathbf {exp} \left\{ -\frac{1}{2} \frac{(1-{\rho _{23}}^2)}{|\Sigma |} \left( \xi _1+\frac{(\rho _{13}\rho _{23}-\rho _{12})d_2+(\rho _{12}\rho _{23}-\rho _{13})d_3}{(1-{\rho _{23}}^2)} \right) ^2 \right\} \mathbf {d}\xi _1 \nonumber \\&\quad =\mathbf {N}\left( \frac{d_1(1-{\rho _{23}}^2)-(\rho _{12}-\rho _{23}\rho _{13})d_2-(\rho _{13}-\rho _{23}\rho _{12})d_3 }{ \sqrt{1-{\rho _{23}}^2} \sqrt{ 1+2\rho _{12}\rho _{13}\rho _{23}-{\rho _{12}}^2-{\rho _{13}}^2-{\rho _{23}}^2 } } \right) \end{aligned}$$

(16)

Next, we explain about LU decomposition method to escape from having above complicated calculations and consider multiple neighbor case. Through above operation, we saw two neighbors case and compare the result with threshold from Eq. (16). Let $\omega _1, \omega _2, \omega _3$ be independent random variables from standardized normal distribution. Let U be upper triangle correlation matrix and L be lower triangle correlation matrix. Temporary, we set firm 3 as target firm and calculate possible default impact from firm 1 and 2. After calculating the threshold based on LU decomposition method, we replace firm number to be consist with above methodology and compare the solutions.

$$\begin{aligned} \Sigma =LU= \left( \begin{matrix} 1 &{} 0 &{} 0 \\ \rho _{12} &{} \sqrt{1-{\rho _{12}}^2} &{} 0 \\ \rho _{13} &{} \frac{\rho _{23}-\rho _{12}\rho _{13}}{\sqrt{1-{\rho _{12}}^2}} &{} \sqrt{ 1-{\rho _{13}}^2-\frac{(\rho _{23}-\rho _{12}\rho _{13})^2}{{1-{\rho _{12}}^2}} } \\ \end{matrix} \right) \left( \begin{matrix} 1 &{} \rho _{12} &{} \rho _{13} \\ 0 &{} \sqrt{1-{\rho _{12}}^2} &{} \frac{\rho _{23}-\rho _{12}\rho _{13}}{\sqrt{1-{\rho _{12}}^2}} \\ 0 &{} 0 &{} \sqrt{ 1-{\rho _{13}}^2-\frac{(\rho _{23}-\rho _{12}\rho _{13})^2}{{1-{\rho _{12}}^2}} } \\ \end{matrix} \right) \end{aligned}$$

(17)

By using above LU decomposition, we obtain correlated random variables $\xi _1, \xi _2, \xi _3$ as below.

$$\begin{aligned}&(\xi _1 \quad \xi _2 \quad \xi _3)=(\omega _1 \quad \omega _2 \quad \omega _3)U \nonumber \\&\xi _1=\omega _1 \nonumber \\&\xi _2=\rho _{12}\omega _1+\sqrt{1-{\rho _{12}}^2}\omega _2 \nonumber \\&\xi _3=\rho _{13}\omega _1+\frac{\rho _{23}-\rho _{12}\rho _{13}}{\sqrt{1-{\rho _{12}}^2}} \omega _2+ \sqrt{ 1-{\rho _{13}}^2-\frac{(\rho _{23}-\rho _{12}\rho _{13})^2}{{1-{\rho _{12}}^2}} }\omega _3 \end{aligned}$$

(18)

When standardized asset of firm 1 and 2 are equal to threshold $d_1$ and $d_2$. Random variables $\omega _1$ and $\omega _2$ become as follows.

$$\begin{aligned}&\omega _1=d_1 \\&\omega _2=\frac{d_2-\rho _{12}d_1}{\sqrt{1-{\rho _{12}}^2}} \end{aligned}$$

We insert these to Eq. (18) and solve $\omega _3$ of inequality by setting left-hand side of Eq. (18) being smaller than $d_3$.

$$\begin{aligned}&d_3>\rho _{13}d_1+\frac{\rho _{23}-\rho _{12}\rho _{13}}{\sqrt{1-{\rho _{12}}^2}} \frac{d_2-\rho _{12}d_1}{\sqrt{1-{\rho _{12}}^2}}+ \sqrt{ 1-{\rho _{13}}^2-\frac{(\rho _{23}-\rho _{12}\rho _{13})^2}{{1-{\rho _{12}}^2}} }\omega _3 \end{aligned}$$

(19)

$$\begin{aligned} \Leftrightarrow&\omega _3<\frac{d_3(1-{\rho _{12}}^2)-(\rho _{31}-\rho _{12}\rho _{32})d_1-(\rho _{32}-\rho _{12}\rho _{31})d_2 }{ \sqrt{1-{\rho _{12}}^2} \sqrt{ 1+2\rho _{31}\rho _{32}\rho _{12}-{\rho _{31}}^2-{\rho _{32}}^2-{\rho _{12}}^2}} \end{aligned}$$

(20)

The right-hand side of Eq. (20) is new threshold for the setting of target firm 3 and neighbor firm 1 and 2. Also, this result is equal to Eq. (16) by replacing firm number accordingly. We showed two neighbors case here and LU-decomposition method enables us to increase the number of neighbors easily.