# Empirical Credit Risk Ratings of Individual Corporate Bonds and Derivation of Term Structures of Default Probabilities

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## Abstract

**:**

## 1. Introduction

#### 1.1. Brief Review of the Literature

#### 1.1.1. GB and Interest Analysis

#### 1.1.2. CB and Credit Risk Analysis

#### 1.2. Overall Summary of Our Paper

- A. Crisk-rating system
- B. Derivation of TSDP for each Crisk-rating class

#### 1.2.1. Crisk-Rating System

#### 1.2.2. Derivation of TSDP for Each Crisk-Rating Class

#### 1.3. Detailed Summary

- (1)
- Definition of CRiPS and its empirical effectiveness
- (2)
- Crisk analysis of R&I credit ratings via CRiPS measures
- (3)
- Crisk-homogeneous grouping by cluster analysis and fixed interval scheme (FIS) via S-CRiPS measure
- (4)
- TSDPs for individual firms, cluster groups with industry categories, and Crisk-rating classes

## 2. Definition of CRiPS and Its Empirical Effectiveness

#### 2.1. GB Pricing Model

_{g}

_{1}< s

_{g}

_{2}< s

_{g}

_{3}< $\cdots $ < s

_{gM(g)}$(g=1,\cdots ,G)$ denote the future times (in years) when the CFs of the $g$th bond are generated, and s

_{gM(g)}is the maturity of the $g$th bond. Then, the GB pricing model is given by

_{g}(s) is a stochastic discount function, which may depend on bond attributes, and the realization of each price P

_{g}is regarded as equivalent to the realization of the whole function {D

_{g}(s): $0\le s\le $ s

_{gM}

_{(g)}}. Function D

_{g}(s) is decomposed into the mean discount function and the stochastic part as

_{1g}= 1, z

_{2g}= s

_{gM}

_{(g)}, and z

_{3g}= c

_{g}with ${w}_{1}=1,\text{\hspace{0.17em}}{w}_{2}=0\text{\hspace{0.17em}}\mathrm{or}\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}{w}_{3}=0\text{\hspace{0.17em}}\mathrm{or}\text{\hspace{0.17em}}1$. The parameters {${\delta}_{ij}$} of each discount function are common to all the bonds for $g=1,\cdots ,G$ and $({w}_{1},{w}_{2},{w}_{3})$ distinguishes the inclusion or exclusion of the attribute variables of maturity z

_{2g}= s

_{gM}

_{(g)}and coupon z

_{3g}= c

_{g}.

- (1)
- the longer the maturity of each bond, the larger the variance of each price;
- (2)
- the larger the difference of maturities of two bonds, the smaller the covariance; and
- (3)
- the closer the two cash flow points, the larger the covariance of the discount factors ${D}_{g}({s}_{gj})$ and ${D}_{h}({s}_{hm})$.

#### 2.2. Investors’ Preferences in the GB Markets and Attribute Effects on Prices

- (1)
- Investors with income motives hold GBs until maturity, such as life insurance companies, pensions, Japanese postal banks, banks, etc.
- (2)
- Investors with trading motives buy and sell GBs for capital gains along changes of TSIR, such as trust funds, hedge funds, and investment banking.

_{M}

_{(g)}, ${\widehat{P}}_{g}^{(0)}$), where the attribute effects are ignored. The M0 prices in the graph are shown to include relatively higher model prices corresponding to higher coupon rates. The graph of M0–M1 in the upper right corner of the figure plots the differences $\{{\widehat{P}}_{g}^{(0)}-{\widehat{P}}_{g}^{(1)}\}$ of M0 and M1 prices against maturities. Looking at the graphs of the differences, it is observed that the maximum and minimum are 28 basis points (bpt) and 22 bpt, respectively. At a maturity of 10 years, it is observed that ${\widehat{P}}_{g}^{(0)}-15\text{}\mathrm{bpt}\le {\widehat{P}}_{g}^{(1)}\le {\widehat{P}}_{g}^{(0)}+15\text{}\mathrm{bpt}$, which implies that ${\widehat{P}}_{g}^{(0)}$ can be higher or lower than ${\widehat{P}}_{g}^{(1)}$ by $\pm 15\text{}\mathrm{bpt}$.

#### 2.3. CRiPS Measure

## 3. Crisk Analysis of R&I Credit Ratings via CRiPS Measures

(1) G1: AAA, AA+, AA, AA−, | (2) G2: A+, A, |

(3) G3: A−, | (4) G4: BBB+, BBB, BBB−, C+. |

#### 3.1. Electric Power Industry

- (1)
- In graph 1, in September 2006, in the middle of the growing economy, the CRiPSs of the CBs were almost perfectly on a straight line, and the term of 10 years ${y}_{k}^{}$ attains the largest (best) value of −1.6 yen among the five cases (graphs), which is larger (better) than August 2010 in graph 5. Around the term of 7 through 10 years, the two divided lines in graph 1 are confirmed to correspond to the Crisk preferences of some issuers for CBs in the electric power industry. The differences of the market Crisk preferences for some specific issuers in August 2010 are identified in terms of TSDPs in Section 5.
- (2)
- In August 2007, when the economy was around the peak of the business cycle, investors who expected a coming downturn of the economy deformed the pattern of the CRiPSs with stronger Crisk preferences and some maturity preferences appearing around the 7-year term (see graph 2). In fact, around the terms of 5–7 years, the CRiPSs in the upper line in graph 2, about −0.7 yen, were better than in graph 1, implying that more of those CBs were bought. In addition, the Crisk preference became stronger for terms of 6–10 years.
- (3)
- The maturity preference continued until the financial crisis period in graph 3. CBs of about 7-year maturity were strongly preferred, though the level of the CRiPS was lower (worse) than in graph 2. What is notable here is that the CRiPSs around 10 years are large enough in absolute value to be close to −3 yen, which is significant relative to those of graphs 1 and 2. The gap between the 7-year and 10-year CRiPSs is as big as 2 yen. This implies that the linearity observed in graph 1 was disturbed by a big crisis.
- (4)
- In graph 4, one year later, the CRiPSs returned to the same shape as in graph 2, though the level is a bit lower (worse), and no issuer preference is observed here.
- (5)
- Under the continuation of bad economies in Japan and world, some maturity preferences seem to appear in graph 5, though no issuer preference can be seen.

#### 3.2. Trading and Metal Industries

## 4. Crisk-Homogeneous Grouping by Cluster Analysis and FIS via S-CRiPS Measure

- (1)
- Three-stage centroid clustering, and
- (2)
- fixed interval filtering with x-yen split rule.

- (1)
- First, we apply a three-stage centroid cluster method to the S-CRiPSs to get 14 Crisk-homogeneous cluster groups (CGs). The first cluster group (CG1) of the CBs that are closest together in the S-CRiPS is found to be the largest group in terms of number of CBs, implying that even three-stage centroid clustering does not sufficiently classify CBs because of the mutual closeness of too many S-CRiPSs.
- (1)
- Alternatively, as an absolute criterion for grouping and rating CBs, we propose the Crisk rating system with a set of fixed intervals filtering Crisks in terms of 10 times S-CRiPS, i.e., 10${\varsigma}_{k}^{}$, where each 10 × S-CRiPS of a CB is assigned to a rating interval in the half line (−∞,0]. It is noted that by the definition of Equation (13), 10${\varsigma}_{k}^{}$ is 10-year-maturity-equivalent CRiPS measure of ${y}_{k}^{}$ and the straight line connecting (0,0) and (${s}_{kM(k)}$, ${y}_{k}^{}$) passes (10, 10${\varsigma}_{k}^{}$). The choice of a set of fixed intervals is made by dividing the half line (−∞,0] by an x-yen split, where x-yen is the basic unit of making intervals for 10${\varsigma}_{k}^{}$s. For Japanese CBs, x = 1 is proposed (below), except for large S-CRiPSs in absolute value, and each CB is empirically rated by the number of fixed intervals. The market price–based categorical rates thus made can be used for investment decision-making and credit portfolio analysis in asset management, since the ratings are universally comparable over all CBs.

#### 4.1. Crisk-Homogenous Groups via Cluster Analysis

- (1)
- All 1545 CBs, including subordinated bonds, are clustered into six groups. The result is given in the upper part of Table 3. Clusters 3, 4, 5, and 6, which are very far away from 1 and 2 in centroid distance (13.86 − 4.67 = 9.19), are recognized as independent clusters and named CG11, CG12, CG13, and CG14. These clusters do not include many CBs; CG11 consists of CBs issued by Promise (BBB+, nonbank) and Daikyo (BBB, real estate), and CG12–CG14 consist of CBs issued by Aiful (CCC+, nonbank) only. Hence, it is difficult to derive the TSDPs for these groups due to a lack of samples. The members of these clusters are excluded in the second stage cluster analysis.
- (2)
- The 1529 remaining CBs, whose CRiPS measures are greater than −4 yen, are again clustered into six groups, and the middle part of Table 3 gives the result. Here again, clusters 3, 4, 5, and 6 are separated from 1 and 2, and form independent clusters CG7, CG8, CG9, and CG10. The centroid distance between clusters 1 and 2 and clusters 3–6 is still as large as 1.83 − 1.03 = 0.80 (yen).
- (3)
- The remaining 1505 CBs, whose CRiPS measures are greater than −1.5, are then clustered into six groups. Though these are relatively indistinguishable, we call them CG1 through CG6 in order of Crisk quality. CG1 still contains 988 CBs. The centroid distance between groups 1 and 2 and 3–6 is as small as 0.458 − 0.317 = 0.141. We stop here, because a further breakdown would not yield meaningfully distinguishable groups.

#### 4.2. Crisk-Rating System with Fixed Interval Scheme for S-CRiPSs

#### 4.3. Comparison of Our Crisk Rating and R&I Rating

_{k}measure. The CBs in R&I rating categories of A+, A, and A− are more broadly scattered over FIS classes. Except for AAA, the modes of all R&I rating distributions over FIS classes are consistent with the order of the Crisk rating classification if the equality is included. However, although the mode of AA belongs to F2, for example, 45% of the 119 CBs with AA rating are assigned to F3, while 27.9 % of CBs with AA− rating are assigned to F2, although the mode belongs to F3.

## 5. TSDPs for Individual Firms, Cluster Groups with Industry Categories, and Crisk-Rating Classes

- (1)
- For some individual firms in the electric power, international distribution (trading), and metal industries, where the firms issue relatively many CBs
- (2)
- For the top seven cluster groups identified above (CG1–CG7)
- (3)
- For industry groups CG1(1), CG1(2), …, CG1(14) obtained by decomposing CG1
- (4)
- For Crisk-rating classes F1, F2, …, F7

#### 5.1. Model for Pricing CBs and Deriving TSDPs

#### 5.2. TSDPs of Individual Firms and Credit-Homogeneous Groups

#### 5.2.1. TSDPs of Individual Firms in the Electric Power Industry

#### Monthly Changes of TSDPs of TEPCO from September 2006 through August 2010

- (1)
- In the time series of 10-year DPs, the maximum DP is 4.8% in February 2009, which is due to the financial crisis that started in September 2008. It is interesting to note that the JGB market was affected the worst in November 2008, though the reason for the time lag of TEPCO’s CBs is not clear.
- (2)
- The period of the first 12 months from September 2006 was in the upturn economy during which TEPCO enjoyed high revenues, and hence the TSDPs are overall lower, with 10-year DPs of about 1.6%. The curves of the TSDPs are not steep.
- (3)
- In the second 12 months starting from September 2007, the TSDPs rose rapidly and the 10-year DP attained 3% in February 2008, when the economy went down.
- (4)
- In September 2008 during the Lehman shock, the 10-year DP was 2.6%, which is a bit amazing because our end-of-month CB price did not respond to the shock on the spot. However, the DPs then increased dramatically to 4.8% in February 2009, and thereafter decreased greatly within five months.
- (5)
- The period August 2009 through August 2010 is somewhat similar to the period in (1) except that the DP curves in this period have flat areas in the middle term, corresponding to Figure 7.

#### 5.2.2. TSDPs of Individual Firms in International Distribution Industry

#### Monthly Changes of TSDPs of Mitsubishi Corp. for September 2006 through August 2010

#### 5.2.3. TSDPs of Individual Firms in Metal (Steel/Nonsteel/Mining) Industry

#### 5.2.4. TSDPs of CG1–CG7

#### 5.2.5. TSDPs of Crisk-Rating Classes

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Covariance Structure of CB Prices in Equation (15)

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**Figure 1.**Generalized least squares variations (GLSVs) defined by ${(\widehat{\psi}/G)}^{1/2}$ plotted over the period September 2005 to August 2010 (end-of-month analysis).

**Figure 2.**F-ratios of testing M0 vs. M1, M0 vs. M1, and M1 vs. M3. Graphs on left are Japanese government bond (JGB) cases for September 2005 through October 2010 and graphs on right are US GB cases for April 2006 through March 2011.

**Figure 4.**Distribution of M3 credit risk price spread (CRiPS) and M0 CRiPS with intervals of 1 yen split (lower graph) and M3 CRiPS with intervals of 0.4 yen split (upper graph).

**Figure 5.**CRiPS-plot of the Japan Rating & Investment Information Center (R&I) credit rating groups in the plane (${s}_{kM(k)}$, ${y}_{k}^{}$).

**Figure 7.**Yearly changes of CRiPS-plots of CBs in the electric power industry (11 companies) for August 2006 to August 2010.

**Figure 8.**CRiPS-plots of credit rating groups in international distribution (trading) and steel/nonsteel/mining (metal) industries.

**Figure 9.**Enterprise-wise CRiPS-plots for firms of the trading and metal industries. Smtm, sumitomo; M, metal; Min, mining; Stl, steel.

**Figure 12.**Term structure of default probabilities (TSDPs) of 10 companies in electric power industry.

**Figure 13.**Time series movements of TEPCO’s TSDPs from September 2006 through August 2010. The 30th month corresponds to February 2009. The upper graph describes 3D movements of the TSDPs, while the lower one represents the time series variations of DPs $\{{p}_{t}(s):t=1,2,\cdots ,48\}$ for each s = 2, …, 10.

**Figure 14.**TSDPs of six firms in the trading industry and table of DPs. MBS, Mitsubishi; SuS, Sumitomo; Mit, Mitsui; ITo, Itochu; MBS, Marubeni; SJT, Sojitz.

**Figure 15.**Time series movements of TSDPs of Mitsubishi Corp. for September 2006 through August 2010. The 30th month corresponds to February 2009.

**Figure 18.**TSDPs of industry groups in CG1. Mtrls/Chm, materials/chemicals; Cnstrctn, construction; Int Dist, international distribution (trading).

I | II | III | IV | |
---|---|---|---|---|

M3 | 0.051 | 0.109 | 0.189 | 0.082 |

M0 | 0.071 | 0.144 | 0.219 | 0.118 |

**Table 2.**Distribution of rated corporate bonds (CBs) in Japan Rating & Investment Information Center (R&I) categories (August 2010). NA, no rating.

AAA | AA+ | AA | AA− | A+ | A | A− | BBB+ | BBB | BBB− | C+ | NA | Total | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

# | 9 | 497 | 119 | 172 | 179 | 152 | 162 | 95 | 52 | 1 | 7 | 100 | 1545 |

% | 0.6 | 54.0 | 8.2 | 11.9 | 12.4 | 10.5 | 11.2 | 6.6 | 3.6 | 0.0 | 0.5 | NA | 100 |

**Table 3.**The 14 groups via three-stage cluster analysis. Max and Min denote the maximum and minimum, respectively, of the 10 × standardized credit risk price spreads (S-CRiPSs) in each cluster. Cluster groups are formed by the three-stage centroid cluster method, first stage in red, second stage in green and third stage in yellow.

1st | 1 | 2 | 3(CG11) | 4(CG12) | 5(CG13) | 6(CG14) | Total # |

# of CBs | 1504 | 25 | 9 | 2 | 4 | 1 | 1545 |

Max | −0.33 | −15.89 | −46.19 | −122.24 | −155.88 | −175.78 | |

Min | −14.80 | −38.09 | −58.54 | −138.54 | −160.19 | −175.78 | |

2nd | 1 | 2 | 3(CG7) | 4(CG8) | 5(CG9) | 6(CG10) | Total |

# of CBs | 1339 | 165 | 7 | 1 | 7 | 10 | 1529 |

Max | −0.33 | −6.20 | −15.89 | −21.65 | −24.71 | −30.48 | |

Min | −6.14 | −14.80 | −18.98 | −21.65 | −28.20 | −38.09 | |

3rd | 1(CG1) | 2(CG2) | 3(CG3) | 4(CG4) | 5(CG5) | 6(CG6) | Total |

# of CBs | 988 | 351 | 80 | 38 | 37 | 10 | 1504 |

Max | −0.33 | −3.03 | −6.20 | −8.26 | −10.51 | −13.67 | |

Min | −3.02 | −6.14 | −8.10 | −10.27 | −13.01 | −14.80 | |

Final Cluster Groups | |||||||

CG1 | CG2 | CG3 | CG4 | CG5 | CG6 | CG7 | |

# of CBs | 988 | 351 | 80 | 38 | 37 | 10 | 7 |

Max | −0.33 | −3.03 | −6.20 | −8.26 | −10.51 | −13.67 | −15.89 |

Min | −3.02 | −6.14 | −8.10 | −10.27 | −13.01 | −14.80 | −18.98 |

CG8 | CG9 | CG10 | CG11 | CG12 | CG13 | CG14 | Total |

1 | 7 | 10 | 9 | 2 | 4 | 1 | 1545 |

−21.65 | −24.71 | −30.48 | −46.19 | −122.24 | −155.88 | −175.78 | |

−21.65 | −28.20 | −38.09 | −58.54 | −138.54 | −160.19 | −175.78 |

Inds | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

CG1 | 24 | 16 | 41 | 30 | 49 | 19 | 44 | 53 | 459 | 124 | 56 | 6 | 1 | 14 | 52 |

CG2 | 18 | 9 | 46 | 23 | 30 | 13 | 27 | 7 | 2 | 81 | 42 | 1 | 18 | 22 | 12 |

**Table 5.**Distribution of CBs under five fixed interval schemes (FISs) for 10${\varsigma}_{k}^{}$. The mode of each FIS is highlighted.

FIS-1 | FIS-2 | FIS-3 | FIS-4 | FIS-5 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

F | 0.5 Yen | # | F | 1 Yen | # | F | M1 Yen | # | F | 1.5 Yen | # | F | 2 Yen | # |

1 | [−0.5,0) | 10 | 1 | [−1,0) | 42 | 1 | [−1,0) | 42 | 1 | [−1.5,0) | 213 | 1 | [−2,0) | 618 |

2 | [−1.0,−0.5) | 32 | 2 | [−2,−1) | 576 | 2 | [−2,−1) | 576 | 2 | [−3.0,−1.5) | 773 | 2 | [−4,−2) | 526 |

3 | [−1.5,−1.0) | 171 | 3 | [−3,−2) | 368 | 3 | [−3,−2) | 368 | 3 | [−4.5,−3.0) | 214 | 3 | [−6,−4) | 186 |

4 | [−2.0,−1.5) | 405 | 4 | [−4,−3) | 158 | 4 | [−4,−3) | 158 | 4 | [−6.0,−4.5) | 130 | 4 | [−8,−6) | 86 |

5 | [−2.5,−2.0) | 223 | 5 | [−5,−4) | 111 | 5 | [−5,−4) | 111 | 5 | [−7.5,−6.0) | 64 | 5 | [−10,−8) | 35 |

6 | [−3.0,−2.5) | 145 | 6 | [−6,−5) | 75 | 6 | [−6,−5) | 75 | 6 | [−9.0,−7.5) | 41 | 6 | (−∞,−10) | 94 |

7 | [−3.5,−3.0) | 81 | 7 | [−7,−6) | 46 | 7 | [−8,−6) | 86 | 7 | [−10.5,−9.0) | 22 | |||

8 | [−4.0,−3.5) | 77 | 8 | [−8,−7) | 40 | 8 | [−11,−8) | 48 | 8 | (−∞,−10.5) | 88 | |||

9 | [−4.5,−4.0) | 56 | 9 | [−9,−8) | 19 | 9 | [−15,−11) | 40 | ||||||

10 | [−5.0,−4.5) | 55 | 10 | [−10,−9) | 16 | 10 | (−∞,−15) | 41 | ||||||

11 | [−5.5,−5.0) | 36 | 11 | (−∞,−10) | 94 | |||||||||

12 | [−6.0,−5.5) | 39 | ||||||||||||

13 | [−7.0,−6.0) | 46 | ||||||||||||

14 | [−8.0,−7.0) | 40 | ||||||||||||

15 | [−9.0,−8.0) | 19 | ||||||||||||

16 | [−10.0,−9.0) | 16 | ||||||||||||

17 | (−∞,−10) | 94 |

**Table 6.**Cross-table of R&I rating groups and Crisk rating classes, where the number in each cell is the percentage ratio relative to the total number of CBs in the rightmost column. The highest percentage ratio in each row is highlighted.

Market Grouping via FIR-3 (Total 1545 CBs) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

R&I | F1 [−1,0] | F2 [−2,−1] | F3 [−3,−2] | F4 [−4,−3] | F5 [−5,−4] | F6 [−6,−5] | F7 [−8,−6] | F8 [−11,−8] | F9 [−15,−11) | F10 (−∞,−15) | Total % | # CBs |

AAA | 0.0 | 22.2 | 77.8 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 100 | 9 |

AA+ | 6.6 | 84.9 | 8.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 100 | 497 |

AA | 0.8 | 52.1 | 45.4 | 1.7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 100 | 119 |

AA− | 3.5 | 27.9 | 66.3 | 1.7 | 0.0 | 0.0 | 0.6 | 0.0 | 0.0 | 0.0 | 100 | 172 |

A+ | 0.6 | 12.8 | 52.0 | 14.0 | 7.8 | 6.1 | 3.4 | 3.4 | 0.0 | 0.0 | 100 | 179 |

A | 0.0 | 0.7 | 13.8 | 46.7 | 9.9 | 3.9 | 7.9 | 8.6 | 8.6 | 0.0 | 100 | 152 |

A− | 0.0 | 0.6 | 8.6 | 25.3 | 27.2 | 11.1 | 9.9 | 6.2 | 2.5 | 8.6 | 100 | 162 |

BBB+ | 0.0 | 0.0 | 1.1 | 5.3 | 20.0 | 14.7 | 27.4 | 9.5 | 9.5 | 12.6 | 100 | 95 |

BBB | 0.0 | 0.0 | 0.0 | 0.0 | 9.6 | 28.8 | 34.6 | 9.6 | 11.5 | 5.8 | 100 | 52 |

BBB− | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 100 | 100 | 1 |

CCC+ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 100 | 100 | 7 |

None | 1.0 | 17.0 | 22.0 | 11.0 | 14.0 | 11.0 | 7.0 | 5.0 | 8.0 | 4.0 | 100 | 100 |

# of CBs | 42 | 576 | 368 | 158 | 111 | 75 | 86 | 48 | 40 | 41 | 100 | 1545 |

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## Share and Cite

**MDPI and ACS Style**

Kariya, T.; Yamamura, Y.; Inui, K.
Empirical Credit Risk Ratings of Individual Corporate Bonds and Derivation of Term Structures of Default Probabilities. *J. Risk Financial Manag.* **2019**, *12*, 124.
https://doi.org/10.3390/jrfm12030124

**AMA Style**

Kariya T, Yamamura Y, Inui K.
Empirical Credit Risk Ratings of Individual Corporate Bonds and Derivation of Term Structures of Default Probabilities. *Journal of Risk and Financial Management*. 2019; 12(3):124.
https://doi.org/10.3390/jrfm12030124

**Chicago/Turabian Style**

Kariya, Takeaki, Yoshiro Yamamura, and Koji Inui.
2019. "Empirical Credit Risk Ratings of Individual Corporate Bonds and Derivation of Term Structures of Default Probabilities" *Journal of Risk and Financial Management* 12, no. 3: 124.
https://doi.org/10.3390/jrfm12030124