Measuring Credit Risk of Individual Corporate Bonds in US Energy Sector

Abstract

In this paper, using the measures of the credit risk price spread (CRiPS) and the standardized credit risk price spread (S-CRiPS) proposed in Kariya’s (A CB (corporate bond) pricing model for deriving default probabilities and recovery rates. Eaton, IMS Collection Series: Festschrift for Professor Morris L., 2013) corporate bond model, we make a comprehensive empirical credit risk analysis on individual corporate bonds (CBs) in the US energy sector, where cross-sectional CB and government bond price data is used with bond attributes. Applying the principal component analysis method to the S-CRiPSs, we also categorize individual CBs into three different groups and study on their characteristics of S-CRiPS fluctuations of each group in association with bond attributes. Secondly, using the market credit rating scheme proposed by Kariya et al. (2014), we make credit-homogeneous groups of CBs and show that our rating scheme is empirically very timely and useful. Thirdly, we derive term structures of default probabilities for each homogeneous group, which reflect the investors’ views and perspectives on the future default probabilities or likelihoods implicitly implied by the CB prices for each credit-homogeneous group. Throughout this paper it is observed that our credit risk models and the associated measures for individual CBs work effectively and can timely provide the market credit information evaluated by investors.

Appendices

Appendix 1

In KGB model, substituting (3.2) into (3.1) yields

$$\begin{aligned}&\sum _{m=1}^{M(g)} {C_g (s_{gm} )\bar{{D}}_g (s_{gm} )} \nonumber \\&\quad =a_g +(\delta _{11} d_{g11} +\delta _{12} d_{g21} +\delta _{13} d_{g31} )+ \cdots +(\delta _{p1} d_{g1p} +\delta _{p2} d_{g2p} +\delta _{p3} d_{g3p} ),\nonumber \\ \end{aligned}$$

(8.1)

where

$$\begin{aligned} a_g =\sum _{m=1}^{M(g)} {C_g (s_{gm} )},&d_{g1j} =\sum _{m=1}^{M(g)} {C_g (s_{gm} )} s_{gm}^j\nonumber \\&d_{g2j} =\sum _{m=1}^{M(g)} {C_g (s_{gm} )s_{gM(g)} } s_{gm}^j, \nonumber \\&\hbox { and } d_{g3j}=\sum _{m=1}^{M(g)} {C_g (s_{gm} )c_g } s_{gm}^j. \end{aligned}$$

(8.2)

Here i in $d_{gij}$ denotes the attribute suffix and j the polynomial order. Thus letting

(8.3)

the model is reduced to a regression model;

(8.4)

where and .

On the other hand, the specification of the stochastic part of the DF is given by

$$\begin{aligned}&{ Cov}(D_g (s_{gj} ),D_h (s_{hm} ))=\sigma ^{2}\lambda _{gh} f_{gh\cdot jm}\hbox { with }f_{gh\cdot jm} =\exp (-\theta \left| {s_{gj} -s_{hm} } \right| )\hbox { and}\nonumber \\&\lambda _{gh} =\left\{ {{\begin{array}{ll} e_{gg} &{} (g=h) \\ \rho e_{gh} &{} (g\ne h) \\ \end{array} }} \right. \quad \mathrm{with} \quad e_{gh} =\exp \left( -\xi \left| {s_{gM(g)} -s_{hM(h)} } \right| \right) , \end{aligned}$$

(8.5)

and hence the covariance matrix of prices is

(8.6)

where $\varphi _{gh} =\sum \nolimits _{j=1}^{M(g)} {\sum \nolimits _{m=1}^{M(h)} {C_g (s_{gj} )C_h (s_{hm} )f_{gh\cdot jm} } }$.

The parameter vector is estimated by generalized least squares (GLS) method and the objective function

is minimized with respect to the unknown parameters (see Kariya and Kurata 2004 for the effectiveness of GLS). First, for given $(\theta ,\rho ,\xi )$, the minimizer of this function with respect to is known to be the GLSE;

(8.7)

and then the marginally minimized function with substitution of is minimized with respect to $(\theta ,\rho ,\xi )$, yielding the GLSE .

Appendix 2

t $=$ 1	0.95	0.05	0	0	t $=$ 11	0.785714	0.214286	0	0
	0.021739	0.978261	0	0		0.102041	0.836735	0.061224	0
	0	0.090909	0.909091	0		0	0.111111	0.888889	0
	0	0	0	1		0	0	0.125	0.875
t $=$ 2	0.974359	0.025641	0	0	t $=$ 12	1	0	0	0
	0.020833	0.958333	0.020833	0		0.12766	0.87234	0	0
	0	0.2	0.8	0		0	0.321429	0.678571	0
	0	0	0	1		0	0	0.428571	0.571429
t $=$ 3	0.923077	0.076923	0	0	t $=$ 13	1	0	0	0
	0.040816	0.938776	0.020408	0		0.12	0.88	0	0
	0	0	1	0		0	0.181818	0.772727	0.045455
	0	0	0	1		0	0	0.25	0.75
t $=$ 4	0.815789	0.184211	0	0	t $=$ 14	1	0	0	0
	0	0.979592	0.020408	0		0	0.979167	0.020833	0
	0	0	1	0		0	0.111111	0.833333	0.055556
	0	0	0	1		0	0	0	1
t $=$ 5	0.903226	0.096774	0	0	t $=$ 15	0.678571	0.321429	0	0
	0.018182	0.890909	0.090909	0		0	0.897959	0.102041	0
	0	0	1	0		0	0	1	0
	0	0	0	1		0	0	0	1
t $=$ 6	0.517241	0.482759	0	0	t $=$ 16	0.894737	0.105263	0	0
	0	0.884615	0.115385	0		0.075472	0.90566	0.018868	0
	0	0.0625	0.875	0.0625		0	0.142857	0.857143	0
	0	0	0	1		0	0	0	1
t $=$ 7	0.666667	0.333333	0	0	t $=$ 17	1	0	0	0
	0.016393	0.803279	0.180328	0		0.037736	0.962264	0	0
	0	0	0.9	0.1		0	0.105263	0.894737	0
	0	0	0	1		0	0	0.2	0.8
t $=$ 8	1	0	0	0	t $=$ 18	0.956522	0.043478	0	0
	0.203704	0.777778	0.018519	0		0.188679	0.792453	0.018868	0
	0	0.206897	0.793103	0		0	0.055556	0.944444	0
	0	0	0	1		0	0	0	1
t $=$ 9	0.681818	0.318182	0	0	t $=$ 19	0.96875	0.03125	0	0
	0.020833	0.875	0.104167	0		0.181818	0.795455	0.022727	0
	0	0.041667	0.833333	0.125		0	0.222222	0.777778	0
	0	0	0.25	0.75		0	0	0	1
t $=$ 10	0.8125	0.1875	0	0	t $=$ 20	1	0	0	0
	0.02	0.9	0.08	0		0.175	0.825	0	0
	0	0.038462	0.884615	0.076923		0	0.266667	0.733333	0
	0	0	0	1		0	0	0.5	0.5

Appendix 3

Assuming the covariance structure in (6.2) for the stochastic discount function yields

(10.1)

where $\lambda _{kl}$ is the same as the one in KGB model.

(10.2)

However a significant difference is that this covariance matrix depends on the unknown regression parameters to be estimated. Of course our objective function to be minimized is of the same form as KGB model;

(10.3)

where the covariance matrix is replaced by the matrix ($\sigma ^{2}\lambda _{kl} \varphi _{kl} )$ in (10.1). To get an approximate minimize, the GLS is applied to (6.2) in a repeated manner. In fact, we set as the initial value in (10.2) to obtain the first GLS estimate from (6.2) and (10.1). Next we insert into (10.2) and get the second GLS estimate from (6.2) and (10.1). Repeating this procedure five times yields our GLS estimate the coefficients of the TSDP in (6.4). The minimized value is . The parameter $\theta$ is always estimated as 0. For the theoretical background for the GLS, the readers are referred to Kariya and Kurata (2004).