Credit Rating Transitions of U.S. Corporate Bonds

Credit Rating Transitions of U.S.Corporate Bonds

S. Figlewski, H. Frydman, W. Liang

The Stern School of Business, New York University

Credit Rating Transitions of U.S.Corporate Bonds – p. 1/24

• Part 1: Corporate Bond Rating Data

• Part 2: Cox Model

• Part 3: Cumulative Incidence Function

• Part 4: Future Research

• Appendix


Part 1: Corporate Bond Rating Data

• Our work is based on Moody’s Corporate Bond Default Database,containing the transition histories from Jan 1, 1981 to Dec 5, 2002.

• Bond Ratings: measure the relative probabilities that bond issuersfail to pay interest or repay principal, i.e. default.

• Major Rating Grades:

Investment Grade Speculative Grade Default Withdrawal

Moody’s Aaa Aa A Baa Ba B Caa Ca C D WR


Example of Transtions


Transitions of Interest

From / To Total A & Baa Ba & B C class (Caa/Ca/C) Default WR

A & Baa 3422 1684 786 5 3 944

Ba & B 4200 641 912 764 397 1486

C class (Caa/Ca/C) 997 4 103 213 508 169


Part 2: Cox Model

• Survival function: S(t) = P (T > t), where T is the time to event.Hazard function for type j event:

λj(t | Zj(t)) = lim∆t→0

P (t ≤ T < t + ∆t, ε = j | T ≥ t, Zj(t))

∆t

where ε is the event type indicator.

• Cox model (1972):

λj(t | Z(t)) = αj(t) exp{Zj(t)′βj}

where αj(t) is an unspecified non-negative baseline and Zj(t) isthe covariate vector. βj can be estimated by

βj = arg max log L(βj)


• Partial likelihood function:

L(βj) = Πni=1Πt>0(

exp{Zij(t)′βj}

∑n

i=1 Yi(t) exp{Zij(t)′βj})∆Nij(t)

where Yi(t) = I(Ti > t−), Nij(t) = I(Ti ≤ t, ε = j) is the countingprocess for type j event of firm i.

• The time t here stands for the duration time, not the calendar time."At-risk" set is defined as the set of firms that haven’t matured orexperienced any event by time t-.

• The coefficient βj for a time-dependent covariate can beinterpreted in this way

exp{βj(Zj(t) + 1)}exp{βjZj(t)}

= exp{βj}


Macroeconomicand Rating History-Specific Factors

• The cumulative effects of macroeconomic covariates are considered(except for the MA3 series): Zt =

∑Kk=1 ρkzt−k∑K

k=1ρk , where K is chosen as 18

months and ρ as 0.88 such that∑K

k=1 ρk

∑

∞

k=1ρk = 90%.

General Econ. Factors Direction of Econ. Measure of Econ. Slack Overall Credit Cond.

interest rate GDP growth deviation from pot. GDP yield spread

unemployment rate change in ind. prod. capacity util. level corp. bond. dflt. rt (MA3)

inflation change in unempl. rt ind. prod. dev. from trend

S&P 500 return

CFNAI (MA3)

• Rating history-specific covariates are as follows.

Initial Rating Class Current Rating Recent Down / Upgrade Age of an issuer

init. Inv. in-Caa downgrade last year log of age

init. C class in-Ca upgrade last year


The Covariates’ Effect onHazard Function

C class → Default

Rating Hist.-Spec. Only Kitchensink Model Best Model

Variable β std.err. p-value β std.err. p-value β std.err. p-value

(1 - init. C)×D.G. to C last yr 0.778 0.127 0.000 0.559 0.135 0.000 0.554 0.134 0.000

in-Caa -2.533 0.185 0.000 -2.549 0.186 0.000 -2.543 0.186 0.000

in-Ca -1.573 0.200 0.000 -1.511 0.203 0.000 -1.512 0.202 0.000

interest rate 0.175 0.086 0.041 0.135 0.039 0.001

unemployment rate -0.158 0.274 0.565

inflation -0.048 0.083 0.561

S&P 500 return 0.163 0.063 0.010 0.141 0.051 0.005

CFNAI (MA3) -0.011 0.245 0.963

GDP growth 0.022 0.442 0.961

change in ind. prod. -0.922 0.713 0.196 -0.669 0.279 0.017

change in unempl. rate -2.239 2.347 0.340

deviation from pot. GDP 0.055 0.195 0.777

capacity util. level 0.144 0.116 0.214 0.195 0.040 0.000

ind. prod. dev. from trend -0.030 0.043 0.487

yield spread 0.711 0.473 0.133 0.926 0.255 0.000

corp. bond. dflt. rate (MA3) 0.218 0.116 0.060 0.221 0.075 0.003

-2 LOG L 5448 5384 5387

AIC 5454 5416 5405


C class → Higher Rating Classes

Rating Hist.-Spec. Only Kitchensink Model Best Model

Variable β std.err. p-value β std.err. p-value β std.err. p-value

init. investment 0.946 0.237 0.000 0.652 0.280 0.020 0.687 0.268 0.010

in-Caa 0.768 0.368 0.037 0.973 0.374 0.009 0.907 0.371 0.015

interest rate 0.026 0.152 0.866

unemployment rate -0.166 0.473 0.726

inflation -0.133 0.195 0.495

S&P 500 return 0.008 0.130 0.953

CFNAI (MA3) -0.176 0.504 0.727

GDP growth 0.359 0.949 0.705

change in ind. prod. 0.801 1.631 0.623

change in unempl. rate -1.994 4.781 0.677

deviation from pot. GDP -1.021 0.410 0.013 -0.551 0.143 0.000

capacity util. level 0.460 0.224 0.040

ind. prod. dev. from trend 0.117 0.095 0.216 0.116 0.040 0.004

yield spread 1.628 0.911 0.074

corp. bond. dflt. rate (MA3) 0.393 0.232 0.089

-2 LOG L 1139 1113 1121

AIC 1143 1143 1129


Part 3:Cumulative Incidence Function

• Practioners want to know more – "What’s the probability that a Cclass firm would default within one quarter or one year?"

• The cumulative incidence function (CIF) for type j event is

CIFj(t) = P (T ≤ t, ε = j) =

∫ t

0

S(u−)dΛj(u)

where Λj(t) =∫ t

0λj(u)du.


Estimate the Cumulative IncidenceFunction

• Non-parametric approach (e.g. Aalen (1978)): use the Kaplan-Meierestimator of S(u) and the Nelson-Aalen estimator of Λj(u) for event type j.

ˆCIF j(t) =

∫ t

0

S(u−)dΛj(u)

• Semi-parametric approach based on the Cox form: use

Λj(u) =

∫ u

0

exp{Zj(s)′β}dAj(s | β)

with the Breslow estimator ofAj(u | β) =

∫ u

0αj(s)ds =

∫ u

0

I(∑n

i=1 Yi(s)>0)∑n

i=1 Yi(s) exp{Zij(s)′βj}dNj(s), and

S(u) = exp{−

J∑

j=1

Λj(u)}.


Compare Non-Parametricwith Semi-Parametric Estimates

**The black line shows the non-parametric estimate of CIF with the orange ones demonstrating 95% pointwise confidence interval. The blueand red line are the semi-parametric estimates of CIF for two spells, from 03/26/1998 to 06/07/2002 and from 06/07/1989 to03/30/1990, respectively. The red one defaulted and carried higher interest rate and S&P 500 return, lower change in ind. prod., yieldspread, deviation from potential GDP and deviation from ind. prod. trend. The default rates are both very high in the two periods. Both spellsstayed in Caa, being downgraded from B, initially rated above C class. Overall, the red spell experienced higher ZD(t)′βD , but lower

ZU (t)′βU than the blue.


Explanatory Power

• Schemper (2000, 2003) considered the problem for the survivalfunction.

• We here first define the expected absolute error as

E|I(T ≤ t, ε = j) − CIFj(t)| = 2 · CIFj(t)(1 − CIFj(t))

At event time tk, let

Q(tk) = 2 · ˆCIF j(tk)(1 − ˆCIF j(tk))

Q(tk | Z) =2

∑n

i=1 Yi(tk)

n∑

i=1

Yi(tk) ˆCIF j(tk | Zi)(1 − ˆCIF j(tk | Zi))

where ˆCIF j(tk) and ˆCIF j(tk|Z) are the non-parametric andsemi-parametric estimates of CIFj , respectively.


• Considering the attenuating effect of censoring, specify the weightat event time tk with dk events as

wG,k =dk

G(tk)

where G is the Kaplan-Meier estimator for censoring, i.e treatingthe censoring as the "event". So the weighted version is

D =

∑

k wG,kQ(tk)∑

k wG,k

, DZ =

∑

k wG,kQ(tk | Z)∑

k wG,k

• The relative gain in explanatory power by using semi-parametricmethod for C class to Default is

V = 1 −DZ

D= 7.95%


The Effect of Time-DependentCovariates on the CIF

• Fine & Gray (1999) regressed directly the effect of the time-fixedcovariates on the CIF using the subdistribution approach:

CIFj(t | Zj) = 1 − exp{−

∫ t

0

α∗

j (u)exp{Z ′

jβ∗

j }du}

where α∗

j (u) and β∗

j are estimated based on a modified "at-risk"set. Unfortunately, their approach doesn’t apply to time-dependentcovariates.


Part 4: Future Research

• Develop a model that can regress directly the impact oftime-dependent covariates on the cumulative incidence function.

• Consider β as a function of time due to structural change over time.


Appendix A: Alternative Cox models

C class → Default C class → Higher Rating Classes

Univariate Model Con. Sign Model Univariate Model Con. Sign Model

Variable β p-value β p-value β p-value β p-value

interest rate 0.048 0.110 0.230 0.000 0.146 0.006

unemployment rate -0.092 0.058 -0.285 0.000 0.335 0.000

inflation 0.118 0.000 0.018 0.837

S&P 500 return -0.073 0.013 0.113 0.085

CFNAI (MA3) -0.412 0.000 0.243 0.160

GDP growth -0.576 0.000 0.024 0.930

change in ind. prod. -0.892 0.000 0.366 0.288

change in unempl. rate 3.245 0.000 -1.372 0.430

deviation from pot. GDP 0.032 0.218 -0.186 0.000 -0.147 0.003

capacity util. level -0.040 0.025 0.000 0.990

ind. prod. dev. from trend -0.007 0.296 -0.037 0.005

yield spread 0.388 0.000 -0.330 0.101

corp. bond. dflt. rate (MA3) 0.215 0.000 0.224 0.000 -0.187 0.038 -0.164 0.056

(1 - init. C)×D.G. to C last yr 0.571 0.000

init. Investment 0.616 0.020

in-Caa -2.522 0.000 0.907 0.015

in-Ca -1.491 0.000


Appendix B: Residuals

• Cox-Snell Residuals

ri = Λi(Ti) =

∫ Ti

0

exp{Zi(t)′β}dA(t | β), i = 1, 2, ..., n

The plot of Nelson-Aalen estimates of ri v.s. ri are as below.


• Martingale Residuals

Mi = Ni(∞) −

∫

∞

0

Yi(t) exp{Zi(t)′β}dA(t), i = 1, 2, ..., n


• Deviance Residuals

di = sign(Mi)

√

2[−Mi − δi log(δi − Mi)], i = 1, 2, ..., n


Appendix C:Asymptotic Variance of the CIF

• The time-fixed covariate version was given by Scheike & Zhang(2003). With some modification, we give the time-dependentversion below.

√n[ ˆCIF j(t) − CIFj(t)] ≈

√n

∫ t

0(1 −

∑

l6=j

CIFl(u))dMj,Λ(u)

−√

nCIFj(t)

∫ t

0

J∑

l=1

dMl,Λ(u)

+√

n

∫ t

0CIFj(u)

∑

l6=j

dMl,Λ(u)

where Mj,Λ(t) = Λj(t) − Λj(t).


By Taylor expansion of Λij(t) w.r.t. β

√n(Λij(t) − Λij(t)) ≈

√nWij,1(t)

where

Wij,1(t) = I−1∑

i

Ui

∫ t

0[Zij(s) exp{Zij(s)

′β} I(∑n

i=1 Yi(s) > 0)∑n

i=1 Yi(s) exp{Zij(s)′βj}

− exp{Zij(s)βj}I(

∑ni=1 Yi(s) > 0)

∑ni=1 Yi(s)Zij(s) exp{Zij(s)

′βj}(∑n

i=1 Yi(s) exp{Zij(s)′βj})2]dNj(s),

and with the help that√

n(βj − βj) ≈√

nI−1∑

i Ui.


Finally,√

n[ ˆCIF j(t) − CIFj(t)] ≈√

n

n∑

i=1

Wij,2(t)

Wij,2(t) =

∫ t

0(1 −

∑

l6=j

CIFl(u))dWij,1(u) − CIFj(t)

∫ t

0

J∑

l=1

dWij,1(u)

+

∫ t

0CIFj(u)

∑

l6=j

dWij,1(u)

and the variance of√

n[ ˆCIF j(t) − CIFj(t)] is consistently estimated by

n

n∑

i=1

(Wij,2(t))2.


Credit Rating Transitions of U.S. Corporate Bonds

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