credit rating transitions of u.s. corporate bonds
TRANSCRIPT
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
Credit Rating Transitions of U.S.Corporate Bonds – p. 2/24
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
Credit Rating Transitions of U.S.Corporate Bonds – p. 3/24
Example of Transtions
Credit Rating Transitions of U.S.Corporate Bonds – p. 4/24
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
Credit Rating Transitions of U.S.Corporate Bonds – p. 5/24
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)
Credit Rating Transitions of U.S.Corporate Bonds – p. 6/24
• 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}
Credit Rating Transitions of U.S.Corporate Bonds – p. 7/24
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
Credit Rating Transitions of U.S.Corporate Bonds – p. 8/24
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
Credit Rating Transitions of U.S.Corporate Bonds – p. 9/24
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
Credit Rating Transitions of U.S.Corporate Bonds – p. 10/24
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.
Credit Rating Transitions of U.S.Corporate Bonds – p. 11/24
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)}.
Credit Rating Transitions of U.S.Corporate Bonds – p. 12/24
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.
Credit Rating Transitions of U.S.Corporate Bonds – p. 13/24
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.
Credit Rating Transitions of U.S.Corporate Bonds – p. 14/24
• 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%
Credit Rating Transitions of U.S.Corporate Bonds – p. 15/24
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.
Credit Rating Transitions of U.S.Corporate Bonds – p. 16/24
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.
Credit Rating Transitions of U.S.Corporate Bonds – p. 17/24
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
Credit Rating Transitions of U.S.Corporate Bonds – p. 18/24
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.
Credit Rating Transitions of U.S.Corporate Bonds – p. 19/24
• Martingale Residuals
Mi = Ni(∞) −
∫
∞
0
Yi(t) exp{Zi(t)′β}dA(t), i = 1, 2, ..., n
Credit Rating Transitions of U.S.Corporate Bonds – p. 20/24
• Deviance Residuals
di = sign(Mi)
√
2[−Mi − δi log(δi − Mi)], i = 1, 2, ..., n
Credit Rating Transitions of U.S.Corporate Bonds – p. 21/24
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).
Credit Rating Transitions of U.S.Corporate Bonds – p. 22/24
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.
Credit Rating Transitions of U.S.Corporate Bonds – p. 23/24
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 – p. 24/24