Corporate Failure Prediction
With Stochastic Covariates Darrell Due Ke Wang First Version: August 30, 2003
Current Version: December 3, 2003
Abstract
We provide maximum likelihood estimators of term structures of
conditional probabilities of bankruptcy over relatively long time hori-
zons, incorporating the dynamics of rm-specic and macroeconomic
covariates. We nd evidence in the U.S. industrial machinery and
instruments sector, based on over 28,000 rm-quarters of data span-
ning 1971 to 2001, of signicant dependence of the level and shape of
the term structure of conditional future bankruptcy probabilities on
a rms distance to default (a volatility-adjusted measure of leverage)
and on U.S. personal income growth, among other covariates. Varia-
tion in a rms distance to default has a greater relative eect on the
term structure of future failure hazard rates than does a comparatively
sized change in U.S. personal income growth, especially at dates more
than a year into the future. We are grateful for support from Moodys, and for comments from Susan Athey,
Richard Cantor, Brad Eron, Jeremy Fox, Peyron Law, Aprajit Mahajan, and especially
Takeshi Amemiya. Graduate School of Business, Stanford University, Stanford, CA 94305-5015, email:
due@stanford.edu, webpage: www.stanford.edu/ due/ Department of Economics, Stanford University, Stanford CA 94305.
email:
kewang@stanford.edu, webpage: www.stanford.edu/ kewang/ Wang gratefully acknowl-
edges support under a grant from Moodys. 1
Introduction
We provide maximum likelihood estimators of term structures of conditional
corporate bankruptcy probabilities. Our contribution over prior work is to
exploit the dependence of failure intensities on stochastic covariates, as well
as the time-series dynamics of the covariates, in order to estimate the likeli-
hood of failure over several future periods (quarters or years). We estimate
our model for the U.S. industrial machinery and instrument sector, using
over 28,000 rm-quarters of data for the period 1971 to 2001. We nd evi-
dence of signicant dependence of the level and shape of the term structure
of conditional future failure probabilities on a rms distance to default (a
volatility-adjusted measure of leverage) and on U.S. personal income growth,
among other covariates. Variation in a rms distance to default has a greater
relative eect on the term structure of future failure hazard rates than does
a comparatively sized change in the business-cycle covariate, U.S. personal
income growth, especially at dates more than one year into the future.
The estimated shape of the term structure of conditional failure probabili-
ties reects the time-series behavior of the covariates, namely leverage target-
ing by rms and mean reversion in macro-economic performance. The term
structures of failure hazard rates are typically upward sloping at business-
cycle peaks, and downward sloping at business-cycle troughs, to a degree
that depends on corporate leverage relative to its long-run target.
In our model, a rms stochastic failure intensity is assumed to depend on
both rm-specic and macroeconomic state variables. Stochastic evolution
of the combined Markov state vector X
t
causes variation over time in a rms
failure intensity
t
= (X
t
). The rm exits for other reasons, such as merger,
acquisition, privatization, or liquidation out of bankruptcy, with an intensity

t
= A(X
t
). The total exit intensity is
t
+
t
.
We specify a doubly-stochastic formulation of the point process for failure
and other forms of exit under which the conditional probability at time t of
corporate survival (from failure or other exit) for s years is
p(X
t
, s) = E e R
t+s
t
((u)+(u)) du
X
t
,
(1)
and under which the conditional probability of failure within s years is
q(X
t
, s) = E
t
+s
t
e R
z
t
((u)+(u)) du z
dz
X
t
.
(2)
1 This calculation of q(X
t
, s), demonstrated in Section 2, reects the fact that,
in order to fail at time z, the rm must survive until time z, avoiding both
failure and other forms of exit, which arrive at a total intensity of (u)+(u).
While, as explained in Section 1.1, there is a signicant prior literature
treating the estimation of one-period-ahead bankruptcy probabilities, for ex-
ample with duration models, we believe this is the rst empirical study of the
conditional term structure of failure probabilities over multiple future time
periods. The sole exception seems to be the practice of certain banks and
dealers in structured credit products of treating the credit rating of a rm
as though a Markov chain, with ratings transition probabilities estimated as
long-term average ratings transition frequencies.
1
It is by now well under-
stood, however, that the current rating of a rm does not incorporate much
of the inuence of the business cycle on failure rates (Nickell, Perraudin,
and Varotto (2000), Kavvathas (2001)), nor even the eect of prior ratings
history (Behar and Nagpal (1999), Lando and Sk�deberg (2002)). There is,
moreover, signicant heterogeneity in the short-term failure probabilties of
dierent rms of the same current rating (Kealhofer (2003)).
We anticipate several types of applications for our work, including (i) the
analysis by a bank of the credit quality of a borrower over various future
potential borrowing periods, for purposes of loan approval and pricing, (ii)
the determination by banks and bank regulators of the appropriate level of
capital to be held by banks, in light of the current state of their loan portfolio,
especially given the upcoming Basel II accord, under which borrower default
probabilities are to be introduced for the purpose of determining the capital
to be held as backing for a loan to a given borrower, (iii) the determination of
credit ratings by rating agencies, and (iv) the ability to shed some light on the
macroeconomic links between business-cycle variables and the failure risks of
corporations. Absent a model that incorporates the dynamics the underlying
covariates, it seems dicult to extrapolate prior models of one-quarter-ahead
or one-year-ahead default probabilities to longer time horizons. While one
could seperately estimate models of xed-horizon failure probabilities for each
of various alternative time horizons, it seems natural and statistically more
ecient to incorporate joint consistency conditions for failure probabilities
at various time horizons within one model.
The conditional survival and default probabilities, p(X
t
, s) and q(X
t
, s),
depend on:
1
See, for example, Due and Singleton (2003), Chapter 4.
2 a parameter vector determining the dependence of the failure and
other-exit intensities, (X
t
) and A(X
t
), respectively, on the covariate
vector X
t
, and
a parameter vector determining the time-series behavior of the un-
derlying state vector X
t
of covariates.
The doubly-stochastic assumption, stated more precisely in Section 2, is
that, conditional on the paths of the underlying state variables determining
failure and other-exit intensities for all rms, these exit times are the rst
event times of independent Poisson processes with the same (conditionally
deterministic) intensity paths.
2
In particular, this means that, given the path
of the state-vector process, the merger and default times of dierent rms
are conditionally independent.
A major advantage of the doubly-stochastic formulation is that it allows
decoupled maximum-likelihood estimations of and , which can then be
combined to obtain the maximum-likelihood estimators of the survival and
failure probabilities, p(X
t
, s) and q(X
t
, s), and other properties of the model,
such as probabilities of joint failure of more than one rm. The maximum
likelihood estimator of the intensity parameter vector is the same as that of
a conventional competing-risks duration model with time-varying covariates,
because of the doubly-stochastic assumption. The maximum likelihood esti-
mator of the time-series parameter vector would depend of course on the
particular specication adopted for the time-series behavior of the state pro-
cess X. Our approach is quite exible in that regard. For examples, we could
allow the state process X to have GARCH volatility behavior, to depend on
hidden Markov chain regimes, or to have jump-diusive behavior. For our
specic empirical application to the U.S. macinery and instrument sector,
we have adopted a simple Gaussian vector auto-regressive specication for
the rm-specic leverage variables and the macroeconomic growth variables,
and we use the conventional maximum-likelihood estimator for the associ-
ated parameter vector . A further advantage of this methodology is that it
allows straightforward maximum-likelihood estimation of the term structure
of failure probabilities, by simply substituting the maximum-likelihood esti-
mators for and into (2). Asymptotic condence intervals for the term
2
One must take care in interpreting this characterization when treating the internal
covariates, those that are rm-specic and therefore no longer available after exit, as
explained in Section 2.
3 structure of future default probabilities can then be obtained by the usual
Delta method, as explained in the appendix.
The doubly-stochastic assumption is overly restrictive in settings for which
failure, or another form of exit by one rm, could have an important direct
inuence on the failure or other-exit intensity of another rm. This inuence
would be anticipated to some degree if one rm plays a relatively large role in
the marketplace of another. Our empirical results should therefore be treated