Macro, industry and frailty effects in defaults: the 2008 credit crisis in perspective

 
Macro, industry and frailty effects in defaults:
                       the 2008 credit crisis in perspective ∗

          Siem Jan Koopman (a,c) André Lucas (b,c) Bernd Schwaab (b,c)
           (a)
                 Department of Econometrics, VU University Amsterdam
                 (b)
                       Department of Finance, VU University Amsterdam
                                     (c)
                                           Tinbergen Institute

                                            June 19, 2009

   ∗
    Corresponding author: Bernd Schwaab, VU University Amsterdam, Department of Finance, De Boele-
laan 1105, 1081 HV Amsterdam, The Netherlands, Tel +31 20 598 8625, Fax +31 20 598 6020, Email:
bschwaab@feweb.vu.nl. We thank Michel van der Wel for comments, Richard Cantor at Moody’s, and
seminar participants at presentations at Humboldt University, Tinbergen Institute, Utrecht University, and
University of Amsterdam, and VU University Amsterdam. We also thank Moody’s to grant access to their
default and ratings database for this research.

                                                    1
Macro, industry and frailty factors of defaults:
          the 2008 credit crisis in perspective
                     S. J. Koopman, A. Lucas and B. Schwaab

                                             Abstract

     In this paper we decompose default risk into its different systematic risk components
     using a new methodological framework. The proposed mixed measurement dynamic
     factor model allows the joint modeling of discrete, non-Gaussian default counts across
     industry sectors and rating classes and continuous, Gaussian macroeconomic and fi-
     nancial covariates. Latent dynamic risk factors may be common to all data due to
     business cycle effects, or specific to subsets of data to capture unobserved default-
     specific (frailty) and industry-specific (contagion) dynamics. In an empirical study
     for US default data between 1971 and 2009 we decompose systematic risk into latent
     macro, frailty, and industry effects. We find that all effects are important to differ-
     ent degrees. Common variation of defaults with business cycle and financial data can
     explain a large part of the systematic variation in defaults. While less important on
     average, a frailty factor is of key importance before and during times of crisis. We also
     find significant differences in the impact of crises on defaults at the sectoral level.

     Keywords: mixed-measurement dynamic factor; frailty-correlated defaults; state space
     methods; dynamic credit risk management.

JEL classification: G21; C33.

                                                  2
1     Introduction

In this paper we decompose default risk into its different systematic risk components using
a new methodological framework. Observed corporate defaults are known to cluster in time.
For example, aggregate US default rates during the 1991, 2001, and 2008 recession periods
are up to five times higher than in the intermediate expansion years. It is also well-known
that default rates depend on the prevailing macroeconomic conditions, see for example Duffie,
Saita, and Wang (2007), Pesaran, Schuermann, Treutler, and Weiner (2006), Metz (2008),
Figlewski, Frydman, and Liang (2008).

    Recent research indicates that conditioning on readily available macroeconomic and firm-
specific information, though important, is not sufficient to fully explain the observed degree
of default clustering. In a seminal study, Das, Duffie, Kapadia, and Saita (2007) reject
the joint hypothesis of well-specified default intensities in terms of observed macro and
firm-specific information and the doubly stochastic (conditional) independence assumption
which underlies virtually all available credit risk models in practice. From this finding, two
important separate strands of literature have emerged.

    First, and most importantly, there appears to be a role for a dynamic unobserved compo-
nent, a ‘frailty’ factor. A frailty factor implies default clustering above and beyond what is
implied by macro data alone. The unobserved component can pick up the effect of omitted
relevant variables in the model, and may capture truly unmeasured effects such as trust
in the accuracy of public accounting information, see Duffie, Eckner, Guillaume, and Saita
(2008). The frailty literature is fairly recent, see Das et al. (2007), McNeil and Wendin
(2007), Koopman, Lucas, and Monteiro (2008), Koopman and Lucas (2008), Duffie et al
(2008), Koopman, Lucas, and Schwaab (2008), and Koopman, Kräussl, Lucas, and Mon-
teiro (2009).

    A second strand of literature points to contagion as a second channel for default clustering
in excess of what is implied by macro data. In the contagion literature, a defaulting firm
weakens other firms with which it has contractual relationships, see e.g. Giesecke (2004).
Default shocks may be amplified through up- and downstream business links, leading to
default dependence at the industry level even after conditioning on macro and frailty factors,
see Lang and Stulz (1992), Lando and Nielsen (2008), and Jorion and Zhang (2007a, 2007b).

                                               3
Despite earlier research in this area, less is known about the relative contribution of the
different sources of systematic default risk for observed default clustering. In particular,
what exactly causes default clustering, and to which extend? To address this question, we
decompose the systematic variation in corporate default counts into its different constituents
as suggested in the literature. We do so by developing a novel methodological framework.
Within this framework, we attribute default rate volatility at the rating and industry level
to macro, frailty, and industry effects.
   The paper makes two main contributions to the literature. First, we propose a new
econometric framework for the joint modeling of systematic default risk and macroeconomic
developments. The key issue is that default events are discrete, whereas macroeconomic and
financial variables are typically modeled as continuous. We thus need to develop a framework
that can accommodate both types of variables simultaneously, while accounting for the fact
that they may be driven by related dynamics. Most models out in the literature either do not
account for frailty effects, or if they do, only allow for a limited number of macro variables
to enter the model, see for example McNeil and Wendin (2007), Duffie, Eckner, Guillaume,
and Saita (2008), and Koopman, Kräussl, Lucas, and Monteiro (2009). Even with a limited
number of macro variables, the observed data is typically treated as unrelated to the frailty
component, and auxiliary models need to be specified for the observed factors to perform out-
of-sample forecasting or conditional risk management exercises, see also Koopman, Lucas,
and Schwaab (2008).
   By contrast to this earlier work, our proposed mixed measurement framework explicitly
captures the (i) joint variation in discrete, non-Gaussian corporate default counts and a large
number of continuous, Gaussian macroeconomic and financial time series data, (ii) default
clustering due to latent frailty risk unrelated to the business cycle, and (iii) unobserved
industry-sector dynamics. The latter may arise as a result of direct default contagion through
business links on the sector level. We demonstrate that these model features lead to a
realistic fit of default rate dynamics at the industry as well as the economy wide level. Our
approach yields an integrated framework for estimation, inference, and forecasting of time
varying corporate default rates. In particular, no auxiliary models for macroeconomic data
are required. Estimation of parameters and latent risk factors is done in a single step.
   As the second contribution of our paper, we decompose default risk into its latent con-

                                              4
stituents. To this purpose we rely on reductions in the estimated Kullback-Leibler (KL)
divergence. The KL divergence is a standard measure of ‘distance’ between distributions
indexed by competing sets of parameters. In our framework, models with an increasing
number of latent factors imply different default rates and default rate volatility. Reductions
in the estimated KL divergence, when scaled with respect to an ideal explanatory model,
allow to assess the relative contribution of each of the systematic default risk components
(macro, frailty, industry) to overall default rate volatility. We find that, on average across
industries and time, about 70%, of total default risk is idiosyncratic, or diversifiable. The
remaining share, about 30%, can be attributed to common variation with the business cycle
and financial data (17%), a frailty factor (6%), and industry-specific developments, which
we interpret as default contagion (8%). The latter sources of risk are systematic in that they
do not vanish as more loans are added to the portfolio. As a result, sources of systematic
risk are of key concern to financial institutions and banking supervision.

   Our reported risk shares vary considerably over industry sectors, rating groups, and time.
For example, we find that the frailty component tends to explain a higher share of default
rate volatility before and during times of crisis. The inclusion of a frailty factor leads to more
probability mass in the right tail of the model-implied portfolio loss distribution. Model-
implied economic capital buffers are higher as a result. We demonstrate that industry-specific
factors are of key importance for a good fit to observed defaults at the industry level, or in
case of industry concentrations in the credit portfolio.

   The remainder of this paper is set up as follows. Section 2 introduces the general frame-
work of the mixed measurement dynamic factor model and presents generic results for es-
timation and inference. We also explain how to use reductions in the KL divergence to
decompose total default rate variability into risk shares. Section 3 applies the model to
mixed default and macroeconomic data and comments on the major empirical findings. Sec-
tion 4 decomposes default rate volatility into its constituents. We comment on implications
for credit risk practitioners in Section 5. Section 6 concludes.

                                                5
2     Mixed measurement dynamic factor model

We propose a dynamic factor model for variables from the exponential family of densities.
Since this class of densities is quite general and can include a range of discrete and con-
tinuous variables, we refer to the model as a mixed measurement dynamic factor model.
Variables associated with densities outside the exponential family can also be included but
for expositional reasons we refrain from further generalities. Also, further generalities are
not necessary for the purpose of our main study.

2.1     Model specification

The mixed measurement dynamic factor model (MMDFM) is based on a set of m dynamic
latent factors that are assumed to be generated from a dynamic Gaussian process. For
example, we can collect the factors into the m × 1 vector ft and assume a stationary vector
autoregressive process for the factors,

                    ft+1 = µf + Φft + ηt ,         ηt ∼ N(0, Ση ),           t = 1, 2, . . . ,                (1)

with the initial condition f1 ∼ N(µ, Σf ). The m × 1 mean vector µf , the m × m coefficient
matrix Φ and the m × m variance matrix Ση are assumed fixed and unknown with the m
roots of the equation |I − Φz| = 0 outside the unit circle and Ση positive definite. The
m × 1 disturbance vectors ηt are serially uncorrelated. The process for ft is initialized by
f1 ∼ N(0, Σf ) where m×m variance matrix Σf is a function of Φ and Ση or, more specifically,
Σf is the solution of Σf = ΦΣf Φ0 + Ση .
    Conditional on a factor path F = { f1 , f2 , . . .}, the observation yi,t of the ith variable at
time t is assumed to come from the exponential family of densities as given by

        yi,t |F ∼ pi (yi,t ; F, ψ),   pi (yi,t ; F, ψ) = exp [yi,t θi,t − bi,t (θi,t ; ψ) + ci,t (yi,t )] ,   (2)

with the signal defined by
                                                      p
                                                      X
                                       θi,t = αi +          λ0i,j ft−j ,                                      (3)
                                                      j=0

where αi is an unknown constant and λi,j is the m×1 loading vector with unknown coefficients

                                                     6
for j = 0, 1, . . . , p. The so-called link function bi,t (θi,t ; ψ) is assumed to be twice differentiable
while ci,t (yi,t ) is a function of the data only. The parameter vector ψ contains all unknown
coefficients in the model specification including those in Φ, αi and λi,j for i = 1, . . . , N and
j = 0, 1, . . . , p. To enable the identification of all entries in ψ, we assume standardized factors
in (1) which we enforce by the restrictions µf = 0 and Σf = I implying that Ση = I − ΦΦ0 .
   Conditional on F, the observations at time t are independent of each other. It implies
that the density of N × 1 observation vector yt = (y1,t , . . . , yN,t )0 is given by

                                                         N
                                                         Y
                                       p(yt |F, ψ) =           p(yi,t |F, ψ).
                                                         i=1

The MMDFM model is defined by the equations (1), (2) and (3). The variables in yt can be
assumed to come from, for example, Gaussian, binary, binomial and Poisson densities.

2.2     Estimation via importance sampling

An analytical expression for the the maximum likelihood (ML) estimate of parameter vector
ψ for the MMDFM is not available. A feasible approach to the ML estimation of ψ is provided
by importance sampling. The parameters are estimated via the direct maximization of the
likelihood function that is evaluated by Monte Carlo integration. A short description of this
approach is given below.
   The observation density function of y = (y10 , . . . , yT0 )0 can be expressed by the joint density
of y and f = (f10 , . . . , fT0 )0 where f is integrated out, that is
                                   Z                           Z
                       p(y; ψ) =        p(y, f ; ψ) d f =            p(y|f ; ψ)p(f ; ψ) d f,          (4)

where p(y|f ; ψ) is the density of y conditional on f and p(f ; ψ) is the density of f . A Monte
Carlo estimator of p(y; ψ) can be obtained by

                                             M
                                             X
                                        −1
                        p̂(y; ψ) = M               p(y|f (k) ; ψ),        f (k) ∼ p(f ; ψ),
                                             k=1

for some large integer M . The estimator p̂(y; ψ) is however numerically inefficient since
most draws f (k) will not support p(y|f ; ψ) for any ψ and k = 1, . . . , K. Importance sampling

                                                          7
improves the Monte Carlo estimation of p(y; ψ) by sampling f from the Gaussian importance
density g(f |y; ψ). We can express the observation density function p(y; ψ) by
                       Z                                                 Z
                           p(y, f ; ψ)                                       p(y|f ; ψ)
           p(y; ψ) =                   g(f |y; ψ) d f = g(y; ψ)                         g(f |y; ψ) d f,   (5)
                           g(f |y; ψ)                                        g(y|f ; ψ)

since f is from a Gaussian density such that g(f ; ψ) = p(f ; ψ) and therefore we have g(y; ψ) =
g(y, f ; ψ) / g(f |y; ψ). In case g(f |y; ψ) is close to p(f |y; ψ) and in case simulation from
g(f |y; ψ) is feasible, the Monte Carlo estimator implied by (5) and given by

                                              M
                                              X
                                         −1     p(y|f (k) ; ψ)
                p̃(y; ψ) = g(y; ψ) M                                 ,       f (k) ∼ g(f |y; ψ),          (6)
                                              k=1
                                                    g(y|f (k) ; ψ)

is numerically much more efficient, see Kloek and van Dijk (1978), Geweke (1989) and Durbin
and Koopman (2001).

   For a practical implementation, the importance density g(f |y; ψ) can be based on the
linear Gaussian approximating model

                                                                            2
                            yi,t = µi,t + θi,t + εi,t ,        εi,t ∼ N(0, σi,t ),                        (7)

                                         2
where mean correction µi,t and variance σi,t are determined in such a way that g(f |y; ψ) is
sufficiently close to p(f |y; ψ). It is argued by Shephard and Pitt (1997) and Durbin and
Koopman (1997) that µi,t and σi,t can be uniquely chosen such that the modes of p(f |y; ψ)
and g(f |y; ψ) with respect to f are equal, for a given value of ψ.

   To simulate values from the importance density g(f |y; ψ), the simulation smoothing
method of Durbin and Koopman (2002) can be applied to the approximating model (7). For
a set of M draws of g(f |y; ψ), the evaluation of (6) relies on the computation of p(y|f ; ψ),
g(y|f ; ψ) and g(y; ψ). Density p(y|f ; ψ) is based on (2), density g(y|f ; ψ) is based on the
                                               2
Gaussian density for yi,t − µit − θi,t ∼ N(0, σi,t ) (7) and g(y; ψ) can be computed by Kalman
filter applied to (7), see Harvey (1989).

   Once the likelihood function can be evaluated for any value of ψ and for a given set
of random numbers from which factors are simulated from g(f |y; ψ), we can maximize the
likelihood function with respect to ψ. The importance sampling method provides the means

                                                       8
for a feasible estimation algorithm.

2.3     Estimation of the factors

Once an ML estimator is available for ψ, the estimation of f can take place, also based on
importance sampling. We have
                                           Z                             Z
                                                                                 p(y|f ; ψ)
                            E(f |y; ψ) =        f p(f |y; ψ) df =            f              .
                                                                                 g(y|f ; ψ)

After some minor algebra, it can be shown that the estimation of E(f |y; ψ) via importance
sampling can be achieved by
                         P        (k)
                           k wk f                     p(y|f (k) ; ψ)
                  f˜ =    P           ,        wk =                  ,           f (k) ∼ g(f |y; ψ),
                             k wk                     g(y|f (k) ; ψ)
        P
where    k   is from k = 1 to M .

2.4     Decomposition of mixed measurement variation

Once the model is estimated, we would like to assess which share of variation in mixed-
measurement exponential family data is captured by different sets of latent factors. To this
purpose we rely on reductions in a standard measure of ‘distance’, or divergence, between
two distributions. The Kullback-Leibler (KL) divergence

                                                Z
                            KL(µ1 , µ2 ) = 2        log [fµ1 (y)/fµ2 (y)] fµ1 (y)dy                    (8)

measures the difference in terms of expected information between two log-densities log fµ1
and log fµ2 . These densities are specified by competing parameter vectors µ1 and µ2 , respec-
tively. We refer to Hastie (1987) and Vos (1991) for details and intuition. By taking the
expectation over y, all values in the support of f are taken into account when calculating
the divergence. The definition (8) compares marginal distributions. This means that un-
observed factors have to be integrated out of their joint density with observed data. This
can be accomplished by Monte Carlo integration based on Importance Sampling, see the
discussion leading to equation (4).

                                                         9
[Insert Figure 1 around here]

    Figure 1 illustrates the intuition underlying our approach to decomposing non-Gaussian
time series variation. Alternative model specifications are graphed into the plane as f 0 , f 1 , ...
These models contain an increasing collection of latent factors. Model f 0 does not contain
any factors. More factors imply a better fit to observed data. The parameters and factors
associated with f 0 , f 1 , ... can be estimated by Monte Carlo maximum likelihood. Once
estimated, parameter and factor estimates imply mean values µ0 , µ1 , .. corresponding to
each observation in y.
    Model f max provides the maximum possible fit. This is achieved when each observation
is captured by its own dummy variable, i.e., there are as many parameters as observations.
While useless for practical purposes, this unrestricted model provides the best fit. The
amount of potentially recoverable information in the data is given by KL(µ0 , µmax ).
    As an example used in Section 4, different model specifications will imply different default
rates and default rate volatility for each combination of industry sector and rating class. For
example, model-implied default probabilities (multiplied by number of trials) may be stacked
in µ0 , µ1 , ... The KL divergence KL(µ0 , µ1 ), when scaled by the potentially recoverable
information KL(µ0 , µmax ), is a measure of the incremental explanatory power of factors f 1
compared to factors f 0 . We refer to Cameron and Windmeijer (1997), who use this approach
to derive related R-squared measures for exponential family regression models. These R-
squared statistics coincide with the familiar R-squared measure in case of Gaussian data,
but also apply to all other exponential family data. Using the same approach, we define
si = KL(µi−1 , µi )/KL(µ0 , µmax ), i = 1, 2, .., as the share of variation explained by factors in
f i not contained in f i−1 .

3     Modeling systematic credit risk

In this section we introduce an integrated empirical model specification that captures three
major sources of systematic default risk. We then apply the model to recent mixed default
and macroeconomic data, and report the major empirical findings.

                                                 10
3.1    A joint model for default and macro risk

We seek to model the joint variation of discrete, non-Gaussian default counts yjt , j = 1, . . . , J
with a large number of continuous, Gaussian business cycle and financial time series data
xit , i = 1, . . . , N . All data is stacked, and subject to common macroeconomic factors.

                                     Yt = (y1t , ..., yJt , x1t , ..., xN t )0 .                  (9)

   We distinguish three sets of risk factors, denoted ftc , ftd , and fti . Factors ftc capture shared
business cycle dynamics in macroeconomic data and default counts. Therefore, factors ftc are
common to all data. Frailty factors ftd are common to default data yt , and independent of
observed macroeconomic data by construction. These factors cause default clustering above
and beyond what is implied by macroeconomic and financial data. Third, firms from the
same industry are exposed to common sector-specific dynamics. These may arise as a result
of default contagion through up- and downstream business links. Latent industry sector
dynamics are captured by factors fti .
   Counts yjt denote the total number of defaults of firms with a certain characteristic
j, such as industry sector and current rating class, during time (t, t + 1]. These defaults
depend on the corresponding number of such firms at risk, kjt . We model yjt as binomial
conditional on all risk factors. For the economic intuition behind the binomial mixture
model, see e.g. McNeil, Frey, and Embrechts (2005, Chapter 9). After conditioning, yjt may
be interpreted as the number of default ‘successes’ in kjt independent Bernoulli trials with
time varying default probability πjt = [1 + e−θjt ]−1 . The logistic transform ensures that πjt
is in the unit interval. Other transforms are also possible.
   The macroeconomic time series xt are modeled as Gaussian with time-varying mean µit
due to exposure to common macroeconomic factors ftc .

                          yjt |ftc , ftd , fti ∼ Binomial(kjt , [1 + e−θjt ]−1 )                 (10)

                          xit |ftc            ∼ Gaussian(µit , σi2 )                             (11)

   Signals θjt can be interpreted as log-odds of time varying event probabilities. The log-

                                                        11
odds vary over time due to variation in systematic risk factors. The signals are specified
as

                                                    0
                                θjt = λ0,yj + βj ftc + γj0 ftd + δj0 fti                        (12)

                               µit = λ0,xi + βi0 ftc                                            (13)

where vectors βj , γj , and δj contain factor loadings associated with risk factors ftc , ftd , and
fti , respectively. Loadings βi refer to Gaussian data. Finally, latent factor dynamics are
given by

                          ft† = Φ† ft−1
                                    †
                                        + η†,t , η†,t ∼ NID(0, I − Φ† Φ0† ),                    (14)

where († ) refers to either (c), (d), (i), and coefficient matrix Φ† is diagonal with coefficients in
the unit interval. This allows for positive serial correlation in the risk factors. The restriction
Σ† = I − Φ† Φ0† implies that factor loadings βj , γj , and δj can be interpreted as the respective
factor standard deviations (volatilities) for firms in cross section j.
     The model (9) to (14) is a mixed measurement dynamic factor model as outlined in
Section 2. The general results for parameter and factor estimation can be applied as a
result.

3.2       Major empirical results

We fit the model to quarterly default and exposure counts obtained from the Moody’s
research database. We focus on defaults from 1971Q1 to 2008Q4. We distinguish d = 1, . . . , 7
broad industry groups, i.e., financials and insurance; transportation; media, hotels, and
leisure; utilities and energy; industrials; technology; and retail and consumer products. We
further consider s = 1, . . . , 4 rating groups, i.e., investment grade Aaa−Baa, and speculative
grade groups Ba, B, Caa−C. Pooling over investment grade firms is necessary since defaults
are rare for this segment.

                                  [Insert Figure 2 around here]

                                                  12
Macroeconomic and financial data is obtained from the St. Louis Fed online database
FRED, see Table 1 for a listing of macroeconomic and financial data. This data enters the
analysis in the form of annual growth rates, see Figure 3 for time series plots.

                                [Insert Table 1 around here]

                                [Insert Figure 3 around here]

   Parameter estimates for the default counts can be observed from Table 2. Estimated
coefficients in the first column combine to fixed effects for each cross-section, according to
λ0,j = λ0 + λ1,dj + λ2,sj . The common intercept λ0 is thus adjusted by specific coefficients
indicating industry sector and rating group, respectively. The two middle columns report
the dynamics (φ) and factor loadings β associated with four common factors ftc . Loading
coefficients differ across rating groups, and tend to be higher for investment grade firms.
This is in accord with the notion that financially healthy firms are more sensitive to business
cycle risk, see e.g. Basel Committee on Banking Supervision (2004).

                                [Insert Table 2 around here]

   Factor loadings γ and δ associated respectively with one frailty factor ftd and six orthog-
onal industry factors fti are given in the last column of Table 2. The frailty risk factor ftd
is, by construction, common to all firms but unrelated to the included macroeconomic data.
Frailty risk is found to be economically large for all firms, but particularly pronounced for
speculative grade firms. Industry sector loadings are highest for the financial, transporta-
tion, and energy sector. These industry level default rates are not well captured by common
risk factors.

                                [Insert Figure 4 around here]

   Estimated factors ftc show clear business cycle dynamics, see Figure 4. The factors are
ordered from the top left to bottom right according to their share of explained variation for
the macro and financial data listed in Table 1.

                                [Insert Figure 5 around here]

                                              13
Figure 5 indicates that the first two common factors in Figure 4 load mostly from labor
market, production, and interest rate data. The factors from bottom panels load mostly
from survey sentiment indicators and price level data. In total, the four factors capture an
average of 24.7%, 22.4%, 11.0%, and 8.0% of the variation in the macro data panel. The
range of explained variation in macro and financial data ranges from about 30% (S&P 500
index returns, fuel prices) to more than 90% (unemployment rate, average weekly hours
index, total non-farm payrolls). At the same time, all common factors ftc tend to load more
into default probabilities of firms rated investment grade rather than speculative grade, see
Table 2.

                                 [Insert Figure 6 around here]

    Figure 6 plots smoothed estimates of the frailty and industry-specific factors. The frailty
factor is high before and during the recession years 1991 and 2001. As a result, the frailty
factor implies additional default clustering in these bad times. On the other hand, the large
negative values before the 2007-09 credit crisis imply defaults that are systematically ’too
low’ compared to what is implied by macroeconomic and financial data. The frailty factor
reverts to its mean level during the 2007-09 credit crisis. Apparently, the current extreme
realizations in macroeconomic and financial variables, and thus factors f c , are sufficient to
account for the high levels of defaults observed during 2007-09.
    Industry factors fti reveal pronounced deviations of sector-specific dynamics from shared
variation. For example, we observe industry-specific default stress for financial firms during
the US savings and loan crises from 1986-1990, and the current crisis in 2007-09. Similarly,
we observe considerably higher default stress for the technology sector following the 2000/01
asset bubble bust, or for the transportation industry following the 9/11 attack.

4     Total default risk: a decomposition
In this subsection we use reductions in the estimated Kullback-Leibler divergence to assess
which share of default rate volatility is captured by an increasing set of systematic risk
factors. Earlier literature on default modeling in the presence of explanatory variables does
not really address this issue.

                                              14
While systematic risk is explicitly captured through latent factors, unsystematic risk is
modeled by the conditionally binomial distributional assumption. We define four risk shares
with respect to the business cycle, frailty, industry-specific, and idiosyncratic component of
total risk as follows

         K(µ0 , µc )                K(µc , µc,d )              K(µc,d , µc,d,i )               K(µc,d,i , µmax )
sbc =                 ,   sf ra =                 ,   sind =                     ,   sidio =                     ,
        K(µ0 , µmax )               K(µ0 , µmax )              K(µ0 , µmax )                    K(µ0 , µmax )

where µ0 are the observation means implied by a model with intercept terms only, signals µ†
are implied by a model specification with factors common to all data, (c), common factors
and a default-specific frailty factor, (c,d), common, frailty, and industry factors, (c,d,i).
Maximum model fit is given by µmax = y.

                                      [Insert Table 3 around here]

   Table 3 reports the estimated risk shares. Pooling over rating and industry groups, and
taking into account default and macroeconomic data over more than 35 years, we find that
about 69% of a firm’s total default risk is idiosyncratic. As a result, this type of risk can
largely be eliminated in a credit portfolio through diversification. The remaining share of
total risk, about 31%, however, does not average out in the cross section. We find that the
largest share of systematic default risk is due to common exposure to a large number of
macroeconomic and financial time series data. This business cycle component is about 17%
of total default risk, or 54% of systematic risk. Business cycle variation, while important, is
not sufficient to account for all default rate variability in the data. Specifically, we find that
about 6% of total default risk, or 20% of systematic risk, is due to an unobserved frailty
factor. Finally, about 8% of total default risk, or 26% of systematic risk, can be attributed
to industry-specific developments such as default contagion through business links.
   Default dynamics differ considerably across industries. For example, firms from the
energy, utilities, or transportation sector are less affected by common macroeconomic and
frailty risk. These industries require an industry-specific factor to capture sectoral dynamics.
We also find that firms rated speculative grade do not appear to have less systematic default
risk than firms rated investment grade. The lower sensitivity towards macroeconomic risk
for the latter group is offset by a higher sensitivity to latent frailty risk.

                                                      15
[Insert Figure 7 around here]

    Figure 7 plots the estimated risk shares over a rolling window of eight quarters. The
estimated risk shares vary considerably over time. While common variation with the business
cycle explains about 17% of total variation on average, this share may be as high as 40%
during the before times of crisis, such as in 1991 and 2007. Similarly, the frailty factor
captures a higher share of systematic default risk before and during times of crisis such as
1990-91 and 2006-07. In the former case, positive values of the frailty factor imply higher
default rates than implied by macroeconomic data during this time. In the latter case, the
significantly negative values of the frailty factor during 2006-07 captures default rates were
lower than what would be expected based on macro data alone. High absolute values of the
frailty factor imply times when the ‘default cycle’ diverges from ‘the business cycle’. Industry
specific effects are important mostly during the late 1980s and 2008. These are times when
e.g. banking specific risk is captured through industry specific factors. The bottom right
graph plots the share of idiosyncratic risk over time. We observe a decrease in idiosyncratic
risk building up to the 2007-09 crisis. Defaults appear to have become more systematic
between 2001 and 2007. Negative values of the frailty risk factor during this time indicates
that default rates were ‘systematically lower’ than what would be expected from macro data.

5     Implications for risk management practise

Most default risk models employed in risk management practise rely on the assumption of
conditionally independent defaults, or doubly stochastic default times. At the same time,
most models do not allow for unobserved (frailty) risk factors and intra-industry (contagion)
dynamics to capture excess default clustering. As reported in Section 4, the latter two
sources account for about 45% of systematic default risk. In this section we explore the
consequences for portfolio credit risk management of leaving out either source of default
clustering. This is of vital interest to practitioners and banking supervision.

                                              16
5.1    The frailty factor

The frailty factor captures about 20% of the common variation in default rates at the industry
and rating level, see Section 4. The presence of a frailty factor may increase default rate
volatility compared to a model without latent dynamics. As a result it may shift the portfolio
credit loss distribution towards more extreme values. This would increase the capital buffers
prescribed by the model. To explore this issue we conduct the following stylized credit risk
experiment.

      Case: A financial institution extends short-term loans to all Moody’s rated US
      firms. Loans are extended at the beginning of each quarter during 1981Q1 and
      2008Q4 at no interest. A non-defaulting loan is re-extended after three months.
      In case of a default only 20% of the principal is recovered. The loan exposure
                                         ³P      ´−1
      to each firm at time t is given by   j kjt     such that the total credit portfolio
      value is 1$ at all times. The institution uses the reduced form model of Section 3
      to set its capital buffers against future losses at a high percentile of the predictive
      loss distribution. Future risk factor realizations need to be forecast out of sample.
      Forecasting is based on a Kalman filter prediction step, using information up to
      time t only.

   The above situation is stylized in many regards. Nevertheless, it allows us to investigate
the importance of macroeconomic, frailty-, and sector-specific dynamics for the out-of-sample
risk management of a diversified loan portfolio.

                                 [Insert Figure 8 around here]

   The top left graph of Figure 8 plots the credit portfolio loss distribution as implied by
actual default data. This distribution can be compared to the loss distribution as implied
by three different specifications of the econometric model (top right to bottom right panel).
Portfolio credit loss distributions for actual portfolios are known to be skewed to the right,
and to exhibit irregular behavior in the right tail. Flat segments or bi-modality can arise due
to the discontinuity in payoffs in case of default. These qualitative features are also found
in the first panel.

                                                17
Comparing the loss distributions in the top panels suggests that common variation with
macro data may not be sufficient to reproduce the thick right-hand tail implied by actual
default data. An additional frailty factor shifts probability mass into the right tail. The
loss distribution implied by the full model is closest to the actual distribution. The full
econometric model therefore appears able to reproduce the positive skewness, excess kurtosis,
and the irregular shape in the right tail. Industry-specific variation in default rates may
cancel to some extent in a diversified portfolio. This may explain the slight difference between
the bottom two graphs.

                                  [Insert Figure 9 around here]

   Figure 9 allows to investigate the model fit to the time variation in aggregate US default
rates. We here distinguish four specifications with (a) no factors, (b) ftc only, (c) ftc , ftd ,
and (d) all factors ftc , ftd , fti . A static model fails to capture the observed default clustering
around recession periods. Latent frailty dynamics given by ftd are required for a good model
fit, in particular during bad times such as the 1991 recession.

5.2    Industry specific risk dynamics

We found that industry-specific variation accounts for about 25% of default rate volatility at
the rating and industry level, see Section 4. Industry-specific factors capture the differential
impact of each crisis on a given sector. For example, banking specific default stress has
been high before and during the 1991 and 2008 recessions, but negligible during the 2001
recession. And while the 2007-09 crisis is particularly stressful for firms from the financial,
manufacturing, and media, hotels, and leisure sector, it is relatively benign on the technology,
energy, and transportation sectors. While industry factors are important to fit industry-
specific defaults rate, the bottom graphs of Figure 9 also indicate that industry-specific
developments average out in the cross-section to some extend, and as a result may matter
less from a portfolio perspective.

                                 [Insert Figure 10 around here]

   A concrete example may be most helpful to see how macro, frailty, and industry-specific
dynamics combine to capture industry-level variation in default rates. Figure 10 compares

                                                 18
the model-implied quarterly default rates for a Ba-rated financial firm with the observed
fractions for all financial firms. We distinguish three model specifications, i.e., common
variation with macro data only, macro and frailty dynamics, and macro, frailty, and industry-
specific factors. Common variation of defaults with macroeconomic and financial data implies
substantial time-variation in implied default rates. For example, implied default rates in bad
times are about five times higher than in good years. The frailty factor captures the general
pattern that defaults are higher before and during the 1991 and 2001 recession than implied
by macro data alone, see Figure 6. Only the first effect is also true for the banking sector. The
industry factor corrects the common dynamics to captures the sector-specific stress during
the years of banking crises 1986-1990, and 2007-09. It also adjusts sector default rates to the
observed low rates during the 2001 recession. Accurate default rates at the industry level
are of importance mainly for short term loan pricing, counterparty risk management, and in
the presence of industry concentrations in the loan portfolio.

6     Conclusion
This paper introduces a novel econometric framework - the mixed measurement dynamic
factor model - for the joint modeling of discrete default counts and continuous macroeconomic
and financial data. Dynamic factors may be common to mixed measurement observations to
capture shared variation, or specific to subsets of data to capture frailty and industry-specific
dynamics.
    In the new framework we decompose total default rate variability into its latent con-
stituents using reductions in the estimated Kullback-Leibler divergence. We distinguish
business cycle, default-specific (frailty), and industry-level dynamics, and explore their con-
tribution to observed default clustering. We find that shared macroeconomic variation ac-
counts for the largest part of default rate volatility at the industry and rating class level.
While less important on average, frailty and industry-specific risk is important during times
of crisis, and to match observed default rates at the industry level.

                                               19
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                                          22
Table 1: Macroeconomic Time Series Data
The table gives a full listing of included macroeconomic time series data xt and binary indicators bt . All
time series are obtained from the St. Louis Fed online database, http://research.stlouisfed.org/fred2/.

               Category                   Summary of time series in category          Total no

              (a) Macro indicators, and   Industrial production index
              business cycle conditions   Disposable personal income
                                          ISM Manufacturing index
                                                                                      5
                                          Uni Michigan consumer sentiment
                                          New housing permits

              (b) Labour market           Civilian unemployment rate
               conditions                 Median duration of unemployment
                                          Average weekly hours index                  4
                                          Total non-farm payrolls

              (c) Monetary policy         Federal funds rate
              and financing conditions    Moody’s seasoned Baa corporate bond yield
                                          Mortgage rates, 30 year
                                          10 year treasury rate, constant maturity    6
                                          Credit spread corporates over treasuries
                                          Government bond term structure spread

              (d) Bank lending            Total Consumer Credit Outstanding
                                          Total Real Estate Loans, all banks          2

              (e) Cost of resources       PPI Fuels and related Energy
                                          PPI Finished Goods
                                                                                      3
                                          Trade-weighted US dollar exchange rate

              (f) Stock market returns    S&P 500 yearly returns
                                          S&P 500 return volatility                   2

                                                                                      22

                                                     23
Table 2: Parameter estimates, binomial part
We report parameter estimates associated with the binomial data. The coefficients in the first column
combine to fixed effects for each cross section, according to rating group and industry sector. The middle
columns refer to common factors ftc and give the respective factor loadings. The last column gives the factor
loadings for frailty factor ftd and industry-specific factors, fti , respectively. Estimation sample is 1971Q1 to
2009Q1.

                    Intercepts λj                Loadings ftm                 Loadings ftd
                    par     val                                               par    val
                                         par       val        par      val
                    λ0      −2.52                                             φd      0.84
                                         φc,1     0.88        φc,3    0.77
                                                                              γIG     0.20
                                         β1,IG    0.30        β3,IG   0.66
                    λf in   −0.24                                             γBa     0.51
                                         β1,Ba    0.20        β3,Ba   0.40
                    λtra    −0.16                                             γB      0.64
                                         β1,B     0.29        β3,B    0.27
                    λlei    −0.17                                             γC      0.42
                                         β1,C     0.19        β3,C    0.19
                    λutl    −0.51
                    λtec    −0.96                                             Loadings fti
                                         φc,2     0.91        φc,4    0.97
                    λret    −0.33                                             δf in   0.64
                                         β2,IG    0.24        β4,IG   0.66
                                                                              δtra    0.74
                                         β2,Ba    0.18        β4,Ba   0.26
                    λIG     −7.13                                             δlei    0.41
                                         β2,B     0.10        β4,B    -0.26
                    λBB     −3.89                                             δutl    0.99
                                         β2,C     0.26        β4,C    -0.02
                    λB      −2.12                                             δtec    0.39
                                                                              δret    0.43

                                                         24
Table 3: A decomposition of total default risk
The table decomposes total, i.e. systematic and idiosyncratic, default risk into four unobserved constituents.
We distinguish (i) common variation in defaults with observed macroeconomic and financial data, (ii) latent
default-specific (frailty) risk, (iii) latent industry-sector dynamics, and (iv) non-systematic, and therefore
diversifiable risk. The decomposition is based on data from 1971Q1 to 2009Q1.

     Data                       Business cycle       Frailty risk Industry-level Idiosyncratic
                                           ftc                 ftd            fti       distr.

     Pooled                               17.0%             6.3%                8.0%            68.7%

     Rating groups:
     Aaa-Baa                              10.5%             0.7%                4.5%            84.3%
     Ba                                    7.5%             5.8%                7.7%            79.1%
     B                                    24.7%             6.8%                7.6%            61.0%
     Caa-C                                13.4%             7.0%                9.5%            70.0%

     Industry sectors:
     Bank                                 10.2%             7.1%              13.7%             68.9%
     Financial non-Bank                    6.5%             3.2%               8.0%             82.3%
     Transportation                       11.9%             6.5%              12.1%             69.5%
     Media                                21.9%             8.1%               6.8%             63.2%
     Leisure                              22.4%             4.3%               4.1%             69.2%
     Utilities                             8.0%             2.8%               3.3%             85.9%
     Energy                               13.1%             8.7%              23.0%             55.3%
     Industrial                           28.5%             8.9%               1.9%             60.7%
     High Tech                            19.6%             6.9%               9.4%             64.0%
     Retail                                9.7%             3.9%               8.2%             78.2%
     Consumer Goods                       17.8%             4.0%               4.4%             73.9%
     Misc                                 10.8%             4.4%               2.6%             82.2%

                                                     25
Table 4: Value-at-Risk exceedances
The table reports the number of times that actual default losses during 1981Q1-2008Q4 exceed the simu-
lated Value-at-Risk levels based on a predictive loss distribution and a given model. We distinguish four
different model specifications with increasingly complex latent dynamics. The calculation of the predictive
loss distribution is out of sample using Kalman filter prediction steps.

                                  Case I: Diversified Portfolio
          V aR1−α         M0: no ft        M1: ftc       M2: ftc , ftd          M3: ftc , ftd , fti
                      # exc % exc # exc % exc # exc % exc                       # exc % exc
            0.80         20 17.9%       12     10.7%    10      8.9%             10       8.9%
            0.90         17 15.2%        8     7.1%      3      2.7%              2       1.8%
            0.95          8 7.1%         5     4.5%      1      0.9%              1       0.9%
            0.99          3 2.7%         1     0.9%      0        0%              0        0%
           0.997          1 0.9%         0      0%       0        0%              0        0%
                           Case II: Financial Industry Concentration
          V aR1−α         M0: no ft        M1: ftc       M2: ftc , ftd          M3: ftc , ftd , fti
                      # exc % exc # exc % exc # exc % exc                       # exc % exc
            0.80         23 20.5%          19 17.0%        21 18.8%                24 21.4%
            0.90         19 20.0%          14 12.5%        15 13.4%                13 11.6%
            0.95         16 14.3%          13 11.6%         7 6.3%                  4 3.8%
            0.99         14 12.5%           7 6.3%          1 0.9%                  1 0.9%
           0.997         12 10.7%           6 5.4%          0 0.0%                  0          0%

                                                    26
Figure 1: Models and reductions in the Kullback-Leibler divergence
The graph shows how reductions in the estimated KL divergence are used to decompose the total variation
in non-Gaussian default counts into risk shares corresponding to increasing sets of latent factors.

                                                  27
Figure 2: Clustering in default data P
The top graph plots (i) the total number of defaults in the Moody’s database j yjt , (ii) the total number
               P                                                                             P         P
of exposures j kjt , and (iii) the aggregate default rate for all Moody’s rated US firms,      j yjt /  j kjt .
The bottom graph plots time series of default fractions yjt /kjt over time. We distinguish four broad rating
groups, i.e., Aaa − Baa, Ba, B, and Caa − C, where each plot contains 12 time series of industry-specific
default fractions.

                        aggregate default counts
           40

           20

                 1970             1975             1980          1985        1990             1995       2000          2005

                        total number of firms
          3000

          2000

          1000

                 1970             1975             1980          1985        1990             1995       2000          2005

                         aggregate default rate
          0.02

          0.01

                 1970             1975             1980          1985        1990             1995       2000          2005

                   Aaa − Baa                                                         Ba
         0.15                                                              0.20

                                                                           0.15
         0.10

                                                                           0.10

         0.05
                                                                           0.05

           1980         1985      1990      1995          2000   2005   2010 1980      1985      1990   1995    2000     2005   2010
         1.00                                                              1.00
                   B                                                                Caa − C

         0.75                                                              0.75

         0.50                                                              0.50

         0.25                                                              0.25

            1980        1985      1990      1995          2000   2005   2010 1980      1985      1990   1995    2000     2005   2010

                                                                          28
Figure 3: Macroeconomic and financial time series data
     The graph contains times series plots of yearly growth rates in macroeconomic and financial data. For a listing of the data we refer to Table 1.

                                Indpro
                                                     0.2    dspi                      napm                       umich                     permit
                         0.1                                                   0.5                       0.25                       0.5

                        −0.1                         0.0                      −0.5                      −0.25                      −0.5
                               1980        2000            1980       2000           1980        2000           1980       2000           1980        2000
                           1    unrate                      uempmed                   AWHI                       payems                    gs10
                                                      1                                                  0.05                      0.25
                                                                              0.05
                           0                          0                                                  0.00
                                                                             −0.05                                                −0.25
                               1980        2000            1980       2000           1980        2000           1980       2000           1980        2000

29
                           2    fedfunds
                                                     0.5    baa                       mortg
                                                                                                          0.2    totalsl                   realln
                                                                               0.5                                                  0.2
                           0                         0.0                       0.0                                                  0.1
                                                                                                          0.0
                               1980        2000            1980       2000           1980        2000           1980       2000           1980        2000
                                ppieng
                                                     0.2    ppifgs                    twexbmth
                                                                                                           5     TSSprd
                                                                                                                                     2     CrdtSprd
                         0.5                                                   0.2
                         0.0                                                   0.0                                                   0
                                                     0.0                                                  −5
                               1980        2000            1980       2000           1980        2000           1980       2000           1980        2000
                                S_P500
                                                     0.3    Vola
                         0.5
                         0.0                         0.1

                               1980        2000            1980       2000
Figure 4: Smoothed Macroeconomic Risk Factors
The figure plots conditional mean estimates for four latent risk factors. These factors are common to all
mixed measurement data, i.e. macros and default counts. We also plot approximate standard error bands
at a 0.05 significance level.

                                                    2.5
           2

                                                    0.0
           0

                                                   −2.5
          −2
           1970     1980      1990     2000     2010 1970      1980     1990      2000     2010
                                                      3

                                                        2
           2
                                                        1

           0                                            0

                                                     −1

          −2                                         −2

           1970     1980      1990     2000     2010 1970      1980     1990      2000     2010

                                                   30
Figure 5: Shares of explained variation in macro and financial time series data
     The figure indicates which share of variation in each time series listed in Table 1 can be attributed to each factor f c . Factors f c are common to the
     (continuous) macro and financial as well as the (discrete) default count data.

31
Figure 6: Smoothed Frailty Risk Factor and Industry-group dynamics
The top graph shows the estimated frailty risk factor, which is assumed common to all default counts. The
second graph plots six industry-specific risk factors along with asymptotic standard error bands at a 0.05
significance level. High risk factor values imply higher expected default rates.

                    4     frailty component

                    3

                    2

                    1

                    0

                   −1

                   −2

                   −3

                   −4
                        1972.5 1975.0 1977.5 1980.0 1982.5 1985.0 1987.5 1990.0 1992.5 1995.0 1997.5 2000.0 2002.5 2005.0 2007.5

          financials (bank and non−bank)                                            transportation
                                                                           2.5
    2.5

    0.0                                                                    0.0

  −2.5
                                                                       −2.5
     1970          1980           1990             2000             2010 1970                 1980             1990                2000   2010
            media, hotels, and leisure                                             utilities and energy
    2.5                                                                    2.5

    0.0                                                                    0.0

  −2.5                                                                 −2.5
     1970          1980           1990             2000             2010 1970                 1980             1990                2000   2010
            technology and telecom                                                 retail and consumer goods
    2.5                                                                    2.5

    0.0                                                                    0.0

  −2.5                                                                 −2.5
    1970           1980           1990             2000             2010 1970                 1980             1990                2000   2010

                                                                        32
Figure 7: Time variation in risk shares
We plot risk shares estimated over a rolling window of eight quarters from 1971Q1 to 2009Q1. Shaded areas
correspond to recession periods as dated by the NBER.

    0.6                                                0.3
            business cycle variation                           frailty factor effect

    0.4                                                0.2

    0.2                                                0.1

     1970        1980             1990     2000    2010 1970         1980              1990   2000   2010
                                                      1.00
            industry−specific effects                          idiosyncratic component

    0.2                                               0.75

                                                      0.50
    0.1
                                                      0.25

     1970        1980             1990     2000    2010 1970         1980              1990   2000   2010

                                                     33
Figure 8: Real vs. model-implied credit portfolio loss distribution
The distribution plots refer to a credit portfolio with uniform loan exposures. The first graph shows the
portfolio loss distribution as implied by actual defaults and exposures in the database. The horizontal axis
measures quarterly loan losses as a fraction of the total portfolio value. The second to fourth panel plot the
portfolio loss distribution as implied by an econometric model with macro factors ftc , macro factors and a
frailty component ftc , ftd , and all factors ftc , ftd , fti , respectively.

                actual data                                 f^c only
                                                      300
          200

                                                      200

          100
                                                      100

                   0.000      0.005   0.010   0.015            0.000        0.005   0.010   0.015

                f^c,d                                       all factors f^c,d,i
          300                                         300

          200                                         200

          100                                         100

                  0.000       0.005   0.010   0.015            0.000        0.005   0.010   0.015

                                                      34
Figure 9: Model fit to observed aggregate default rate
Each panel plots the observed quarterly default rate for all rated firms against the default rate implied by
different model specifications. The models feature either (a) no factors, (b) only macro factors f c , (c) macro
factors and a frailty component f c , f d , and (d) all factors f c , f d , f i , respectively.

             model implied, no factors         actual rate
   0.015                                                                           model implied f^c         actual rate
                                                                       0.015

   0.010                                                               0.010

   0.005                                                               0.005

      1980   1985      1990        1995      2000        2005   2010        1980   1985      1990       1995        2000    2005    2010

             model implied f^c,d         observed rate                             model implied, f^c,d,i
   0.015                                                               0.015

   0.010                                                               0.010

   0.005                                                               0.005

      1980   1985      1990        1995       2000       2005   2010        1980   1985       1990          1995     2000    2005    2010

                                                                       35
Figure 10: Quarterly time-varying default intensities for financial firms
We plot smoothed estimates of quarterly time-varying default probabilities for a Ba rated financial firm.
We distinguish a model with (i) common variation with macro data only, (ii) macro factors and a frailty
component, and (iii) macro factors, frailty component, and industry-specific factors, respectively. The model-
implied quarterly probabilities are graphed against the observed default fractions for all financial firms.

              f^m,d            f^m
              f^m,d,i          observed rate
 0.0175

 0.0150

 0.0125

 0.0100

 0.0075

 0.0050

 0.0025

     1980               1985              1990       1995            2000            2005            2010

                                                     36
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