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Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Dynamic Factor Models With Macro, Frailty, and
Industry Effects for U.S. Default Counts: The Credit
Crisis of 2008
Siem Jan Koopman , André Lucas & Bernd Schwaab
To cite this article: Siem Jan Koopman , André Lucas & Bernd Schwaab (2012) Dynamic Factor Models With Macro, Frailty, and Industry Effects for U.S. Default Counts: The Credit Crisis of 2008, Journal of Business & Economic Statistics, 30:4, 521-532, DOI: 10.1080/07350015.2012.700859
To link to this article: http://dx.doi.org/10.1080/07350015.2012.700859
Accepted author version posted online: 29 Jun 2012.
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Supplementary materials for this article are available online. Please go tohttp://tandfonline.com/r/JBES
Dynamic Factor Models With Macro, Frailty,
and Industry Effects for U.S. Default Counts:
The Credit Crisis of 2008
Siem Jan K
OOPMANDepartment of Econometrics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam and Tinbergen Institute, 1082 MS Amsterdam, The Netherlands ([email protected])
Andr ´e L
UCASDepartment of Finance, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands, Tinbergen Institute and Duisenberg School of Finance, 1082 MS Amsterdam, The Netherlands ([email protected])
Bernd S
CHWAABFinancial Research Division, European Central Bank, Frankfurt, Germany ([email protected])
We develop a high-dimensional, nonlinear, and non-Gaussian dynamic factor model for the decomposi-tion of systematic default risk condidecomposi-tions into latent components for (1) macroeconomic/financial risk, (2) autonomous default dynamics (frailty), and (3) industry-specific effects. We analyze discrete U.S. corporate default counts together with macroeconomic and financial variables in one unifying framework. We find that approximately 35% of default rate variation is due to systematic and industry factors. Ap-proximately one-third of this systematic variation is captured by the macroeconomic and financial factors. The remainder is captured by frailty (40%) and industry (25%) effects. The default-specific effects are particularly relevant before and during times of financial turbulence. We detect a build-up of systematic risk over the period preceding the 2008 credit crisis. This article has online supplementary material.
KEY WORDS: Credit portfolio models; Default risk; Financial crisis; Frailty-correlated defaults; State space methods.
1. INTRODUCTION
We decompose default risk of U.S. firms into different sys-tematic components using a high-dimensional and partly non-linear, non-Gaussian dynamic factor model. Systematic default rate variation, also known as default clustering, constitutes one of the main risks in the banking book of financial institutions. It is established that such clustering is empirically relevant. For example, aggregate U.S. default rates during the 1991, 2001, and 2008 recession periods are up to five times higher than in intermediate expansion years. It is also known that default rates depend on the prevailing macroeconomic conditions (see, e.g., Pesaran et al.2006; Duffie, Saita, and Wang2007; Figlewski, Frydman, and Liang2008; Koopman et al.2009).
The common dependence of corporate credit quality on macroeconomic conditions is not the only explanation provided in the literature for default clustering. 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 rate variation. Das et al. (2007) rejected the joint hypothesis of (1) well-specified de-fault intensities in terms of observed macroeconomic and firm-specific information, and (2) the doubly stochastic independence assumption that underlies many credit risk models that are used
The views expressed in this article are those of the authors and they do not necessarily reflect the views or policies of the European Central Bank or the European System of Central Banks.
in practice. From this finding, two important separate strands of literature have emerged, focusing on frailty-correlated defaults and contagion.
In a frailty model, the additional variation in default intensities is captured by a latent dynamic process, or an unobserved com-ponent (see Das et al.2007; McNeil and Wendin2007; Koopman and Lucas2008; Koopman, Lucas, and Monteiro2008; Duffie et al.2009). This frailty factor captures default clustering above and beyond what can be explained by macroeconomic variables and firm-specific information. The unobserved component can capture effects of omitted variables in the model as well as other effects that are difficult to quantify (see Duffie et al. 2009). Contagion models, by contrast, focus on the phenomenon that a defaulting firm weakens other firms with which it has busi-ness links (see Giesecke2004; Giesecke and Azizpour2008). Adverse contagion effects may dominate potentially offsetting competitive effects at the intra-industry level (see, e.g., Lang and Stulz1992). Lando and Nielsen (2009) screened hundreds of default histories for evidence of direct default contagion. Their results suggest that domino style contagion is a minor concern. Indirect spillovers, for example through a fire sale of industry-specific assets, may nevertheless still explain some de-fault dependence at the industry level beyond what is induced
© 2012American Statistical Association Journal of Business & Economic Statistics
October 2012, Vol. 30, No. 4 DOI:10.1080/07350015.2012.700859
521
by shared exposure to macroeconomic/financial and default-specific (frailty) factors (see Jorion and Zhang2007,2009).
It is not known to what extent the three different explana-tions (macro, frailty, and industry effects) for default clustering interconnect. In particular, it is not yet clear how to measure the relative contribution of the different sources of systematic default risk to observed default clustering. This question is fun-damental to our understanding and modeling of default risk. Lando and Nielsen (2009) discussed whether default clustering can be compared with asthma or the flu. In the case of asthma, occurrences are not contagious but depend on exogenous back-ground processes such as air pollution. On the other hand, the flu is directly contagious. Frailty models are, in a sense, more related to models for asthma, while contagion models based on self-exciting processes are similar to models for flu. Whether one effect dominates the other empirically is therefore highly relevant for the appropriate modeling framework for risk in the banking book.
We decompose the systematic variation in corporate defaults into different constituents within a high-dimensional and partly nonlinear, non-Gaussian dynamic factor model. Within this modeling framework, we let changes in default rates at the rating and industry level be attributed to the macro, frailty, and industry effects, simultaneously. The estimation of these dynamic factors and other model parameters is carried out by Monte Carlo max-imum likelihood methods. The implementation details of the estimation methods are presented.
The attractive feature of our framework is threefold. First, it allows us to combine typical Gaussian time series (such as macroeconomic variables, business cycle indicators, financial market conditions, and interest rates) with discrete time series such as default counts. From a practical point of view, such an integrated framework for both defaults and macro covariates proves to be very convenient if the empirical model is also used for determining adequate economic capital buffers and in a stress-testing exercise. For example, it is straightforward to simulate future values of the joint factor process. Second, and in contrast to earlier approaches, we can include a substantive number of macroeconomic/financial variables to account for the macroeconomic conditions. Earlier empirical studies have relied on a small number of explanatory variables, say five or less. Third, our new framework allows for an integrated view on the interaction between macro, frailty, and industry factors by treating them simultaneously rather than in a typical two-step estimation approach. Macro and credit risk developments are endogenous, and can be forecast jointly without recourse to auxiliary models. Inference is preserved, which is usually problematic in a two-step approach.
Our estimation results indicate that defaults are more related to asthma than to flu: the common factors to all firms (macro and frailty) account for approximately 75% of the systematic default clustering. It leaves industry (and thus possibly conta-gion) effects as a substantial secondary source of credit portfolio risk. To quantify these contributions to systematic default risk, we introduce a pseudo-R2 measure of fit based on reductions in Kullback-Leibler (KL) divergence. The KL divergence is a standard statistical measure of “distance” between distributions and reduces to the usualR2in a linear regression model. Its use
is appropriate in a context where there are both discrete (default
counts) and continuous (macro variables) data. We find that on average across industries and time, 66% of total default risk is idiosyncratic and therefore diversifiable. The remainder 34% is systematic. For subinvestment grade firms, one-third of system-atic default risk can be attributed to shared exposure to com-mon macroeconomic and financial conditions. For investment grade firms, this percentage is as high as 60%. The remaining share of systematic credit risk is driven by a default-specific (frailty) factor and industry-specific factors (in approximately equal proportions). The frailty component cannot be diversified in the cross-section, whereas the industry effects can only be diversified to some extent.
Our reported risk shares vary considerably over industry sec-tors, rating groups, and time. For example, we find that the frailty component tends to explain a higher share of default rate variation before and during times of crisis. In particular, we find systematic credit risk building up in the years 2002–2008, lead-ing up to the financial crisis, when default activity was much lower than suggested by macro-financial data. The framework may thus also provide a tool to detect systemic risk build-up in the economy. Tools to assess the evolution and composition of latent financial risks are urgently needed at macro-prudential policy institutions, such as the Financial Services Oversight Council (FSOC) for the United States, and the European Sys-temic Risk Board (ESRB) for the European Union.
In addition, we investigate the possible missing sources of default rate variation that may be captured by the frailty factor. Interestingly, we find a positive correlation between changes in our estimated frailty factor and proxies for tightening lending standards. Together with other pieces of evidence, this find-ing suggests that the frailty component may be able to capture changes in credit supply conditions and the ease of credit ac-cess. Credit supply and credit risk are clearly connected: it is hard to default if available credit is plentiful. Conversely, even solvent firms can get into financial distress if credit is heavily rationed. Changes in the ease of credit access are typically hard to quantify empirically as they may depend on many develop-ments that are also hard to measure, such as changing activity in the securitization market or changes in banks’ business models. Still, when combined, all these factors may have a systematic and economically significant impact on the loss experience of diversified credit portfolios.
In relation to the 2008 crisis period, the following findings from our empirical study are relevant. First, significant frailty effects imply that default and business cycles do not coincide. They have diverged significantly and persistently in the past, and most recently during the run-up to the 2008 credit crisis. Such a decoupling may indicate a credit bubble, in particular if in addi-tion credit quantity growth is unusually high and bank lending standards are low. Second, stressing the usual macro-financial covariates may not be sufficient to assess financial stability con-ditions in a stress test. Systematic default rate increases also depend on additional latent systematic risk factors. These may need to be stressed just as the observed systematic risk factors are stressed. An admittedly incomplete understanding of latent credit risk sources does not change the fact that they matter em-pirically. Finally, while there is a long tradition in central banks of analyzing creditquantities over time and comparing them to fundamentals (such as tracking the private-credit to GDP
ratio), the tracking of creditriskconditions and its composition has received less to no attention at all. The new econometric methodology developed in this study is a versatile framework for making the latter operational.
The remainder of this article is organized as follows. Section
2introduces our general methodological framework. Section3
presents our core empirical results, in particular a decomposition of total systematic default risk into its latent constituents. We comment on implications for portfolio credit risk in Section4. Section5concludes.
2. A JOINT MODEL FOR DEFAULT, MACRO, AND INDUSTRY RISK
The key challenge in decomposing systematic credit risk is to define a factor model structure that can simultaneously handle normally distributed (macro variables) and nonnormally dis-tributed (default counts) data, as well as linear and nonlinear factor dependence. The factor model we introduce for this pur-pose is a Mixed Measurement Dynamic Factor Model, or in short, MiMe DFM. In the development of our new model, we focus on the decomposition of systematic default risk. However, the model may also find relevant applications in other areas of finance. The model is applicable to any setting where different distributions have to be mixed in a factor structure.
In our analysis, we consider the vector of observations given by
yt =(y1,t, . . . , yJ,t, yJ+1,t, . . . , yJ+N,t)′, (1)
with time indext =1, . . . , T. The first J elements of yt are
default counts. We count defaults for different rating groups and industries. As a consequence, the firstJelements ofytcontain
discrete, nonnegative values. The remainingN elements ofyt
contain macro and financial variables that are typically taken as Gaussian random variables. We assume that the default counts and the macro and financial time series data are subject to a set of dynamic factors. Some of these factors may be common to all variables inyt. Other factors may only affect a subset of the
elements inyt.
2.1 The Mixed Measurement Dynamic Factor Model
We distinguish macro, frailty, and industry factors; these common factors are denoted as fm
t , fdt, and fti, respectively.
The factors fm
t capture shared business cycle dynamics in
macroeconomic variables and default counts. Therefore, fac-torsfm
t are common to all variables inyt. Common frailty
fac-torsfd
t are default-specific, that is, common to default counts
(y1,t, . . . , yJ,t) only and independent of observed
macroeco-nomic and financial data by construction. By not allowing the frailty factors to impact the macro series yj,t for j =
J+1, . . . , J +N, we effectively restrictftd to capture default clustering above and beyond what is implied by macroeconomic and financial factorsftm. The third set of factorsftiaffects firms in the same industry. Such factors may be particularly relevant in specific industries such as the financial industry. In addi-tion, our estimated industry frailty factors may partly capture contagion-driven default clustering in specific industries.
We gather all factors into the vectorft=(ftm′,f d′
t ,f i′
t )′, which
we model as a dynamic latent (unobserved) vector variable. We adopt an autoregressive dynamic process for the latent factors,
ft=ft−1+ηt, t =1,2, . . . , T , (2)
where the coefficient matrix is assumed a diagonal matrix and with them×1 disturbance vectorηt ∼NID(0,η),which
is serially uncorrelated. The standard stationarity conditions ap-ply to ft. To complete the specification of Equation (2), we
specify the initial factor byf1∼N(0,f) wheref is the
un-conditional variance matrix offt and the solution of the matrix
equationf =f′+η. We below setf =Ito identify
the respective factor loadings.
More elaborate dynamic processes forft can also be
consid-ered. The autoregressive structure in Equation (2) allows the components offt to be sticky. The macroeconomic factorsftm
evolve slowly over time and capture business cycle effects in both macro and default data. The credit climate and industry default conditions are represented by persistent processes for fd
t andftithat typically capture the clustering of defaults during
high-default years.
Conditional on ft, the first J elements of yt are modeled
as binomial random variables with parameters kj,t and πj,t,
for j =1, . . . , J, wherekj,t is the number of firms and πj,t
is the probability of default in the industry-rating group j at time t. Exposures kj,t are counted at the beginning of each
time periodt, and are held fixed during this period. (For more details on the conditionally binomial model see, e.g., McNeil, Frey, and Embrechts 2005, chap. 9.) Frey and McNeil (2002) showed that almost all available industry credit risk models, such as Creditmetrics, Moody’s KMV, and CreditRisk+, can be presented as conditional binomial models. The lastNelements of yt are, conditional onft, distributed as independent normal
random variables with means µj,t and fixed variancesσj2 for
j =J+1, . . . , J +N.
The density specifications for default counts and macro vari-ables are given by
yj,t|ft ∼Bin(πj,t, kj,t), for j =1, . . . , J,
yj,t|ft ∼N(µj,t, σj2), for j =J +1, . . . , J +N,
(3)
where Bin refers to the Binomial density andN to the normal density. The support of probability πj,t is restricted
bet-ween 0 and 1. We enforce this by considering the logit trans-formation
The auxiliary variableπ∗
j,t and the mean of the normal density
µj,tare linear functions of the factorftand given by
πj,t∗ =λj+β′jf
In this particular specification of the MiMe DFM, we can measure the relative contributions of macro, frailty, and indus-try risk to general default risk. The factors infm
t capture
gen-eral developments such as business cycle activity, lending con-ditions, and developments in financial markets. The auxiliary variable for default probability in Equation (4) partly depends on macro factors, but also depends on frailty riskftdand industry fti factors. The specifications in Equations (4) and (5) are key to our empirical analysis where we focus on studying whether macro dynamics explain all systematic default rate variation, or whether and to what extent frailty and industry factors are also important.
For the identification of the factor loadings, we require stan-dardized factors ft. We therefore restrict the variance matrix
to f =I and hence the variance matrix of the disturbance
vector ηt in Equation (2) becomes η =I−′. The
es-timation of the variances σ2
j can be circumvented since we
assume that our macroeconomic variables are standardized. This is common practice in the macroeconomic forecasting literature (see, e.g., Stock and Watson 2002). We then have var(yj,t)=β′jfβj +σj2=1 and henceσ
The estimation of the constantsλj and the factor loadingsβj,
γj′ andδj′, forj =1, . . . , J +N andj′=1, . . . , J, together
with the diagonal elements ofin Equation (2) is carried out by the method of Monte Carlo maximum likelihood. For this purpose, we collect all unknown coefficients in a parameter vectorψ and numerically maximize the likelihood function
p(y;ψ)= in Equation (6) is not known analytically, and we therefore rely on numerical methods. The likelihood function in Equation (6) can be approximated efficiently via Monte Carlo integra-tion using the method of importance sampling (see Durbin and Koopman2001, Part II). After the numerical maximization of the Monte Carlo estimate of Equation (6) has produced the es-timate ofψ, we can estimate the unobserved macro, frailty, and industry factorsft and their standard errors using Monte Carlo
methods as well. The online Appendix provides further details.
2.2 Decomposition of the Default Count Variation
Given the estimated parameters and risk factors, we may wish to assess which share of variation in default data is captured by the different risk factors. For this purpose we adopt a pseudo-R2
measure that is discussed by Cameron and Windmeijer (1997). It is based on the Kullback-Leibler divergence measure and is defined as
to signals that are composed of different selections of factorsfm t ,
ftd, andfti in Equations (4) and (5). The KL(θ1,θ2) divergence
in Equation (7) measures the distance between the log-densities logpθ1 and logpθ2 with respect to the density of the model
with signal θ1. For example, if the densities in Equation (7)
are normal, KL(θ1,θ2) measures the increase in the sum of
squared residuals of model withθ2in relation to model withθ1.
In our set of models,pθ(y) refers to both normal and binomial
distributions.
The pseudo-R2 is defined as the proportional reduction in
variation of default rates due to the inclusion of additional fac-tors, that is,
R2(θ)=1− KL(θmax,θ)
KL(θmax,θna)
, (8)
where KL is defined in Equation (7). The signalθna does not
depend on any factor and consists of the constantsλj only, for
j =1, . . . , J +N. The model with signal θmax provides the
maximum possible fit as it contains a separate dummy variable for each observation. The model contains as many parameters as observations. While this unrestricted model is not useful for practical purposes, it does provide a benchmark for the max-imum possible fit.Figure 1 illustrates the KL measures from which we can compute the pseudo-R2 measures. We distin-guish several alternative model specifications indicated by their signalsθna, θm,θmd, and θmdi, which contain an increasing
collection of latent factors. The models withθm,θmd, andθmdi
cumulate the macroftm, frailtyfdt, and industryfti factors, re-spectively.
The value ofR2(θ) is by construction between 0 and 1. The relative contribution from each of our systematic credit risk factors is measured as the increase in the pseudo-R2value when
moving fromθmviaθmd toθmdi. The remainder increase from
θmdi to θmax can be qualified as idiosyncratic risk. Model fit
can be computed for subsets of data as well. In that case, only subsets ofθ(·)are compared to each other. This is possible since
all data conditional on their respective signals are assumed to be independent in the cross-section and over time. We refer to Cameron and Windmeijer (1997) for a clear motivation of using KL-basedR2statistics.
3. EMPIRICAL FINDINGS FOR U.S. DEFAULT AND MACRO DATA
3.1 Data
We study the quarterly default and exposure counts obtained from the Moody’s corporate default research database for the pe-riod 1971Q1 to 2009Q1. We distinguish seven industry groups (fin: financials and insurance; tra: transportation; lei: media, ho-tels, and leisure; utl: utilities and energy; ind: industrials; tec: technology; and ret: retail and consumer products) and four rating groups (investment grade Aaa–Baa, and the speculative grade groups Ba, B, Caa–C). We have pooled the investment grade firms because defaults are rare for this segment. It is assumed that current issuer ratings summarize the available in-formation about a firm’s financial strength. This may be true only as a first approximation. However, rating agencies take into account a vast number of accounting and management
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 models with increasing sets of latent factors (Mna, Mm, Mmd, Mmdi, and Mmax).
information, and provide an assessment of firm-specific infor-mation that is comparable across industry sectors. While we focus on issuer ratings as available to us from Moody’s, rat-ings could alternatively be constructed from mapping EDF and CDS data into rating bins where available. In these cases, how-ever, limited sample sizes are typically an issue. Finally, other factors may help in explaining firm-specific default risks, such as distance-to-default measures, trailing stock returns, and ac-counting information (leverage). However, improving risk pre-diction for single-name credit risks is not the aim of this study. Ratings are taken into account mainly because macro, frailty, and industry effects (in terms of factor loadings) may be differ-ent for financially healthy and financially weaker firms. Ratings are likely to be sufficiently accurate for that purpose.
Figure 2presents aggregate default fractions and disaggre-gated default data. We observe a considerable time variation in aggregate default fractions. The disaggregated data reveals that
defaults cluster around recession periods for both investment grade and speculative grade rated firms.
Macroeconomic and financial data are obtained from the St. Louis Fed online database FRED. We consider macro-financial covariates that are typically also stressed in a macro stress test, as indicated in the literature by, for example, CEBS (2010) and Tarullo (2010). These usually involve business cycle measure-ments (such as real production and income data), labor market conditions (such as the unemployment rate and hours worked), short- and long-term interest rates (term structure) and credit spreads, as well as stock market returns and volatilities. We re-fer to the online Appendix for a detailed listing of the macro data covariates and corresponding time series plots. The macro covariates enter the analysis in the form of annual growth rates, with the exception of stock market volatility and the term struc-ture, which are not transformed. Data are available at a quarterly frequency.
1980 1990 2000
0.01 0.02
aggregate default rate
1980
1990
2000
2010
0.25
0.50
0.75
1.00
observed default fractions, Aaa−Baa observed default fractions, Ba observed default fractions, B observed default fractions, Caa−C
Figure 2. Clustering in default data. The left panel presents the aggregate default rate for all Moody’s rated U.S. firms,jyj,t/
jkj,t, for
j=1, . . . , J from 1971Q1 to 2009Q1. The right panel plots the disaggregated default fractionsyj,t/ kj,t over the same time. It distinguishes
four broad rating groups Aaa–Baa, Ba, B, and Caa–C. Each rating group contains industry-specific default fractions (which may often be zero). Shaded areas correspond to NBER U.S. recession periods.
Table 1. Parameter estimates (for the binomial series)
Interceptsλj Loadingsftm Loadingsf
d t
par val t-stat par val t-stat par val t-stat
λ0 −2.52 7.61 1 0.89 2.61 γI G 0.16 0.75
β1,I G 0.57 1.05 γBa 0.52 2.15
λfin −0.21 0.82 β1,Ba 0.30 0.55 γB 0.72 4.80
λtra −0.02 0.07 β1,B 0.26 0.72 γC 0.43 5.92
λlei −0.17 0.75 β1,C 0.17 0.57
λutl −0.76 2.02 Loadingsfti
λtec −0.09 0.64 2 0.91 2.21 δfin 0.73 2.96
λret −0.32 1.68 β2,I G −0.01 −0.06 δtra 0.61 1.85
β2,Ba 0.05 0.10 δlei 0.41 2.62
λI G −6.95 15.30 β2,B 0.12 0.39 δutl 0.98 3.87
λBB −3.92 14.11 β2,C 0.24 0.97 δtec 0.41 2.56
λB −2.22 11.21 δret 0.40 2.71
3 0.77 1.64
β3,I G 0.70 1.86
β3,Ba 0.34 0.86
β3,B 0.24 1.08
β3,C 0.18 1.45
4 0.94 2.99
β4,I G 0.53 0.80
β4,Ba 0.41 0.65
β4,B −0.08 −0.43
β4,C 0.14 0.37
NOTE: We report parameter estimates associated with the binomial part of our model. The coefficients in the first column combine to fixed effects according toλj=λ0+λ1,rj+λ2,sj;
the common interceptλ0is adjusted to take into account a fixed effect for the rating group and industry sector. The middle column reports autoregressive coefficientsk(thekth diagonal
element ofin Equation (2) fork=1,2,3,4) and loading coefficients inβjon four common macro factorsfmt. The last column reports the loading coefficientsγjon the frailty factor
fd
t and loadingsδjon the respective industry-specific risk factorsfti. The estimation sample is from 1971Q1 to 2009Q1.
3.2 Model Specification
Each time series of default countsyj,t, forj =1, . . . , J, is
modeled as a binomial sequence. Its log-odds ratioπ∗
j,t is given
by Equation (4), where the scalar coefficientλjis a fixed effect,
loading vectorsβj,γj, andδj correspond to macro factorsftm,
to frailty factorsfd
t, and to industry factorsfti, respectively.
Since the cross-section is high-dimensional due to the com-bination of rating classes and industry specifications, we follow Koopman and Lucas (2008) in reducing the number of param-eters by restricting the coefficients in the following additive structure:
¯
χj =χ0+χ1,sj +χ2,rj, χ =λ,β,γ,δ, (9)
where ¯χj represents the combined effect, χ0 is the baseline effect, χ1,s is the industry-specific deviation, and χ2,r is the rating group deviation. The deviations of all industry and rating groups cannot be identified simultaneously given the presence of the baselineχ0, such that some deviation parameters need to be set to zero. Likelihood ratio tests can be used to assess whether specific effects are significant. For example, we im-pose the restriction that β1,s=γ1,s=0, for all industries s,
after noticing that the joint test for this null hypothesis is sta-tistically insignificant. Industry-specific effects from common factors are now captured by the industry-specific factors. In ad-dition, we do not distinguish between ratings when assessing the impact of industry-specific factors, that is,δ2,r =0 for allr. We
impose this restriction because defaults are particularly rare at the disaggregated industry level. This renders the parameterδ2,r
difficult or impossible to estimate. Using the above parameter specification, we combine model parsimony with the ability to still investigate a rich set of hypotheses empirically given the data at hand. In particular, our specification allows for consider-ably more flexibility than specifications in the current empirical literature.
For the selection of the factor structure, we rely on likelihood-based information criteria (IC). Due to space constraints, we report these criteria in the online Appendix. We compare stan-dard criteria such as likelihood, AIC, BIC, and compute also the panel information criteria by Bai and Ng (2002). All information criteria are estimated from the complete model specification. According to information criteria, two to four macro factors are appropriate to summarize the information content in the macro panel. We include four macro factors in our final specification to be conservative. Allowing for one frailty factor is standard in the literature (see McNeil and Wendin 2007; Duffie et al.
2009; Azizpour, Giesecke, and Schwenkler2010), and appears sufficient to capture deviations of credit from macro conditions when modeling defaults. Finally, we allow for six industry-specific factors driving defaults for firms in the broad industry groups mentioned earlier.
3.3 Parameter and Risk Factor Estimates
Parameter estimates associated with the default counts are presented inTable 1. Estimated coefficients refer to a model specification with macroeconomic, frailty, and industry-specific
1970 1975 1980 1985 1990 1995 2000 2005 2010 −4
−3 −2 −1 0 1 2 3 4
frailty factor, with 0.95 SE band
1970 1980 1990 2000 2010
−2.5 0.0 2.5
financials (bank and non−bank)
1970 1980 1990 2000 2010
−2.5 0.0 2.5
transportation
1970 1980 1990 2000 2010
−2.5 0.0 2.5
media and leisure
1970 1980 1990 2000 2010
−2.5 0.0
2.5 utilities and energy
1970 1980 1990 2000 2010
−2.5 0.0 2.5
technology
1970 1980 1990 2000 2010
−2.5 0.0 2.5
retail and consumer goods
Figure 3. Frailty risk factor and industry-group dynamics. The first panel presents the estimated conditional mean of the frailty risk factor. This risk factor is common to all default counts. The final six panels present six industry-specific risk factors along with asymptotic standard error bands at a 95% confidence level. High risk factor values imply higher expected default rates.
factors. Parameter estimates in the first column combine to fixed effects for each cross-sectionj as given in Equation (9). The second column reports the factor loadings β associated with four common macro factors fm
t . Factor loadingsγ andδ are
given in the last two columns ofTable 1.
The macro factor loadings tend to be larger for investment grade firms; in particular, their loadings associated with factors 1, 3, and 4 are relatively large. This finding confirms that finan-cially healthy firms tend to be more sensitive to business cycle risk, as suggested, for example, in the literature by Basel Com-mittee on Banking Supervision (2004). The loadings inγ are associated with a single common frailty factorfd
t. The frailty
risk factorfd
t is, by construction, common to all firms, but
unre-lated to the macroeconomic data. Frailty risk is relatively large
for all firms, but particularly pronounced for speculative grade firms. The loadings in δ are for six orthogonal industry fac-tors fi
t. Industry sector loadings are highest for the financial,
transportation, and energy and utilities sector.
Figure 3presents conditional mean estimates of the frailty and industry-specific factors. We refer to the online Appendix for conditional mean estimates of the macroeconomic risk factors. The frailty factor is highbeforeandduringthe recession years 1991 and 2001. As a result, the frailty factor implies additional default clustering in these times of stress. On the other hand, the large negative values before the 2007–2009 credit crisis imply defaults that are systematically “too low” compared to what is implied by macroeconomic and financial data. The frailty fac-tor reverts to its mean level during the 2007–2009 credit crisis.
Investment grade default signal, for industrial firms IG default signal implied by ftm only
1985 1990 1995 2000 2005
−2 0 2
Investment grade default signal, for industrial firms
IG default signal implied by ftm only Speculative grade default signal for industrial firms SG default signal implied by f
t m only
1985 1990 1995 2000 2005
−2.5 0.0 2.5
Speculative grade default signal for industrial firms SG default signal implied by ftm only
Figure 4. Estimated default signals. The left and right panels plot the time variation in default signalsθj,tfor firms rated investment grade
and speculative grade, respectively. In each panel, the share of variation implied by the macro factorsfm
t is indicated.
Apparently, the extreme realizations in macroeconomic and fi-nancial variables during 2008–2009 are sufficient to account for the levels of observed defaults.
Both Das et al. (2007) and Duffie et al. (2009) asked what effects are captured by the frailty factor. Our estimate inFigure 3
suggests that the frailty factor may partly capture the outward shift in credit supply due to a high level of asset securitization activity and a lowering of lending standards during 2005–2007. Put starkly, it is hard to default if one is drowning in credit. Conversely, in 2001 and 2002, the frailty factor may capture adverse shocks to credit supply due to the disappearance of creditor’s trust in accounting information in response to the Enron and Worldcom scandals. In these times, even solvent firms can get into illiquidity problems if they are credit rationed. These credit supply effects are likely to be important for defaults, but also difficult to measure, for instance, because only the intersection of credit supply and demand is observed, and not credit supply itself.
We now present two additional pieces of evidence for the claim that our estimated frailty effects in part capture changes in the ease of credit access for credit constrained firms. First,
Figure 4 presents the estimated composite default signalsθj,t
for investment grade industrial firms (Aaa–Baa) and respective speculative grade firms (Ba–C). For investment grade firms, the default clustering implied by the “observed” macro-financial risk factors is sufficient. For the speculative grade group,
how-ever, frailty effects imply additional default clustering dur-ing the late 1980s (savdur-ings and loan crisis), and help capture the very low default rates for bad risks in the years leading up to the financial crisis. This demonstrates that frailty ef-fects are more important for financially weaker, and thus more credit constrained firms. Investment grade firms load on frailty to a relatively low extent (see also the parameter estimates inTable 1).
Second, Figure 5 relates changes in frailty effects to changes in bank lending standards (BLS) from the U.S. Se-nior Loan Officers (SLO) survey on lending practices, see (www.federalreserve.gov/boarddocs/snloansurvey). We con-sider the net percentage of banks that reported a tightening of credit standards for new commercial and industrial loans to large- and medium-sized enterprises. Lending standards are considered an imperfect but mainly accurate measure of credit supply, as demand conditions may also play a role (see, e.g., Lown and Morgan 2006; Maddaloni and Peydro 2011). Fig-ure 5shows a positive correlation between past net changes in lending standards on new loans and changes in estimated frailty effects. The intuition is that providing credit to increas-ingly worse borrowers pushes default activity beyond what is implied by macro-financial covariates that are usually employed in credit risk modeling. The correlation is otherwise consistent with the notion that frailty effects may capture specific omitted variables related to changes in credit supply, which are hard to
changes in frailty factor, yoy
reported net tightening of bank lending standards
1990 1995 2000 2005
−25 0 25 50
75 changes in frailty factor, yoy reported net tightening of bank lending standards changes in lending standards, yoy × changes in frailty, yoy
−1.0 −0.5 0.0 0.5 1.0
−25 0 25 50
75 changes in lending standards, yoy × changes in frailty, yoy
Figure 5. Frailty and bank lending standards. The left panel compares changes in the frailty factor with the net percentage of U.S. financial firms that have reported a tightening of lending standards in the Senior Loan Officer’s survey. The right panel reports the corresponding scatterplot. The respective correlation coefficient isρ=0.32; the slope of the regression line is statistically significant at a 1% significance level.
Table 2. A decomposition of total default risk
Business cycle Frailty risk Industry-level Idiosyncratic
Data fc
t ftd fti distr.
Pooled 11.4% 13.9% 8.6% 66.1%
(33.6%) (40.9%) (25.4%)
Rating groups:
Aaa–Baa 10.4% 1.1% 6.4% 82.1%
(58.0%) (6.3%) (35.7%)
Ba 7.1% 7.5% 6.2% 79.2%
(34.0%) (36.0%) (30.0%)
B 12.5% 22.3% 7.0% 58.2%
(30.0%) (53.2%) (16.8%)
Caa–C 12.3% 8.9% 12.3% 66.5%
(36.7%) (26.5%) (36.8%)
Industry sectors:
Bank 5.4% 11.9% 18.8% 63.8%
Financial nonbank 5.0% 5.3% 9.2% 80.5%
Transportation 7.4% 13.7% 18.8% 60.1%
Media 10.6% 19.9% 8.8% 60.8%
Leisure 15.7% 11.1% 2.6% 70.7%
Utilities 1.1% 4.9% 10.7% 83.3%
Energy 24.0% 8.7% 18.0% 49.3%
Industrial 16.3% 23.1% – 60.7%
Technology 17.2% 11.0% 12.5% 59.3%
Retail 6.7% 9.6% 10.4% 73.2%
Consumer goods 4.6% 18.4% 1.3% 75.7%
Misc 4.5% 13.2% 1.4% 80.9%
NOTE: This table decomposes total (systematic and idiosyncratic) default risk into four unobserved constituents. We distinguish (1) common variation in defaults with observed macroeconomic and financial data, (2) latent default-specific (frailty) risk, (3) latent industry-sector dynamics, and (4) nonsystematic, and therefore diversifiable risk. The decomposition is based on our factor and parameter estimates using data from 1971Q1 to 2009Q1. The risk shares are obtained as increments in theR2measure, Equation (8), as discussed in Section
2.2, and refer to total (systematic and idiosyncratic) default risk. Risk shares may refer to subsets of data; model fit is then compared for data from these subsets. Numbers in parentheses are shares of systematic risk.
measure empirically. The correlation seems to break down dur-ing the crisis when many other concerns are likely to have driven bank lending behavior and lending standards. Finally, frailty may also capture contagion effects across industry sectors. We refer to Azizpour, Giesecke, and Schwenkler (2010) who devel-oped a framework to disentangle contagion dependence from frailty.
Industry factorsfi
tcapture deviations of industry-specific
con-ditions from what is implied by purely common variation. Two alternative explanations come to mind. First, industry-specific dynamics may capture the industry-specific response of firms to common shocks. In that case, industry factors can be inter-preted as industry-specific frailty effects. For example, different recessions have had different impacts on a given industry over time. Second, intra-industry contagion through, for example, supply chain links may be an important source of dependence even after conditioning on common factors. Lando and Nielsen (2009) stated that it is very difficult to find evidence for di-rect default contagion in the Moody’s database. They base their conclusion on studying the qualitative summaries of individual firms’ default histories. Further data analysis suggests that di-rect contagion plays no role, while some weaker evidence for indirect contagion through balance sheet covariates exists. We come to a similar conclusion. For example, the default stress for technology firms in 2001–2002 is clearly visible in the esti-mated industry-specific risk factor, but is most likely not due to contagion, but to the burst of the earlier asset bubble. Similarly,
the 9/11 shock to the airline industry is visible as a brief spike in the transportation sector at the time, and difficult to interpret as contagion. As a result, the statistical and economic significance of factorsfi
t is more likely due to industry-specific
heterogene-ity rather than domino-style contagion dynamics. We accept, however, that no more rigorous distinction can be made based on the available data. We again refer to Azizpour, Giesecke, and Schwenkler (2010) for a first step to empirically distinguish common factors from contagious spillovers.
3.4 Total Default Risk: A Decomposition
We use the pseudo-R2 measure as explained in Section2.2
to assess which share of default rate variation is captured by an increasing set of systematic risk factors. We are the first to do so in detail and investigate the changing composition of systematic default risk over time.
Table 2 reports the estimated risk shares. By pooling over rating and industry groups, and by taking into account default and macroeconomic data for more than 35 years, we find that approximately 66% of a firm’s total default risk is idiosyncratic, and 8.6% is industry related. The idiosyncratic risk (and to some extent the industry risk as well) can be eliminated in a large credit portfolio through diversification. The remaining share of risk, approximately 25%, does not average out in the cross-section and is referred to as systematic risk. We find that for finan-cially healthy firms (high ratings) the largest share of systematic
default risk is due to the common exposure to macroeconomic and financial time series data. This common exposure can be regarded as the business cycle component. It constitutes ap-proximately 58% of systematic risk for firms rated investment grade, and 30% –37% for firms rated speculative grade. The business cycle variation is not sufficient to account for all de-fault rate variation in the data. Specifically, our results indicate that approximately 14% of total default risk, which is 41% of systematic risk, is due to an unobserved frailty factor. Frailty risk is low for investment grade firms (6%), but substantially larger for financially weaker firms (from 26% for Caa to 53% for B rated firms). Finally, approximately 9% of total default risk, or 25% of systematic risk, can be attributed to industry-specific developments.
Table 2indicates how the estimated risk shares vary across rating and industry groups. The question whether firms rated investment grade have higher systematic risk than firms rated speculative grade is raised, for instance, by the Basel committee (see Basel Committee on Banking Supervision2004), and credit risk practitioners. Empirical studies employing a single latent factor tend to confirm this finding (see McNeil and Wendin2007; Koopman and Lucas2008). In contrast to earlier studies, the last column ofTable 2indicates that speculative grade firms do not have less systematic risk than investment grade firms. This finding can be traced back to two sources. First, the frailty factor loads more heavily on speculative grade firms than investment grade firms. Second, some macro risk factors load on low rating groups also (seeTable 1).
Figure 6presents time series of estimated risk shares over a rolling window of eight quarters. We also plot estimates for a rolling window of four and 16 quarters, which yields similar
results. The estimated risk shares vary considerably over time. While common variation with the business cycle explains ap-proximately 11% of total variation on average, this share may be as high as 40%, for example in the years leading up to 2007. Similarly, the frailty factor captures a higher share of systematic default risk before and during times of crisis such as 1990–1991 and 2006–2007. In the former case, positive values of the frailty factor imply higher default rates that go beyond those implied by macroeconomic data. In the latter case, the significantly neg-ative values of the frailty factor during 2006–2007 imply lower default rates than expected from macroeconomic data only. High absolute values of the frailty factor imply times when systematic default risk diverges from business cycle developments as rep-resented by the common factors. Industry-specific effects have been important mostly during the late 1980s and 2001–2002. These are periods when banking-specific risk and the burst of the technology bubble are captured through industry-specific factors, respectively.
The bottom right graph ofFigure 6presents the share of id-iosyncratic risk over time. We observe a gradual decrease in idiosyncratic risk building up to the 2007–2009 crisis. Defaults become more systematic between 2001 and 2007 due to both macro and frailty effects. Negative values of the frailty risk factor during these years indicate that default rates were “systemati-cally lower” than what would be expected from macroeconomic developments. Again, this underlines the need for default risk models that include other risk factors above and beyond stan-dard observed macroeconomic and financial time series. Such factors pick up rapid changes in the credit climate that might not be captured sufficiently well by observed risk factors. We find that frailty effects are economically important in addition to
1970 1980 1990 2000 2010
0.1 0.2 0.3 0.4
0.5 business cycle variation, window of 4 quarters
8 quarters 16 quarters
1970 1980 1990 2000 2010
0.2 0.4
frailty factor effect, window of 4 quarters 8 quarters
16 quarters
1970 1980 1990 2000 2010
0.1 0.2 0.3
industry−specific effects, window of 4 quarters 8 quarters
16 quarters
1970 1980 1990 2000 2010
0.25 0.50 0.75 1.00
idiosyncratic component, window of 4 quarters 8 quarters
16 quarters
Figure 6. Time variation in risk shares. We present risk shares estimated over a rolling window from 1971Q1 to 2009Q1. Rolling window sizes are 4, 8, and 16 quarters. Shaded areas correspond to recession periods as dated by the NBER.
loss density/histogram actual data loss density, fm
loss density, fm,fd loss density, fm,fd,fi
0.000 0.005 0.010 0.015 0.020 50
100 150 200
loss density/histogram actual data loss density, fm
loss density, fm,fd loss density, fm,fd,fi
predictive portfolio loss 2009, in % of total, fm predictive portfolio loss 2009, in % of total, fm,fd
predictive portfolio loss 2009, in % of total, fm,fd,fi
0.0 2.5 5.0 7.5 10.0 12.5 15.0 0.1
0.2 0.3
predictive portfolio loss 2009, in % of total, fm predictive portfolio loss 2009, in % of total, fm,fd
predictive portfolio loss 2009, in % of total, fm,fd,fi
Figure 7. Real versus model-implied credit portfolio loss distribution. We present distribution plots for a credit portfolio with uniform loan exposures to Moody’s rated firms. The left panel presents the unconditional loss distribution as implied by historical quarterly defaults and firm counts in the database. The horizontal axis measures quarterly loan losses as a fraction of portfolio value. The left panel also presents the unconditional loss densities as implied by models with macro factorsfm
t , macro factors and a frailty componentfmt ,ftd, and all factorsftm,ftd,
fi
t, respectively. Positive probabilities of a negative portfolio loss are due to the (Gaussian Kernel) smoothing of the histogram. The right panel
presents three simulated predictive portfolio loss densities for the year 2009, conditional on macro and default data until end of 2008, for different risk factor specifications. The horizontal axis measures annual losses as fractions of portfolio value.
being statistically significant. We address the economic impact of frailty and industry factors further in Section4.
4. IMPLICATIONS FOR RISK MANAGEMENT
Many default risk models that are employed in day-to-day risk management rely on the assumption of conditionally inde-pendent defaults, or doubly stochastic default times (see Das et al.2007). At the same time, most models do not allow for unobserved risk factors and intra-industry dynamics to capture excess default clustering. We have reported in Section3.4that frailty and industry factors often account for more than half of systematic default risk. In this section, we explore the conse-quences for portfolio credit risk when frailty and industry factors are not accounted for in explaining default variation. This is of key importance for internal risk assessment as well as external (macro-prudential) supervision: If shared effects are missing, standard portfolio credit risk models may tend to be wrong all at the same time.
The left panel inFigure 7contains the credit portfolio loss distribution implied by actual historical default data and a port-folio of uniform exposures to all Moody’s rated U.S. firms that are active at a given time. For simplicity, we assume a stressed loss-given-default of 80%. The loss density is therefore a hori-zontally scaled version of historically observed loss rates. This distribution can be compared with the (unconditional) loss dis-tribution implied by three different specifications of our model from Section2. Portfolio loss densities for actual loan portfolios are known to be skewed to the right and leptokurtic (see, e.g., McNeil, Frey, and Embrechts2005, chap. 8). Flat segments or bimodality may arise due to the discontinuity in recovered prin-cipal in case of default. These qualitative features are confirmed in the left panel ofFigure 7but only in a mildly manner.
By comparing the unconditional loss distributions in the left panel ofFigure 7, we find that the common variation obtained from macroeconomic data is in general not sufficient to repro-duce the thick right-hand tail implied by actual default data. In particular, the shape of the upper tail of the empirical
distribu-tion is not well reproduced if only macro factors are used. The additional frailty and industry factors shift some of the prob-ability mass into the right tail. The loss distributions implied by these models are closer to the actual distribution. The full model is able to reproduce the distributional characteristics of default rates, such as the positive skewness, excess kurtosis, and an irregular shape in the upper tail. It indicates that our extended model with multiple factors provides a better description of the observed data.
The right-hand panel ofFigure 7plots the simulated predictive credit portfolio loss densities for the year 2009, conditional on data until the end of 2008, as implied by different model specifications. We refer to the online Appendix for the detailed results of our simulation study. Similarly to the unconditional case, the frailty factor shifts probability mass from the center of the distribution into the upper tail. Simulated risk measures are higher as a result. For the plotted densities, the simulated 99th percentile shifts out from about 6.24% to 8.34% of total portfolio value, which is an increase by more than 33%. Predicted annual mean losses are comparable at 2.96% and 2.71%, respectively. This demonstrates that frailty effects are economically important in addition to being statistically significant.
5. CONCLUSION
We have presented a new decomposition framework for sys-tematic default risk. By means of a dynamic factor analysis, we can measure the contribution of macro, frailty, and industry-specific risk factors to overall changes in default rates. In our study of defaults for U.S. firms, we found that approximately one-third of default rate variation at the industry and rating level is systematic. The systematic default rate variation can be fur-ther decomposed into macro- and frailty-driven. The part due to dependence on common macroeconomic conditions and fi-nancial activity ranges from about 30% for subinvestment up to 60% for investment grade companies. The remaining share of systematic credit risk is captured by frailty and industry risk. These findings suggest that credit risk management at the
portfolio level should account for observed macro drivers of credit risk as well as for unobserved risk factors. In particular, standard portfolio credit risk models that account for macroeco-nomic dependence only leave out a substantial part of systematic credit risk.
We have given further empirical evidence that the composi-tion of systematic risk varies over time. In particular, we observe a gradual build-up of systematic risk over the period 2002–2007. Such patterns can be used as early warning signals for financial institutions and supervising agencies. If the degree of systematic comovement between credit exposures increases through time, the fragility of the financial system may increase and prompt for an adequate (re)action. Interestingly, the frailty component appears more important for lower grade companies in periods of stress, which is precisely the time when such companies might be expected to experience more difficulties in rolling over debt. In addition, changes in our estimated frailty factor appear positively correlated with tightening lending standards. This suggests that frailty factors may capture changes in economic conditions which are hard to quantify, but impact the quality and default experience of bank portfolios in an economically significant way.
Our results have a clear bearing for risk management at finan-cial institutions. When conducting risk analysis at the portfolio level, the frailty and industry components cannot be discarded. This is confirmed in a risk management experiment using a styl-ized loan portfolio. The extreme tail clustering in defaults cannot be captured using macro variables alone. Additional sources of default variation such as frailty and contagion need to be iden-tified to capture the patterns in default data over time.
ACKNOWLEDGMENTS
We thank the editor and the referees for their insightful com-ments. The article has improved considerably as a result. We also thank Arthur Korteweg (discussant at 2011 AFA meet-ing in Denver), Kasper Roszbach (discussant at Atlanta Fed’s day ahead conference), Alexandre Adam (discussant at 2009 CREDIT conference in Venice), and participants at 2009 SoFiE credit risk conference in Chicago. We finally thank seminar par-ticipants at European Central Bank, Swedish National Bank, Tinbergen Institute, and University of Amsterdam, as well as Robert Engle, Kay Giesecke, Philipp Hartmann, David Lando, and Michel van der Wel. Andr´e Lucas thanks the Dutch National Science Foundation (NWO) for financial support.
[Received June 2011. Revised May 2012.]
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