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Ultimate Recovery Mixtures
Abstract
We propose a relatively simple, accurate and flexible approach to fore-
casting the distribution of defaulted debt recovery outcomes. Our approach
is based on mixtures of Gaussian distributions, explicitly conditioned on bor-
rower characteristics, debt instrument characteristics and credit conditions
at the time of default. Using Moody’s Ultimate Recovery Database, we show
that our mixture specification yields more accurate forecasts of ultimate re-
coveries on portfolios of defaulted loans and bonds on an out-of-sample basis
than popular regression-based estimates. Further, the economically inter-
pretable outputs of our model provide a richer characterization of how con-
ditioning variables affect recovery outcomes than competing approaches. The
latter benefit is of particular importance in understanding shifts in the rel-
ative likelihood of extreme recovery outcomes that tend to be realized more
frequently than observations near the distributional mean.
JEL Classification: G17, G21, & G28
Keywords: Bankruptcy, Ultimate Recovery, Loss Given Default, Credit
Risk, Mixtures of Distributions, Defaulted Debt
1. Introduction
The economic value of debt in the event of default is a key determinant
of the default risk premium required by a lender and the regulatory capital
Preprint submitted to Journal of Banking and Finance October 29, 2013
charged to limit exposure to losses. The pricing of default risk insurance
(CDS contracts) and the emergence of distressed debt as an investment class
add further incentive to better understand the distribution of payoffs in the
event of default.1 Adding to market-driven incentives, Basel II and III pro-
vide regulatory incentives to the development of recovery models in financial
institutions adopting an advanced internal ratings based (IRB) approach to
computing capital requirements.
Recognizing the importance of capturing the behavior of recoveries in
the event of default to quantitative models of credit risk, recent years have
seen a wave of research from academics and industry professionals seeking to
document the key empirical features of observed recovery outcomes. While
payoffs to debt holders in the event of default depend on the interplay of
many factors, often idiosyncratic, notable empirical regularities from prior
research are evident.
1. Recovery distributions tend to be bimodal, with recoveries either very
high or low, implying as Schuermann (2004)2 observes, that the concept
of average recovery is potentially very misleading.
2. Collateralization and degree of subordination are the key determinants
of recovery on defaulted debt. The value of claimants subordinate to
the debt at a given seniority, known as the Debt Cushion, also seems
1Altman and Kuehne (2013) estimate the face and market values of distressed anddefaulted debt in the U.S. over time. At the end of 2012 their estimates of the size ofthe market exceeded $1.15 trillion face value and $678 billion market value with over 200institutions investing in such securities.
2Schuermann’s work provides an excellent review of the empirical features of recoverieswhile Altman et al. (2005) combine a theoretical review as well as important aggregate-level empirical findings.
2
to matter. The analysis of Keisman and Van de Castle (1999) suggests
that all else equal, the larger the Debt Cushion, the higher the expected
recovery outcome.
3. Recoveries tend to be lower in recessions and other periods when the
rate of aggregate defaults is high. Altman et al. (2005) demonstrate
an association between default rates and the mean rate of recovery
whereby up to 63% of the variation in average annual recovery can be
explained by the coincident annual default rate. Further, Frye (2000)
shows that a 10% realized default rate results in a 25% reduction in
recoveries relative to its normal year average.
4. Industry matters. Acharya et al. (2007) suggest that macroeconomic
conditions do not appear to be significant determinants of individual
bond recoveries after accounting for industry effects. More recently,
Jankowitsch et al. (2012) find that the type of default, seniority of
the bond and industry are as important determinants of recovery as
balance sheet ratios motivated by structural credit risk models, macro
variables and transaction cost variables.
5. Variability of recoveries is high, even intra-creditor-class variability,
after categorization into sub-groups. For example, Schuermann (2004)
notes that senior secured bond investments have a flat distribution –
indicating that recoveries are relatively evenly distributed from 30% to
80%.
Clearly, the empirical features of historical recoveries suggest the need for
caution in applying popular (parametric) tools of inference – such as OLS
regressions and calibrated Beta distributions. While OLS regression models
3
provide simple, intuitive summaries of data relationships, they make strong
assumptions about the conditional distribution of recovery outcomes and
focus attention on variation in the mean. Alternatively, Beta distributions
calibrated to historical data are used in many commercial models of portfolio
risk to characterize the distribution of loss outcomes.3 While Beta distribu-
tions offer a simple, parsimonious way of capturing a very broad range of
distributional shapes over the unit interval, Servigny and Renault (2004) ob-
serve that they cannot accommodate bi-modality, or probability masses near
zero and unity – important features of empirical recovery distributions.
While stylized models and a growing body of empirical evidence reveal
much about the important influences on debt recovery outcomes, they also
serve to highlight the challenges inherent in building a quantitative model
to account for: characteristics specific to the defaulted instrument, borrower
characteristics, macroeconomic conditions at the time of default, and the
idiosyncrasies of recovery distributions’ shape. Building on insights from
empirical research and the findings of recent studies documenting the relative
merits of non-parametric and regression based approaches, we present in
this paper a novel approach to modeling recoveries on defaulted debt using
mixtures of Gaussian distributions.
More specifically, our paper makes three contributions to the literature.
First, we present an approach to modeling recovery distributions that retains
the flexibility of non-parametric methods while providing transparency with
3Portfolio Manager (Moody’s KMV), Portfolio Risk Tracker (Standard and Poor’s) andCreditManager (MSCI Inc.) [formerly CreditMetrics (J.P. Morgan)] are all based on theassumption that losses in the event of default are described by a Beta distribution.
4
respect to the economic sources of variation in recovery outcomes. Second,
we estimate and evaluate the out-of-sample performance of our model using
Moody’s ultimate recovery database spanning a 25 year sample period end-
ing in 2011. As noted by Bastos (2010) and Qi and Zhao (2011), very few
studies to date have evaluated the predictive performance of alternative mod-
eling methodologies. While they present tests of non-parametric approaches
relative to regression-based alternatives, neither of the studies consider semi-
parametric models. Third, our model provides further clarity on the role and
importance of economic influences on recovery outcomes.
The remainder of our study proceeds as follows. We provide in Section
2 an overview of recent approaches to recovery modeling and an overview of
the approach proposed in this paper. In Section 3 we describe the ultimate
recoveries data used in this study and we detail the econometric approach in
Section 4. We report model estimates and comparative performance metrics
in Section 5 and summarize our findings in Section 6.
2. Recovery Modeling Approaches
Recent studies have investigated the forecasting performance of non-
parametric estimation approaches relative to a variety of parametric regres-
sion specifications. Using loss data on defaulted Portuguese bank loans,
Bastos (2010) finds that non-parametric regression trees tend to outperform
parametric regression-based forecasts over shorter (annual) horizons. Simi-
larly, using a larger US sample of defaulted loans and bonds, Qi and Zhao
(2011) find that forecasts based on regression trees and neural networks out-
perform those of parametric regression models. Importantly, they attribute
5
the success of non-parametric models to their ability to accommodate non-
linear associations between debt recoveries and continuous conditioning vari-
ables. Similarly, recent work by Loterman et al. (2012) underscores the
importance of models that incorporate non-linearities in predictive relations.
In demonstrating the predictive properties of non-parametric techniques
relative to regression models the studies by Bastos (2010) and Qi and Zhao
(2011) also serve to highlight the potential shortcomings of the approaches.
Qi and Zhao (2011) acknowledge a basic criticism of neural networks, namely,
that they do not provide any insight to the economic relationships underpin-
ning the forecasts. While regression trees are more transparent and intuitive
they can become unwieldy in size and incorporate relationships that are dif-
ficult to reconcile with a-priori expectations.
In modeling Moody’s data on ultimate recoveries between 1985 and 2008,
Qi and Zhao (2011) build a tree with 342 splits. While the regression trees
built using the much smaller dataset employed by Bastos (2010) contain be-
tween 1 and 3 splits only, they suggest a primary role for loan size as a
driver of expected recovery outcome – a strong finding that appears specific
to the data used in the study. More recently, Bastos (2013) suggests that en-
sembles of regression trees, obtained through varying the estimation sample,
outperform trees estimated using a single historical data set.
Given the empirical properties of recoveries and the relative merits of
regression-based and non-parametric modeling techniques, we present in this
paper a simple semi-parametric approach based on mixtures of distributions.
Our approach is flexible enough to capture the distinctive features of recov-
ery distributions while providing insight to the economic relationships from
6
which predictions are derived. Instead of trying to force-fit a parametric
distribution, we adopt a Bayesian perspective and model the distribution of
recoveries using mixtures of Gaussian distributions.4 By taking the appro-
priate probability weighted average of Gaussian components, we are able to
accommodate the unusual defining features of such distributions. By explic-
itly modeling the assignment of recovery outcomes mixture components using
an ordered probit regression we accommodate non-linearities in the relation
between continuous conditioning variables and recovery outcomes suggested
in earlier work.5
Similar to Hu and Perraudin (2002), we commence by transforming ulti-
mate recoveries r from the unit interval to the real line such that
y = Φ−1(r), (1)
where Φ denotes the standard Normal CDF and y is the transformed
recovery. To compute the inverse of the Normal CDF we make a small
adjustment to values of r in cases where r = 0 or r ≥ 1. If r ≥ 1 then the
observation is replaced with a value of 1− ǫ, and if r = 0 we replace it with
a value of ǫ. We set the adjustment parameter ǫ = 10−9.
The second step of our approach rests on the assumption that (trans-
4Recent work by Hagmann et al. (2005), Hlawatsch and Ostrowski (2011) and Zhangand Thomas (2012) present alternative semi-parametric approaches to modeling recoverieson defaulted debt. We discuss the benefits of our approach to these alternatives in Section4.3.
5For example, regressions reported in Altman et al. (2005) suggest a non-linear relationbetween aggregate recoveries and the contemporaneous default rate.
7
formed) recovery outcomes y can be thought of as draws from a distribution
g(y) of unknown functional form. While the form of g(y) is not known, we
set out to approximate it using a weighted combination of standard densities
f(y|θj) such that:
g(y) =m∑
j=1
pjf(y|θj) (2)
where p1 + . . .+ pm = 1, and the standard densities f(y|θ1), . . . , f(y|θm)
form the functional basis for approximating g(y). In our application, the
m densities f(y|θj) are chosen to be Gaussian with parameters θj . Robert
(1996) observes that such mixtures can model quite exotic distributions with
few parameters and with a high degree of accuracy.
The shape of the target distribution g(y) depends on the number of com-
ponent distributions, the parameters of the components and the probability
of drawing from each. The interdependence of the mixture parameters neces-
sitates their simultaneous estimation. Fortunately, there are well established
techniques to solve such problems in both Bayesian and maximum likelihood
frameworks. Taking a Bayesian perspective, we use the Markov Chain Monte
Carlo (MCMC) technique of Gibbs Sampling.6 As will be described later,
our choices regarding the number of mixture components are guided by in-
formation criteria, and the economic properties of the resultant estimates. In
Section 4 we provide a full description of the basic model and its elaboration
to account for the effects of conditioning information – that is, known or
6Casella and George (1992) provide a description of the algorithm .
8
hypothesized determinants of recovery outcomes.
3. Data Description
We use discounted ultimate recoveries from Moody’s Ultimate Recovery
Database provided by Moody’s of New York. Moody’s database provides
several measures of the value received by creditors at the resolution of default
– usually upon emergence from Chapter 11 proceedings. Moody’s estimate
of the discounted value of ultimate recovery is our choice of the measure of
the economic value accruing to a creditor at the time of default. Moody’s
calculates discounted ultimate recoveries by discounting nominal recoveries
back to the last time interest was paid using the instrument’s pre-petition
coupon rate. The database includes US non-financial corporations with over
$50m debt at the time of default. The sample period covers obligor defaults
from April 1987 to late 2011, covering 4,720 debt instruments, of which 60%
are bonds.
Table C.1
Table C.1 summarizes key features of the ultimate recoveries data with
reference to selected facility characteristics and industry classifications.7 The
data is roughly evenly divided between observations with and without col-
lateral and 40% of facilities are in default for less than a year, and less than
20% for more than two. Broadly speaking, seniority, collateralization and
industry classification are reflected in mean and median recovery rates as
7Refer to Keisman et al. (2011) for a description of how recoveries in the databasebehave at an aggregate level.
9
one may expect. However, as observed by Schuermann (2004) and in other
prior work, characteristic-based subsets continue to exhibit high degrees of
variability: for example, the standard deviation of recoveries within most
industry and seniority-based subsets are in the 34-42% range.
Figures C.1 & C.2
Figure C.1 summarizes the distribution of ultimate recoveries on loans and
bonds pre and post transformation with the inverse Gaussian CDF (1). The
transformation yields in each case a distribution with three distinct modes
– albeit with far fewer loan observations in the lower extreme (as would be
expected). Figure C.2 provides a sampling of recovery histograms for various
sub-categories of exposures that illustrate two general features of the data.
First, the bi-modality of the recovery distribution varies according to sub-
group – the contrast between high and low debt-cushion exposures being
the most dramatic example.8 Second, degrees of multi-modality (such as the
case of senior secured bonds) and a high degree of uncertainty between modes
are also observable in sub-groups. These observations are broadly consistent
with those of Schuermann (2004).
4. Econometric Framework: Some Elaboration
As noted, we commence by assuming a Gaussian form for the approximat-
ing densities f(.) in (2) modeling the data y using a probability pj weighted
mixture of m Gaussian likelihoods:
8The (proportional) value of claimants subordinate to the debt at a given seniority,known as the ‘Debt Cushion’.
10
φ(y|α, σ, p) =1
(2π)N
2
N∏
i=1
{
m∑
j=1
pjσj
exp
[
−1
2σ2j
(yi − αj)2
]
}
(3)
where αj is the mean of mixture component j and its standard deviation
σj .9 The sample size is N .
Confronted with the likelihood (3), one can follow the specification in
Koop (2003) and adopt proper, but minimally informative, conjugate priors
on the parameters α, σ and p and estimate the joint posterior of all param-
eters using the MCMC technique of Gibbs sampling. However, in order to
accomplish this two problems must be addressed.
First, there are no directly observable data to estimate the probability
weights pj. Second, there exists an identification problem in that multiple
sets of parameter values are consistent with the same likelihood function.10
Fortunately, there are established solutions to both problems. The identifi-
cation problem is circumvented by way of a labeling restriction. We follow
Koop (2003) in imposing the restriction that αj−1 < αj for j = 2 . . .m.
While there is nothing special about this particular restriction (in the sense
that restrictions on other parameters can equivalently solve the identification
problem), it facilitates interpretation of the Gibbs output.
The solution to the problem of not observing data with which to estimate
pj involves a well-established technique called data augmentation. If one were
to observe an indicator variable eij taking on a value of 1 when observation
9For the sake of clarity we suppress wherever possible time and facility/firm subscripts.Unless stated otherwise, all analysis is on data pooled in time series and cross section.
10Refer to page 255 of Koop (2003) for elaboration and an example.
11
i is an outcome drawn from mixture component j, and zero otherwise, then
the likelihood (3) could be written as:
φ(y|α, σ, e) =1
(2π)N
2
N∏
i=1
{
m∑
j=1
eijσj
exp
[
−1
2σ2j
(yi − αj)2
]
}
(4)
and estimation would follow easily.11 However, since we do not observe
indicator flags associating observations with mixture components, we rely on
the decomposition described in Robert (1996) to generate them as part of the
sampling scheme.12 Specifically, the latent data is generated based on draws
from a Multinomial distribution. Conditional on the data and parameters
of the mixture components (αj , σj), the latent data draw associated with
each observation is an m−vector of indicator variables wherein one of the
indicators is non-zero. In particular, a value of 1 in position j associates the
observation with mixture component j. The probability of an observation
being so assigned to mixture component j on any particular draw of the
sampling scheme depends on the relative likelihood of it being observed as
an outcome of the particular mixture component.
More generally, we use the Markov Chain Monte Carlo technique of
Gibbs sampling to generate draws from the joint posterior distribution of
mixture parameters α, σ, e, p|y using the marginal posterior of each param-
eter, conditional on all other parameters. In the current case, we would
need to cycle through the following steps many times: 1. Draw α|σ, p, e, y;
11Robert (1996) notes that this re-expression is possible when the likelihood is from anexponential family.
12Refer to equation 24.7 in Robert (1996).
12
2. Draw σ|α, p, e, y; 3. Draw p|α, σ, e, y; 4. Draw e|α, σ, p, y 5. Back
to step 1 – conditioning each new parameter draw on the most recently
drawn values of the other parameters. Having cycled through the steps
many times, and having discarded an initial set of ‘burn-in’ draws, the G
remaining draws are treated as outcomes from α, σ, e, p|y, the joint posterior
distribution of model parameters. Draws from α, σ, e, p|y enable inference
and prediction from a Bayesian perspective. For example, the computation
p(ypred|y) ≈ 1G
∑G
g=1 p(yp|α[g], σ[g], p[g], e[g], y) yields a numerical estimate of
the predictive likelihood p(ypred|y).
4.1. Inferring the Effects of Conditioning Information
While the simple specification (4) does not explicate the influence of re-
covery determinants, the outputs of Gibbs sampling can be used to infer
the effects of conditioning information on the probability of realizing recov-
ery outcomes from each of the component distributions. Recall from the
likelihood in equation (4) that each observation i is associated with a mix-
ture component j by way of the indicator variable eij . Each iteration of the
Gibbs sampler involves drawing from the conditional posterior of the indica-
tor variables eij for each i = 1 . . .N exposure, thus providing the information
required to compute the probability (mixing) weights associating particular
portfolios of exposures with each mixture component. Suppose (for example)
that we are interested in modeling this distribution of recoveries on subor-
dinated debt. Further, suppose that the debt exposures i ∈ Q denote the
sub portfolio of interest – recoveries on subordinated debt. Then, pQj, the
mixing weight for portfolio Q associated with component j, can be estimated
from the Gibbs output using
13
pQj =
G∑
g=1
e[g]Qj
n(Q)G(5)
where eQj denotes all eij such that i ∈ Q, G is the total number of post
burn-in iterates from the Gibbs sampler, and n(Q) is the number of obser-
vations in Q. Using equation (5) we can compute the mixing probabilities
for a portfolio of subordinated debt exposures as the proportion of non-zero
indicator variables sampled for each component j.
Inferring the effects of conditioning information on mixture weights using
equation (5) is appealing insofar as it does not involve strong assumptions
about the form of the relationship between conditioning variables and re-
covery outcomes. However, the approach is best suited to applications with
relatively few categorical conditioning variables. To overcome the latter lim-
itation we augment the model with a latent variable regression to account
explicitly for the dependence of mixture weights on predictive information.
4.2. Parametric Conditioning
If one has reason to believe that the component means αj in equation
(4) are linearly related to conditioning or predictive information, the model
is easily extended to take the form of a mixture regression model. As in
the current case however, there may be reason to consider such linearity
assumptions overly restrictive, or at the very least there are little a-priori
grounds for such restrictions. We thus take a different approach and allow
for the dependence of mixture assignment probabilities on determinants of
14
recovery outcomes.13
Taking a 3-component mixture as an illustrative example, we re-specify
the likelihood 4 as follows:
p(yt|xt−1, θ, z∗t ) = φ(yt;α1, σ
21)
I(c0<z∗t≤c1)φ(yt;α2, σ
22)
I(c1<z∗t≤c2)
φ(yt;α3, σ23)
I(c2<z∗t≤c3) (6)
where I is an indicator variable taking on a value of 1 or 0, depending on
the value of the latent variable z∗ relative to the cut-points c1 . . . c3. The cut-
points c0 = −∞, c1 = 0, and c3 = +∞ are set to enable unique identification,
and as before, α1 < α2 < α3. The notation θ is shorthand for the set of
likelihood parameters.
The latent variable z∗ is linear in the conditioning information xt−1, ob-
servable prior to the realization of yt:
z∗t = β0 + β1xt−1 + ǫt, ǫtiid∼ N [0, 1] (7)
Together, equations (6) and (7) constitute a conditional mixture of nor-
mals wherein the mixture assignments depend on an ordered probit model.
We provide in Appendix A the specific form of the priors and associated
conditional posteriors required to implement a Gibbs sampling scheme in-
corporating steps to draw the β regression coefficients, the latent variable z∗
13Of course, such dependence does not preclude the former (a possibility we are consid-ering in current work).
15
and the unrestricted cut point(s).
4.3. Comparison to Semi-Parametric Alternatives
As noted earlier, recent work by Hagmann et al. (2005), Hlawatsch and
Ostrowski (2011) and Zhang and Thomas (2012) also suggest semi-parametric
approaches to modeling distributions of recoveries on defaulted debt.
Hagmann et al. (2005) model the density of recoveries using a Beta dis-
tribution and scaling discrete recovery outcomes by a non-parametrically
estimated ‘correction factor’. The correction factor is based on the kernel
density estimate of the distribution of recoveries transformed using a Beta
cumulative distribution function. While they demonstrate that the approach
affords far greater flexibility in capturing the features of industry-level re-
coveries, the authors acknowledge that the approach is limited insofar as the
semi-parametric density estimator can only take values of 0 or +∞ at the
boundaries – a property inherited from the original (starting) beta density.
This limitation is of significance when, as is often the case, substantial con-
centrations are observed at the distributional boundaries of zero recovery and
full recovery. By estimating mixture models of recoveries transformed by an
inverse Gaussian CDF, we avoid the inherent limitations of Beta distributions
in accommodating probability masses at the distributional boundaries.
More recent work by Hlawatsch and Ostrowski (2011) and Zhang and
Thomas (2012) consider mixture based approaches. The former models simu-
lated defaulted debt losses with mixtures of Beta distributions in a maximum
likelihood framework, while the latter models consumer credit losses using
regression analysis on discrete subgroups, partitioned using non-parametric
classification trees. The diversity of these recent approaches notwithstand-
16
ing, they share a basic limitation insofar as each relies on classification of the
data into distinct subgroups based on characteristics of the exposures and are
thus limited by the curse of dimensionality. Our current approach circum-
vents this limitation by explicitly modeling mixture component assignments
in an ordered probit regression – thus enabling the use of many discrete and
continuous conditioning variables. Given the number and nature of variables
thought to affect recovery outcomes, this is an important advantage if one
wants to simultaneously use all available conditioning information, and, to
understand the marginal effects of included variables.
5. Results
Given that a primary objective of our empirical analysis is to estimate
and evaluate the predictive performance of our proposed conditional mixture
model, we note from the outset our split of the overall sample into 2,307
estimation observations comprising the sample up to and including 2001,
and 2,413 test observations cover the period beginning 2002 to the end of
2011. As such, our results are based on true out of sample data. Unless
stated otherwise, our discussion of the model focuses on estimates derived
using data from the pre-2002 estimation period – the estimates used for out-
of-sample forecast evaluation. However, we also report full sample estimates
for purposes of comparison and to illustrate the robustness of the parameter
estimates.
Another important feature of our modeling is the comprehensive set of
conditioning information included in each forecast. We select our set of con-
ditioning variables to capture what appear to be important influences on re-
17
covery outcomes in light of the empirical literature, and, to ensure broad con-
sistency with related studies of forecasting models. Each conditional model
includes indicator variables to capture: instrument type and seniority, the
rank of the claim, whether or not the debt is collateralized, and an indi-
cator or whether the defaulting borrower belongs to the ‘Utilities’ industry
classification.14 Also included are continuous variables providing an alter-
native measure of security subordination and a summary measure of credit
conditions at industry level: Debt Cushion and an industry-level Default
Likelihood Index (DLI) based on the Merton (1974) framework respectively.
Debt Cushion, as suggested by Keisman and Van de Castle (1999), is
a facility-level metric that captures not only the rank of debt in capital
structure, but the degree of its subordination as a proportion of total claims.
They present evidence to show that Debt Cushion categories are associated
with extreme recoveries. Our use of industry-level default expectations is
motivated by a mixture of theory and empirical evidence. The theoretical
work of Frye (2000) implies a negative association between the probability
of default and recovery outcomes. The empirical estimates of Altman et al.
(2005) are consistent with Frye’s theory to the extent that realizations of
default are used in place of expectations. Industry DLI is thus intended
as a composite measure of expectations incorporating macroeconomic and
industry-specific effects.
Specifically, to summarize industry level expectations of credit conditions
we use estimates of default likelihood derived from equity markets in the
Merton (1974) framework. Altman et al. (2011) demonstrate empirically the
14The separation of Utilities follows Altman and Kishore (1996).
18
predictive value of industry level default-likelihood indices – consistent with
the notion that such market-based measures reflect a broad range of infor-
mation about credit conditions. As described in Appendix B, we aggregate
firm-level measures of default likelihood estimated using the approximation
derived by Bharath and Shumway (2008).
5.1. Mixture Model Estimates
To determine the appropriate number of mixture components we used
a common set of conditioning variables to estimate 2, 3, 4 and 5 compo-
nent specifications using all pre-2002 (estimation sample) observations. We
computed three model selection metrics: the Akaike information criterion
(AIC), the Bayesian information criterion (BIC) and the Hannan-Quinn in-
formation criterion (HQC). When evaluated at the posterior mean of the
parameter draws, all three model selection criteria overwhelmingly favor a
four-mixture conditional specification. Accordingly, Table C.2 summarizes
the properties of the ordered mixture components: the recoveries implied by
the component means, the within-component variability of recoveries and the
posterior probability weights.
Table C.2
Several aspects of the results in Table C.2 are noteworthy. To begin
with, the extreme mixture components are, in effect, degenerate. Outcomes
drawn from the first or fourth mixture components exhibit very low variabil-
ity – implying no variation from either zero or full recovery (respectively).
The second and third mixture components on the other hand imply differing
19
forms of uncertainty with respect to recovery outcomes. Table C.2 docu-
ments the impact of standard deviation variations in the Gaussian mixture
components comprising the distribution of transformed recoveries yt, and
it is immediately apparent that the impact of deviations from the mean is
asymmetric. More generally, the asymmetric and variable response of re-
covery outcomes to deviations from within component means, reflecting the
non-linearity of the inverse-Gaussian data transformation, is documented in
Figure C.3. Clearly the third mixture component implies a set of recovery
outcomes much preferable, from the investor’s perspective, to that of the
second.
Finally, we note that the findings in Table C.2 are robust in the sense that
the modal values and variability of parameters obtained from the estimation
period appear very close to the corresponding estimates based on the full
sample.
Figure C.3
The probability of realizing recovery outcomes from each of the mixture
components varies according to characteristics of the exposure, the borrower
and industry conditions at the time of default, as captured by the ordered
probit regression model (7). Table C.3 presents the posterior mean and
variability of the regression model using conditioning variables to capture
the security type, the degree of subordination, collateralization and indus-
try distress conditions at the time of default. Like the parameters of the
mixture components, the marginal posterior distributions of the regression
parameters appear robust to the period of estimation insofar as the full and
20
estimation period based measures of parameter mean and variability appear
closely matched. The latter observation is, however, subjective without a
clearer understanding of the economic impact of variations in the parame-
ters of interest.
Table C.3
To gauge the economic effects implied by the ordered probit regression
the sign and magnitude of the coefficients must be interpreted with reference
to the values of the cut-points c1, c2 and c3, and in noting that the base
case underlying the estimates in Table C.3 is a high quality exposure: a term
loan. While it is possible to compare the direction of relationships and the
associated uncertainty reflected in the posterior variance of the parameters
to a-priori expectations, it is easiest to get a sense of the marginal effects of
variation in conditioning variables through studying their impact on predic-
tive outcomes. Table C.4 summarizes the results of such a comparison by
documenting the impact of industry distress conditions on two hypothetical
exposures: a senior secured bond with a high Debt Cushion, and uncollat-
eralized junior debt that ranks lower than third in the order of claimant
precedence. The example exposures are denoted as High Quality (HQ) and
Low Quality (LQ) respectively. ‘Normal’ conditions denote a period where
industry default expectations, as measured by Industry DLI, is at its median
value from a historical perspective.
Table C.4
Table C.4 summarizes the properties of the predictive densities associated
with the example exposure classes generated at the posterior mean of the
21
latent variable regression parameters. That is, the posterior mean assignment
probabilities reported in the bottom panel of Table C.4 translate into the
distributions approximated by the kernel density estimates plotted in Figure
C.4. High quality exposures under normal conditions (the solid line plot)
are projected to exhibit a skewed, essentially unimodal recovery distribution
with a large probability mass at full recovery.15 A contrasting extreme is
afforded by the multimodal distribution of low quality debt recoveries under
the same ‘normal’ conditions (dotted line plot). The conditional mixture
assignment probabilities reported in the first and third columns of Table C.4
account for these effects.
Next, for purposes of comparison, we define ‘Distress’ to mean a situa-
tion where the Industry DLI takes on a value equal to the 90th percentile
of its historical distribution. So defined, Figure 4 illustrates the effects on
the predictive distributions associated with high and low quality exposures
associated with the parameters in Table C.4.
Recoveries on high quality exposures are only affected to the extent that
the posterior probability weight shifts from the fourth to the third mixture
component – consistent with a slightly diminished possibility of a full re-
covery. The effect of the same industry distress on low quality exposures is
marked by a shift of the probability weight on the fourth mixture component
to the first and second. Accordingly, as per the dashed plot with dots in
15The projected recovery distribution obtained from the mixture model is restricted tothe unit interval, however, the kernel density plots in Figure 4 are not restricted to theunit interval simply to enable an easy comparison of the four cases under considerationin a single plot. The unrestricted density estimates provide a neat visual summary of theimportant dynamics.
22
Figure C.4, the probability mass in the lower (upper) mode of the recovery
distribution increases (decreases) markedly. At the same time, the distribu-
tion associated with the high quality exposure (solid line with dots) is far
less sensitive to industry distress.
To summarize: while in both cases the mean recovery is affected by indus-
try distress, the nature of the effect on distributional shape is quite different
in each case. The means reveal only a very small part of the story. For exam-
ple, in the event of industry-level distress, the relative likelihood of recoveries
realised by investors from the extreme mixture components shifts markedly
and in distinctive ways across seniorities and industry groupings.
Figure C.4
Before finishing with our example, it is worth noting that the predicted
dynamics are consistent with a hypothesis suggested by Carey and Gordy
(2004), namely, that the distribution of losses given default (LGD) shifts to
the right in good years relative to bad years. They also suggest that a higher
proportion of bad LGD firms may file for bankruptcy in high default years
while less-than-bad LGDs may not be significantly affected.
The differential responses of high and low quality debt recoveries to in-
dustry distress conditions at the time of default also serve to illustrate the
challenge of generalizing the marginal effects of conditioning variables on the
distribution of recovery outcomes. Continuous conditioning variables affect
the overall shape of the recovery distribution in accordance with the char-
acteristics of the defaulted exposures. However, to provide a more general
characterization of marginal effects, Figures C.5 and C.6 illustrate how in-
dustry DLI and Debt Cushion interact to affect the median and lower tail
23
of ultimate recovery distributions associated with senior secured bonds and
senior unsecured bonds respectively.
Figures C.5 & C.6
Comparing first Panel (a) of Figures C.5 and C.6, the impact of Debt
Cushion on median recoveries on both senior secured and senior unsecured
exposures is shown to be non-linear and of great importance beyond a par-
ticular cut-off point – beyond which median recoveries decline quite precipi-
tously with diminishing Debt Cushion. All else equal, the higher the industry
default likelihood at the time of default, the greater the sensitivity of me-
dian recoveries to the level of Debt Cushion. All else equal, the unsecured
exposure is more sensitive to Debt Cushion.
Comparing Panel (b) of Figure C.5 to that of Figure C.6 shows that the
10th percentile of the recovery distributions on senior secured and senior
unsecured bonds respond non-linearly to variations in Debt Cushion, again,
changing in accordance with the level of industry default likelihood. While
these findings accord with economic intuition in general terms, they serve to
show that the marginal effects of conditioning variables on the quantiles of
recovery distributions must be considered and quantified on a case by case
basis.
5.2. Predictive Performance
To gauge the benefit of the conditional mixture specification in forecast-
ing ultimate recoveries, we conduct an out-of-sample, out-of-time simulation
experiment employing two popular parametric models of recovery, as well
as a non-parametric regression tree. We consider an Inverse Gaussian (IG)
24
regression, wherein the dependent variable is transformed according to equa-
tion (1). As noted earlier, Hu and Perraudin (2002) model recoveries by way
this transformation, and Qi and Zhao (2011) use such a model to benchmark
non-parametric approaches.
Second, we also use an IG regression with a Beta transformation – a fea-
ture of Moody’s Loss Calc 2.0 developed by Gupton and Stein (2005). The
second (IG-B) regression approach involves fitting a Beta distribution to the
recovery data and computing the cumulative probabilities of the recoveries
under the fitted Beta distribution prior to the inverse Gaussian transfor-
mation. In effect, the IG-B regression approach models the dependence of
cumulative probabilities of recoveries on conditioning information under the
assumption that recoveries are Beta distributed.
Third, we estimate a non-parametric regression tree (Reg Tree) – a data
driven technique in which conditioning information is used to partition ob-
served recovery outcomes into sub-groups exhibiting minimal within group
variation. Estimation yields a hierarchical classification table in which the
predicted recovery assigned to an exposure is set equal to the mean recovery
of the sub-group to which it is assigned based on characteristics of the bor-
rower, the exposure, and economic conditions at the time of default. Bastos
(2010) introduces and exemplifies the technique in modeling losses on bank
loan exposures.
Tables C.5 & C.6
We summarize estimates of the benchmark regressions in Tables C.5 and
C.6. The regression coefficients are consistent with expectations in terms of
25
coefficients’ sign and significance. The explanatory power of the regressions
(adjusted R2 values just under 50%) is also consistent with similar models
estimated by Acharya et al. (2007), Qi and Zhao (2011) and others.
Using the conditioning information common to the regression and mix-
ture models, we estimate a regression tree using Matlab’s RegressionTree.fit
function. At its default settings, the estimation algorithm yields a tree with
701 nodes based on the pre-2002 sample. Adjusting the minimum number
of observations from the default value of 1 to 25 observations per leaf yields
a tree with 127 nodes – less complex than the tree obtained by Qi and Zhao
(2011) over a slightly longer estimation interval using a somewhat different
set of conditioning information. All reported predictive results are based on
a tree estimated with a minimum leaf size of 25.
We document the predictive performance of the models by way of a styl-
ized application to modeling the distribution of recoveries associated with
randomly drawn portfolios of defaulted exposures according to the following
resampling procedure:
1. All observations from 2002-2011 are included in the test pool.
2. We draw a random sample of 100 recoveries on defaulted loans and
bonds from the test pool and compute the ultimate recovery on an
equally-weighted portfolio of the selected exposures. This value is
stored as an outcome of the empirical loss distribution. Each expo-
sure has a $1.00 face value.
3. We compute the characteristic based forecast of the recovery outcome
for each exposure based on the IG and IG-B regressions and the re-
gression tree. We then aggregate each set of forecasts and store the
26
portfolio-level loss based on each model. In the case of the conditional
mixture, we evaluate the mixing probabilities at the posterior mean of
the mixture assignment regression parameters and then draw an out-
come from the mixture in accordance with the fitted probability point
estimates.16 The recovery outcomes drawn from the mixture are also
aggregated to portfolio level and stored.
4. Steps 2-3 are repeated 50,000 times.
The exposures comprising the randomized test portfolios are independent
of the observations used for estimation, that is, there is no overlap between
the estimation and test samples in terms of the identity of the obligors or the
time of observation. As such, this procedure constitutes an out-of-sample,
out-of-time test of the models.
Table C.7
The results of the resampling experiment, presented in Table C.7, show
that the mixture-based calculations out-perform the parametric and non-
parametric benchmarks on out-of-sample basis. Of the two regression based
benchmarks, the IG specification seems to work best on an out-of-sample
basis – outperforming the IG-B regression throughout the range of realized
portfolio-level recovery outcomes. Both regression benchmarks are however
out-performed the more flexible models - the semi-parametric mixture model
and the non-parametric regression tree.
16We also fix the mixture distribution parameters at their posterior mean values. Esti-mation risk is simply accounted for by drawing parameter values from the Gibbs sampleroutputs.
27
With the exception of the extreme left tail of the portfolio recover distri-
bution, the mixture model forecasts out-of-sample portfolio recoveries more
precisely than the regression tree. From the 10th through to the 90th per-
centile of the portfolio recovery distribution the mixture model’s predictive
errors are approximately one half to one third the magnitude of those from
the regression tree in proportional terms. Overall, the Root Mean Squared
Error (RMSE) and the Mean Absolute Error (MAE) derived from the mix-
ture model are approximately 9% lower than those from the regression tree.
As noted earlier, the predictive results for the regression tree are based
on a minimum leaf size of 25. In evaluating the performance of the regression
tree we varied the minimum leaf size from 1 (the Matlab default setting) to
5, 10, 25, 50, 75 and 100. Consistent with Qi and Zhao (2011), we found
that the predictive performance of the regression tree is quite sensitive to the
specification of minimum leaf size in the current context. Of the specifications
we tested, those with minimum leaf sizes larger or smaller than 25 yield
larger forecast errors. – comparable to or larger than those obtained from
parametric regression models. For example, estimates of the regression tree
with a minimum leaf size of 5, the size found to optimal by Qi and Zhao
(2011), yield forecasts that tend to underestimate portfolio recoveries with
larger proportional forecast errors than those of the parametric regression
models. In light of these findings, the Reg Tree forecast results in Table
C.7 represent an optimistic indication of the true out of sample performance
obtainable using regression trees in real time.17
17However, recent work by Bastos (2013) suggests that the out of sample performance ofregression trees may be improved through the use of a bootstrap aggregation (“Bagging”)
28
Consistent with recent studies of non-parametric approaches to modeling
defaulted loan and bond recoveries, our out-of-sample performance evalua-
tion underscores the benefit of accommodating non-linearities in the relation
between recovery outcomes and conditioning information. Our results fur-
ther suggest that mixture models offer superior forecast accuracy without
reliance on out-of-sample data dependent ‘tuning’ of modeling choices.
6. Conclusion
We present in this paper a new approach to modeling the distribution
of recoveries on defaulted loans and bonds using mixtures of distributions.
We take a Bayesian perspective and model (transformed) ultimate recoveries
using a mixture of Gaussian distributions wherein the mixing probabilities
are explicitly conditioned on borrower characteristics, debt features and the
economic conditions prevailing at the time of default.
Our empirical findings suggest that our formulation delivers predictive
recovery distributions that adapt to the conditioning variables in ways that
are consistent with expectations based on prior empirical studies, and, that
our methodology outperforms parametric regression-based alternatives used
in empirical research and industry models of recovery (or LGD), as well as a
non-parametric (regression tree) benchmark. While recent empirical studies
have advocated the benefits of non-parametric approaches, a key benefit
of our approach lies in its flexibility and transparency with respect to the
economic sources of variation in predictive outcomes.
technique.
29
Our current methodology is readily adaptable to existing models of de-
fault, and readily extensible in its use of conditioning information. Recent
work by Qi and Zhao (2011) suggests that the source of non-parametric mod-
els performance advantage over the parametric regression based approaches
lies in solely in their ability to accommodate non-linear relationships between
recoveries and certain continuous conditioning variables. The latter obser-
vation suggests that the performance of our current mixture specification
may benefit from the added flexibility of allowing for dependence between
the mixture component means and borrower or facility level categorical vari-
ables.
30
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33
Appendix A. Model and Notation
We assume that the density of transformed recoveries yt takes the follow-
ing form:
p(yt|xt−1, θ, z∗t ) = φ(yt;α1, σ
21)
I(c0<z∗t≤c1)φ(yt;α2, σ
22)
I(c1<z∗t≤c2) (A.1)
. . . φ(yt;αm, σ2m)
I(cm−1<z∗t≤cm),
z∗t = β0 + β1xt−1 + ǫt, ǫtiid∼ N [0, 1]. (A.2)
Equation (6) is an m−mixture formulation wherein the φ(.) components
of the mixture are normal and the restriction α1 < α2 < . . . < αm implies
that the means are ordered. The parameter θ is shorthand notation for all
parameters of the model, xt−1 denotes conditioning information observable
prior to yt, and z∗t is a latent variable. The notation I(.) denotes an indicator
function that has a value of 1 when the associated condition is true, and zero
otherwise.
The probabilistic assignment of observations to mixture components k =
1, 2, . . . , m depends on the outcome of z∗t relative to a set of cut-points
c0 . . . cm. We set the cut-points c0 = −∞, c1 = 0, and cm = +∞ to enable
unique identification. Specifically, letting yt denote the mixture component
to which an observation at time t is assigned, the following mapping applies:
−∞ < z∗t ≤ 0 : It = 1,
34
0 < z∗t ≤ c2 : It = 2,
......
cm−1 < z∗t ≤ +∞ : It = m. (A.3)
Note that It is the mixture component to which observation yt is assigned.
Given the process (A.2) for z∗t and the cut-points c0 . . . cm, we model the
assignment of observations to mixture components as an ordered probit –
conditional on observing yt, the (discrete) assignment of each observation.18
Generalizing the analysis in Koop et al. (2007), we choose the following
forms of priors for the remainder of the parameters:
α ∼ N [α, Vα]I(α1 < α2 < . . . < αm)
σ2i ∼ IG(ai, bi), i = 1, 2, . . .m
[β0 β1]′ ∼ N [µβ, Vβ]
The resultant posteriors follow.19
α1|θ−α1, y, x ∼ TN(−∞, α2)[Dαk
dαk, Dαk
]
α2|θ−α2, y, x ∼ TN(α1, α3)[Dαk
dαk, Dαk
]
......
18See Koop et al. (2007) for an exposition of a 2-mixture case with fixed cut-points.19The notation αj |θ−αj
is shorthand for ‘αj conditional on all parameters θ apart fromαj ’. In general terms: θ−param is ‘all parameters apart from param’.
35
αm|θ−αm, y, x ∼ TN(αm−1,∞)[Dαk
dαk, Dαk
] (A.4)
with Dαk= (ηk/σ
2k + V −1
αk)−1, dαk
=∑T
t=1 ztkyt/σ2k + V −1
αkµαk
and ηk =∑
t ztk.
z =
z11 z12 zim
z21 z22 z2m...
......
zT1 zT2 zTm
with ztk ≡ I(ck−1 < z∗t ≤ ck) for k = 1, 2, . . . , m.
β0, β1|θ−β0,β1, r ∼ N(Dβdβ, Dβ) (A.5)
with Dβ = (X ′X + V −1β )−1, dβ = X ′z∗ + V −1
β µβ, z∗ = [z∗1 . . . z
∗T ]
′, x =
[x0 . . . xt−1]′, and X = [ι x].
σ2k|θ−σ2
k
, y ∼ IG
[
ηk/2 + ak, [b−1k + 0.5
∑
t
ztk(yt − αk)2]
]
(A.6)
We model the mixture probabilities as an ordered probit, drawing the
latent z∗ as follows,
p(z∗t |θ−z∗t, yt) = TN(−∞,0][β0 + β1xt−1, 1], if It = 1
36
= TN(0,c2][β0 + β1xt−1, 1], if It = 2
...
= TN(cm−1,+∞)[β0 + β1xt−1, 1], if It = m (A.7)
where
Pr(It = 1|θ, y, x) =[1− Φ(β0 + β1xt−1)]φ(y;α1, σ
21)
Π
Pr(It = 2|θ, y, x) =[Φ(β0 + β1xt−1)− Φ(β0 + β1xt−1 − c2)]φ(y;α2, σ
22)
Π
Pr(It = 3|θ, y, x) =[Φ(β0 + β1xt−1 − c2)− Φ(β0 + β1xt−1 − c3)]φ(y;α3, σ
23)
Π...
...
Pr(It = m|θ, y, x) =Φ(β0 + β1xt−1 − cm−1)φ(y;αm, σ
2m)
Π(A.8)
such that Φ(.) denotes the CDF of a standard normal and Π = [1−Φ(β0+
β1xt−1)]φ(y;α1, σ21)+[Φ(β0+β1xt−1)−Φ(β0+β1xt−1−c2)]φ(y;α2, σ
22)+ . . .+
Φ(β0 + β1xt−1 − cm−1)φ(y;αm, σ2m).
Following the Koop et al. (2007) analysis of the ordered probit model, a
flat prior on the cut-point ck for 1 < k < m implies:
ck|r, θ−ck ∼ U [l, u], (A.9)
with
37
l = max {ck−1,max z∗t |It = k}
u = min {ck+1,min z∗t |It = k + 1}
Some notes on computational matters:
1. In the absence of non-sample information prior parameters are set to
be uninformative. Current computations reflect Vαk= 1, 000 ∀k and Vβ
is a diagonal matrix with entries of 1,000 on the main diagonal. Prior
means are set to zero.
2. Unless stated otherwise, all reported in this study are based on 50,000
iterations of the Gibbs sampler after 5,000 initializing (burn-in) iter-
ations were discarded. Estimation of the full model over the entire
sample in Matlab takes approximately 45 minutes on a 2011 Macbook
laptop.
3. While model parameters converge rapidly to the target distribution
with the possible exception of the cut-points. As noted in Koop et al.
(2007), cut-points in models of this form may exhibit slow convergence.
Judicious choice of starting values helps a lot so a little experimentation
can go long way.
4. Qi and Zhao (2011) suggest that the choice of adjustment parameter ǫ,
applied prior to the inverse Gaussian transformation, may materially
affect model performance. Our work to date does not suggest this to
be case in the context of the conditional mixture model.
38
Appendix B. Default Likelihood Estimates
We compute industry level default likelihoods (Industry DLI) according
to the following procedure.
1. We utilize all observations meeting our data requirements from the
Wharton Research Data Services merged CRSP-COMPUSTAT database.
We assume a one quarter lag between financial statement information
and market data.
2. We assume the market value of debt to the face value of total liabilities
F and compute:
σD = 0.05 + 0.25σE (B.1)
where σE is the 90-day (rolling) equity return volatility of the industry
to which the firm belongs (as per the 17 Fama-French industry portfolio
classifications) and E is the market value of equity.
Asset volatility σA is computed as:
σA =E
E + FσE +
F
F + E(0.05 + 0.25σE) (B.2)
Equations (B.1) and (B.2) are the ‘naıve’ estimates of debt and asset
volatility respectively, as provided in Bharath and Shumway (2008).
3. Using σD, σA, and the prior year’s industry level-equity return as a
proxy for firm-level assets’ physical drift rate we compute annual default
probabilities in Merton (1974) framework. We use lagged values of the
median default likelihood, so estimated, as our measure of industry-
level default conditions or Industry DLI.
39
Appendix C. Tables and Figures
40
Table C.1: Discounted Ultimate Recoveries 1988-2011
Standard deviation is ‘Std’ and ‘IQR’ is interquartile range.Mean Median Std IQR 10% 90% N
Pooled 0.592 0.645 0.392 0.797 0.021 1 4720Term Loans 0.754 1 0.334 0.511 0.198 1 908
Revolvers 0.852 1 0.265 0.22 0.365 1 984
Bonds 0.449 0.360 0.379 0.727 0.004 1 2828
Collateral=Yes 0.768 1 0.318 0.473 0.209 1 2405Collateral=No 0.409 0.275 0.377 0.692 0 1 2315
Distressed Exchange 0.546 0.545 0.393 0.838 0.012 1 3983Bankruptcy 0.995 1 0.031 0 1 1 57
Default & Cure 0.821 1 0.285 0.328 0.299 1 648
Other Restructure 0.909 1 0.132 0.193 0.787 1 32
Days in Default < 1yr 0.562 0.543 0.39 0.802 0.021 1 19021yr < Days in Default < 2yr 0.526 0.536 0.395 0.86 0.008 1 1222
2yr < Days in Default < 3yr 0.478 0.405 0.399 0.865 0.005 1 4423yr < Days in Default 0.63 0.672 0.38 0.671 0.019 1 442
Senior Secured 0.635 0.651 0.34 0.755 0.193 1 591
Senior Subordinated 0.294 0.163 0.335 0.497 0 0.823 507
Senior Unsecured 0.486 0.444 0.375 0.76 0.014 1 1284Junior or Subordinated 0.274 0.14 0.343 0.447 0 0.972 446
Food 0.692 0.952 0.4 0.575 0.008 1 114
Mining 0.623 0.562 0.346 0.689 0.196 1 44Oil 0.545 0.5 0.369 0.781 0.058 1 215
Clothes, Textiles, Footware 0.625 0.638 0.345 0.673 0.156 1 163
Consumer Durables 0.605 0.686 0.396 0.808 0.031 1 170Chemicals 0.698 1 0.373 0.652 0.1 1 94
Drugs, soap, perfume, tobacco 0.594 0.605 0.422 0.86 0.096 1 17
Construction and Materials 0.584 0.611 0.399 0.746 0.01 1 222Steel 0.551 0.569 0.41 0.907 0 1 143
Fabricated Products 0.709 0.883 0.376 0.515 0.015 1 40
Machinery 0.624 0.672 0.375 0.752 0.094 1 267Automotive 0.657 0.806 0.385 0.587 0.007 1 160
Transport 0.517 0.518 0.362 0.781 0.037 1 363Utilities 0.864 1 0.259 0.113 0.364 1 278
Retail 0.54 0.51 0.403 0.849 0.014 1 528
Financial 0.564 0.572 0.417 0.832 0.007 1 118Other 0.561 0.626 0.397 0.847 0.009 1 1784
41
Table C.2: Mixture Components
Mean and standard deviation (Std) of of mixture components 1 . . . 4. ‘Sample Mean Pr’is the posterior mean of the conditional assignment probabilities based on the respectiveestimation samples.
ComponentEstimation Sample 1 2 3 4Mean -5.06 -1.23 0.15 5.20Std 0.16 0.79 0.77 0.00Recovery at Mean 0.00 0.11 0.56 1.00Mean + SD 0.00 0.33 0.82 1.00Mean - SD 0.00 0.02 0.27 1.00Sample Mean Pr 0.06 0.24 0.34 0.37
ComponentFull Sample 1 2 3 4
Mean -4.99 -1.29 0.11 5.20Std 0.22 0.77 0.78 0.00Recovery at Mean 0.00 0.10 0.54 1.00Mean + SD 0.00 0.30 0.81 1.00Mean - SD 0.00 0.02 0.25 1.00Sample Mean Pr 0.06 0.21 0.36 0.37
42
Table C.3: Conditional Mixing Probabilities
Posterior estimates of the latent variable regression z∗t = β0 + β1xt−1 + ǫt parameters.‘Coeff’ is the posterior mean of the parameter, and ‘Std’ is the posterior standarddeviation. ‘Debt Cushion’ is the proportion of total debt ranking below an instrument.‘Rank’ indicators denote the instrument rank with a null case of 1. ‘Collateral’ is anindicator of collateralization with a null case of NO. The security type indicator null caseis a term loan. Lagged Industry DLI is the industry level default likelihood observablea month prior to default. ‘Utilities’ is an indicator of whether the defaulting firm isclassified as a utility with a null case of NO. NOTE: c1 = 0, c2 = 1.21 and c3 = 2.57. Theposterior standard deviations of c2 and c3 are 0.15 and 0.21 respectively.
Estimation Full SampleCoeff Std Coeff Std
Intercept 1.81 0.14 1.54 0.21Debt Cushion 1.78 0.12 1.78 0.09Rank 2 -0.31 0.07 -0.19 0.05Rank 3 -0.55 0.10 -0.45 0.07Rank ≥ 4 -0.76 0.14 -0.72 0.09Collateral (Yes) 0.43 0.12 0.48 0.09Revolver 0.29 0.09 0.32 0.06Senior Secured Bond -0.20 0.11 -0.29 0.07Senior Subordinated -0.40 0.15 -0.39 0.11Senior Unsecured -0.05 0.14 0.12 0.10Junior or Subordinated -0.38 0.15 -0.41 0.11Lagged Industry DLI -0.51 0.07 -0.16 0.05Utilities 1.87 0.13 1.26 0.09
43
Table C.4: Industry Distress Effects
The top two panels of this table summarize the mean, standard deviation and percentilesof the implied recovery distributions in 4 cases. High quality (HQ) debt in ‘Normal’conditions and ‘Distress’ conditions, and, low quality debt under ‘Normal’ and ‘Distress’conditions. The debt characteristics and the definitions of ‘normal’ and ’distress’ aredescribed in Section 5.1. p1 - p4 are the posterior mean mixture assignment probabilitiesconditional on characteristics and conditions.
HQ, Normal HQ, Distresss LQ, Normal LQ, DistressMean 0.84 0.76 0.44 0.33Std 0.28 0.32 0.35 0.32Percentile 10 0.36 0.23 0.02 0.00Percentile 25 0.76 0.51 0.11 0.05Percentile 50 1.00 1.00 0.39 0.23Percentile 75 1.00 1.00 0.73 0.56p1 0.00 0.00 0.04 0.09p2 0.02 0.06 0.32 0.43p3 0.27 0.39 0.50 0.41p4 0.71 0.55 0.14 0.07
44
Table C.5: Inverse Gaussian Regression
The data transformation underlying this model is described in Section 5.2. ‘Debt Cushion’is the proportion of total debt ranking below an instrument. ‘Rank’ indicators denote theinstrument rank with a null case of 1. ‘Collateral’ is an indicator of collateralization witha null case of NO. The security type indicator null case is a term loan. Lagged IndustryDLI is the industry level default likelihood observable a month prior to default. ‘Utilities’is an indicator of whether the defaulting firm is classified as a utility with a null case of NO.
Model 1 Coeff St Error T Value PIntercept 0.54 0.03 16.98 0.00Debt Cushion 0.38 0.03 13.91 0.00Rank 2 -0.09 0.02 -5.32 0.00Rank 3 -0.15 0.03 -6.08 0.00Rank ≥ 4 -0.21 0.03 -6.12 0.00Collateral (Yes) 0.08 0.03 2.65 0.01Revolver 0.05 0.02 2.59 0.01Senior Secured Bond -0.09 0.02 -3.53 0.00Senior Subordinated -0.19 0.04 -5.29 0.00Senior Unsecured -0.07 0.03 -2.03 0.04Junior or Subordinated -0.15 0.04 -4.12 0.00Lagged Industry DLI -0.13 0.02 -7.42 0.00Utilities 0.38 0.02 15.52 0.00
R2 0.47 N 2307
45
Table C.6: Beta Transformation Regression
The data transformation underlying this model is described in Section 5.2. ‘Debt Cushion’is the proportion of total debt ranking below an instrument. ‘Rank’ indicators denote theinstrument rank with a null case of 1. ‘Collateral’ is an indicator of collateralization witha null case of NO. The security type indicator null case is a term loan. Lagged IndustryDLI is the industry level default likelihood observable a month prior to default. ‘Utilities’is an indicator of whether the defaulting firm is classified as a utility with a null case of NO.
Model 2 Coeff St Error T Value PIntercept 0.55 0.09 6.19 0.00Debt Cushion 1.25 0.08 16.32 0.00Rank 2 -0.13 0.05 -2.73 0.01Rank 3 -0.27 0.07 -3.86 0.00Rank ≥ 4 -0.39 0.10 -4.07 0.00Collateral (Yes) 0.32 0.08 3.92 0.00Revolver 0.18 0.06 3.19 0.00Senior Secured Bond -0.27 0.07 -3.92 0.00Senior Subordinated -0.30 0.10 -3.01 0.00Senior Unsecured -0.03 0.09 -0.29 0.77Junior or Subordinated -0.26 0.10 -2.55 0.01Lagged Industry DLI -0.34 0.05 -7.02 0.00Utilities 1.22 0.07 17.88 0.00
R2 0.46 N 2307
46
Table C.7: Conditional Out-of-Sample, Out-of-Time Portfolio Recovery Prediction
This table summarizes the features of the simulated portfolio recovery distributions basedon the resampling experiment described in Section 5.2. ‘Outcomes’ is the distribution ofactual data-based portfolio recoveries. ‘IG Reg’ is the distribution of recovery outcomesobtained from the IG regression in Table C.5 and ‘IG-B Reg’ is the corresponding set ofresults based on the IG-B regression in Table C.6. ‘Reg Tree’ refers to the distributionof recovery forecasts obtained from the regression tree estimated as described in Section5.2..The top number in each cell of the top two panels is expressed in terms of the recoveryamount, and the number below in italics is the prediction error (where applicable).RMSE is root mean-squared error and MAE is mean absolute error.
IG Reg IG-B Reg Mixture Reg Tree Actual
Mean 64.19 65.69 62.29 62.71 61.99
Std 3.89 3.11 3.87 4.12 3.79
Percentiles
1% 55.04 58.39 53.25 53.05 53.00
3.85% 10.16% 0.47% 0.09%
5% 57.10 60.55 55.87 55.89 55.74
2.44% 8.63% 0.24% 0.27%
10% 59.180 61.683 57.284 57.430 57.13
3.58% 7.96% 0.26% 0.52%
25% 61.59 63.58 59.69 59.97 59.45
3.61% 6.96% 0.41% 0.88%
50% 64.23 65.71 62.31 62.77 62.02
3.56% 5.96% 0.47% 1.21%
75% 66.85 67.79 64.92 65.51 64.54
3.58% 5.04% 0.59% 1.51%
90% 69.13 69.66 67.24 67.96 66.87
3.38% 4.17% 0.55% 1.63%
IG Reg IG-B Reg Mixture Reg Tree Average
RMSE 4.37 5.01 3.75 4.11 8.59MAE 3.52 4.17 2.99 3.28 7.76
47
0 0.2 0.4 0.6 0.8 10
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Figure C.1:
Histograms of bond and loan recoveries data pre and post inverse Gaussian transformation.
48
0 0.2 0.4 0.6 0.8 10
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Figure C.2:Histograms of ultimate recovery data by representative sub-categories.
49
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
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Mixture Component 4Mixture Component 1Mixture Component 2Mixture Component 3
Figure C.3:
Mapping transformed mixture components to recovery outcomes in terms of deviationsfrom the mean of the component distributions. All parameters are set equal to theirposterior mean values.
50
−0.5 0 0.5 1 1.5
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Figure C.4:
Kernel density estimates of predictive distributions obtained from a 4-component mixturein four scenarios: High quality (HQ) debt in ‘Normal’ conditions (solid line) and ‘Distress’conditions (solid line with dots), and, low quality debt under ‘Normal’ (dashed line) and‘Distress’ conditions (dashed line with dots). The debt characteristics and the definitionsof ‘normal’ and ’distress’ are described in Section 5.1.
51
00.1
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Figure C.5: Senior Secured Bonds
Senior Secured Bonds, rank 1. Panel (a) plots the median ultimate recovery based oncombinations of industry DLI and Debt Cushion, and Panel (b) the 10th percentile ofthe same distributions. The reported estimates are based on 100,000 draws from the4-component mixture specifications documented in Tables C.2 and C.3.
52
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Debt CushionIndustry Default Likelihood
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(b) Marginal Effects on Tail Recovery
Figure C.6: Senior Unsecured Bonds
Senior Unsecured Debt: Rank 2. Panel (a) plots the median ultimate recovery basedon combinations of industry DLI and Debt Cushion, and Panel (b) the 10th percentileof the same distributions. The reported estimates are based on 100,000 draws from the4-component mixture specifications documented in Tables C.2 and C.3.
53
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