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

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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

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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

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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

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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

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Histograms of bond and loan recoveries data pre and post inverse Gaussian transformation.

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Figure C.2:Histograms of ultimate recovery data by representative sub-categories.

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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.

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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.

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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.

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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.

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