An Empirical Study of Exposure at Default
Michael Jacobs, Jr.^{1}
Office of the Comptroller of the Currency
Draft: August 2008
J.E.L. Classification Codes: G33, G34, C25, C15, C52.
Keywords: Exposure at Default, Loan Equivalent Exposure, Recoveries, Default, Loss Given Default, Financial Distress, Bankruptcy, Restructuring, Credit Risk
Abstract
In this study we empirically investigate the determinants of and build a predictive econometric model for exposure at default (EAD) using a sample of defaulted firms having revolving credits, at one time having S&P or Moody’s rated debt. We extend prior empirical work (Araten et al 2001, Asarnow 1994) by considering alternative determinants (i.e., borrower accounting, industry / macroeconomic and debt / equity market determinants) of EAD risk, in addition to the traditional factors of credit rating, utilization and tenor. Various measures of EAD risk are derived and compared (the loan equivalentLEQ, credit conversionCCF and exposure at defaultEADF factors). We build a multiple regression model in the generalized linear class for these and examine the comparative rank ordering and predictive accuracy properties of these measures, in which EADF (CCF) performs best (worst). We find weak evidence of countercyclicality in EAD. While we find EAD risk to decrease with default risk, with risk rating having some explanatory power, utilization has the strongest inverse relation. We also find EAD risk reduced for greater leverage, liquidity, more debt cushion; and increased for greater company size, higher collateral rank of the loan or more bank debt in the capital structure of the defaulted obligor. The models are validated rigorously through resampling experiment in a rolling outoftime and outofsample experiment.
1. Introduction and Summary
Committed revolving credit facilities offer borrowers an option to draw funds up to specified limits as dictated by changing circumstances. A great interest in analyzing revolving lines of credit (or unfunded commitments) stems from the unique characteristics relative to other traditional credit or fixed income products, which have implications for pricing and credit risk management. Revolving lines appeal to a clientele of borrowers with particular financing strategies (e.g., working capital needs or CP backup for investment grade obligors). There is an attractive return profile from investor’s perspective, as interest and fees on the drawn, and annual fees on the undrawn amount, lead to a high allin return on invested capital. There is the opportunity to invest in high quality borrowers not otherwise in the loan market. Finally, there is potential relief on regulatory capital relative to similar investments^{2}. However, these instruments present a great challenge in valuation and risk management, as estimates of revolving credits facility’s expected usage at default of the borrower need to be formed. We term this quantity the exposure at default (EAD), or the loan equivalent exposure, and recognize this to be a key parameter in estimating expected loss and credit risk capital for unfunded commitments. A related quantity is the loan equivalency factor (LEQ), which is the proportion of the undrawn commitment that is drawn down upon in the event of default. Financial institutions have a great interest in estimating such quantities from their internal histories, to parameterize credit risk models, as well as to satisfy regulatory requirements^{3}. In this study we shall build upon a limited practitioner literature (Asarnow and Marker [1995], Araten and Jacobs [2001]) by empirically estimating EADs from publicly available data, and relating these to a set of variables predictive of realized EAD.
As noted in the previous literature, there exists an adverse selection problem in the context of revolving commitments: if a borrower’s credit quality improves, the ability to pay down or negotiate better pricing increases, while when deteriorating there is the tendency to drawdown on the unused portion of the commitment. Various risk mitigants are often in place to cope with this. There are upfront fees to deter prepayment, not the practice for most other credit products. Covenants are often in place to obtain amendment fees or improved pricing if borrower’s credit quality worsens. Finally, stepup pricing, or higher undrawn (drawn) spreads fees for lower credit quality (increased usage), help deter this.
In most applications, it is assumed that the current outstanding is outstanding at default (i.e., the LEQ factor may not be negative) and that the current commitment will not be higher at default (i.e., the LEQ may not exceed 100%), but this is not supported any the available empirical evidence. Furthermore, typically banks assume that EAD does not depend on factors other than an obligor’s credit rating and perhaps the time horizon under consideration and in some cases there may be only a single EAD assumption by broad product type, but there is limited evidence to support this. It may be the case that alternative variables may be predictive of the EAD (e.g., expected LGD, current usage, size of commitment, borrower financials, etc.)
As defined herein, the credit conversion factor (CCF) factor is the multiple of the current utilized amount, while the loan equivalent exposure (LEQ) is the proportion of the current undrawn amount that is expected to be outstanding at the time of default, respectively. CCF has the potential advantage of Incorporating not requiring a positive unused amount at the point of observation prior to default, which is particularly advantageous in certain products for which systems are unlikely to report this or uncommitted lines (e.g., letters of credit). Alternatively, one may estimate a factor to be applied to the total committed amount, which we call the exposure at default factor (EADF). LEQs, CCFs and EADs may be estimated either directly or indirectly. The direct method starts from a universe of defaulted facilities and traces utilization and credit quality changes going back in time. The problem in this is paucity of data for investment grade obligors. The indirect method studies changes in utilization rates with changes in credit quality for the entire universe of revolving credits, extrapolating to the better risk ratings. A hybrid of these approaches combines information in migrations short of default with observations of default. While in this study we restrict attention to the direct approach, in line with Basel requirements, one can imagine extensions beyond the defaulted universe. And while. Also, while one may imagine more general approaches to measuring EAD risk that better Incorporate the joint dynamics of utilization and commitment as borrowers approach default, in this study we limit ourselves to these three traditional approaches.
The exercise of empirically inferring EAD risk measures is fraught with practical, as well as conceptual, challenges. There is the question of defining the optimal estimator of a quantity that may have different meanings depending upon the context. The sampling methodology may also be subject to discretion. The issue of the treatment of outliers and extreme nonnormality is pronounced in this setting, which calls into question the validity of standard statistical inference and predictive econometric techniques. Various ad hoc methods of dealing with the latter (e.g., Winsorizing or capping / flooring of estimates^{4}) lead to further doubts about the efficiency and consistency of this methodology.
This paper will proceed as follows. In Section 2 we review the relevant literature. Section 3 examines and mathematically characterizes different ways to quantify EAD risk, the LEQ, CCF and EADF factors. In Section 4 we propose and econometric framework that is suitable for estimation of the various EAD risk measures so derived, the beta link generalized linear (BLGLM) model. We describe the data in Section 5 through various summary statistics and measures of association between EAD risk measures and potential drivers. The results of the estimation are discussed in Section 6, including an outofsample and outoftime validations of the estimated regression models. Finally, Section 7 concludes and provides directions for future research.
2. Review of the Literature
In this section we review the existing literature on EAD. This has been for the most limited to internal bank and trade journal studies. The earliest known study is an analysis conducted by Chase Manhattan Bank (1994), with the assistance of Oliver Wyman Mercer. Drawdowns are studied on 104 revolving credit facilities downgraded in the period 12/9112/93. The analysis is divided into three parts: 6 month commitments (“shortterm”), 1year commitments (“longterm”) and a blend of the two (more specifically, an average for the even years). LEQs are directly estimated for facilities for which defaults are observed in the sample, for speculative ratings (BB and below). However, for investment grade commitments, the migration method is used, which extrapolates factors for better ratings from worse with less time to default using estimated cutback and drawdown rates. Estimated LEQ factors were found to increase with risk rating and tenor.
Asarnow et al (1995) analyze utilization patterns, on a monthly basis for revolving commitments in the period 1/8712/93, for companies having an S&P rating history. Results by subgrade are not statistically meaningful due to thin data. Utilization rates by rating are computed for 84 months and averaged by rating category. An unpublished version of this study analyzed empirical LEQs based upon subset of 50 facilities downgraded to BB/B or worse in the 19911993 period.^{5} LEQs are extrapolated to the better risk grades due to lack of investment grade downgrades, and not averaged across facilities, but across quarterly total used and unused for each rating category. Unlike the Chase study, estimated LEQs are found to decrease with increasing credit quality.
Araten et al (2001) study direct estimates of LEQ factors for 1,021 observations (408 facilities of 399 borrowers) at a quarterly frequency in the period 1Q954Q00. Given the set of all revolving commitments and advised lines eventually having facility grades rated (accruing) substandard or worse^{6}, they track the rating history and usage.^{7} The sampling methodology involves stepping back in time from the point of default, calculating the LEQ as change in usage to default relative to unused at a given point in time, at either 1 year intervals or at the time of a grade change.^{8} The main result of this analysis is estimated LEQs increasing with timetodefault and with diminished credit risk (i.e., better risk grades.) A pronounced increase in estimated LEQs with tenor was found, with one and five year revolving credits having averages of 32% and 72%, respectively. The decrease in LEQ by grade (worsening credit quality) was found to be milder: 62% for BBB and better, 48% for BBB to B+, and 27% for B and worse. The overall average is 43.4%, with relatively high 41.4% standard deviation, and a “Barbell” shaped distribution with significant point masses at 0% & 100%. The latter distributional feature is largely an artifact of the truncation of the LEQ estimates, as calculated LEQs greater (less) than 100% (0%) were capped (floored) at 100% (0%). There is a lack of statistical differentiation by other demographics: lending organization, commitment type or size, geography or industry. LEQs are found to decrease with percent usage, but this is highly correlated with risk rating. However, lower LEQs are found for Advised Lines – 17% for one year, but having a similar pattern by grade and timedefault. The array of statistical and conceptual issues encountered in this study include outliers, high volatility (on the order of the mean), lack of statistical significance by risk drivers, paucity of data at the investment grade, default definition, sensitivity of estimates to small unused, nonnormality, judgmental recoding and data management.
The JP Morgan Chase 2003 update to 2001 LEQ study^{9} constitutes a major effort to extend as well as improve upon the methodology and analysis. The main extension is an attempt to measure the “true” default date, as proxied for by nonaccrual or the incurrence of a chargeoff.^{10} This includes 2 more years of data (20012002), resulting in about 50% more observations. The principle difference with the previous published study is s flatter “term structure” of LEQs (i.e., less pronounced in crease in the estimated factors with timetodefault). Had the new results been applied, there would have been a slight increase in LEQ for one year and investment grade revolvers. However, based upon changes to allocated credit capital, overall results were deemed not materially different than original study to warrant changes factors used in capital allocation and credit systems.
As noted previously, beyond trade journals and internal models, there is no known work of this type in the academic literature to date, but there is some notable related work. A recent working paper circulating in the regulatory channels, Moral (2006), reviews different methodologies and proposes an optimal framework (from the regulatory viewpoint) for estimating EAD factors. An academic paper that is most similar to this line of research, Sufi (2005) empirically examines use of bank lines of credit to corporations, using annual 10K filing data. The author finds that the flexibility afforded firms by use of unfunded commitments creates a moral hazard problem, which is mitigated by banks imposing strict covenants, and lending to borrowers with historically high profitability. Most recently, Loukoianova et al (2007) develop a theoretical pricing model for contingent credit lines (CCLs), widely used in bank lending and instrumental in the functioning of shortterm capital markets, which are closely related to the instruments considered herein. The authors use a financial engineering approach to analyzing the structure of simple CCLs, applying various derivative pricing methods, and discuss issues in the hedging of CCL portfolios.
3. Mathematical Depiction of EAD Risk Measures: LEQ, CCF and EADF Factors
Define the notation:


t, T : current time, fixed horizon (maturity or ex post calendar time of default)

τ : random time of default

X_{t} = vector of obligor or facility characteristics (e.g., risk rating, product type, financial ratios)

^ : superscript denotes an estimate

UTIL_{t} : dollar utilization (or used) at time t

AVAIL_{t} : dollar availability (or commitment, limit) at time t

EAD_{t,T}(EAD_{T}^{f}) : time t expected dollar exposure (EAD factor) at horizon T

LEQ_{t,T}^{f} : time t expected loan equivalency factor for horizon T
CCF_{t,T}^{f }: time t expected credit conversion factor for horizon T
Dollar EAD is the expected utilization at the time of default:
(3.1)
Traditionally the dollar EAD is estimated through an LEQ factor that is applied to the current unused:
(3.2)
The LEQ factor is the expected portion of the unused drawn down upon in the event of default:
(3.3)
Note that by the properties of conditional expectation . The LEQ for a given factor can be estimated by observations of changes in utilization over unused to default:
(3.4)
An alternative approach estimates a credit conversion factor (CCF) to be applied to the current commitment:
(3.5)
The CCF is simply the expected gross percent change in the total commitment:
(3.6)
CCF can be estimated by averaging the observed percent changes in commitment:
(3.7)
Alternatively, model expected dollar EAD as the expected availability at default:
(3.8)
The dollar EAD factor may be factored into the product of the current utilization and an EAD factor:
(3.9)
The EAD factor is the expected gross percent change in availability:
(3.10)
The EAD factor may be estimated as the average product of the changes in USAGE and AVAIL:
(3.11)
3.1 A Generalized Model for EAD Estimation: An Optimization Framework
The approaches outlined above follow closely what has been the common practice in financial institutions thus far, the development of a factor that can be applied to some component of a revolving commitment, for example the outstanding balance or the limit. In this section we outline an alternative approach to this problem, which potentially could lead to the development of such factors, but which has certain desirable properties. This discussion follows closely Moral (2006), who comprehensively surveys various practices among banks, and issues of relevance to bank regulators, in the estimation of EAD. First, following Levonian (2007), let us take a step back from the various specifications of the previous section, and consider the problem wherein the expectation of EAD is an unspecified function of current availability AVAIL, outstanding balance USAGE and a vector of risk drivers X:
(3.1.1)
We can then think of a regression model in which we try to match realized to forecast EAD on a dollar basis, and then back out an estimate of an LEQ or CCF factor. Moreover, we could test parameter restrictions that would give rise to these. As a concrete example, we may think of a linear, generalized linear (see section 4) or nonlinear regression framework. Let us denote the forecast error in predicting EAD by ε. Following Moral (2006), we may seek to minimize a loss function L(.) of this forecast error:
(3.1.2)
Where the covariates have been collected into a vector and the function g(.) is the object of interest, in this context an LEQ or CCF factor, and P represents the physical measure. Moral (2006) proposes the deviation in the quantile of the regulatory capital requirement as an appropriate metric, in that such can be asymmetric, as in that context one would care more about under than overestimating the EAD. So in the context of supervision, we get the problem:
(3.1.2)
Where CPD() is the conditional probability of default and LGD() the conditional lossgivendefault, which in all generality can depend upon the same risk drivers as EAD, and the loss function is of the form:
(3.1.3)
Assuming that PD and LGD are independent, and casting the problem in terms of LEQ estimation, so that , one obtains:
(3.1.4)
Moral (2006) demonstrates two interesting and profound facts about this problem. First, the solution to (3.1.4) is equivalent to a quantile regression estimator (Koenker and Bassett, 1978) of the dollar change in usage to default, , on the risk drivers Y_{t}:
(3.1.5)
Where denotes the quantile of the empirical distribution P^{*} such that . Second, if we adopt the loss function (3.1.3), then if we are interested in imposing the constraint (for example, Advanced IRB), then imposing this constraint on the estimation using raw data is equivalent to an unconstrained estimation in which we restrict ourselves to realizations of LEQs meeting this constraint (i.e., censored data). Said different, problem (3.1.4) with the added constraint is equivalent to the problem:
(3.1.6)
This is an attractive alternative to “collaring” LEQ estimates, since the bounding mechanism is optimal using raw data as well as using data restricted in that way, while on the other hand “collaring” does not share this property. We will term this estimator the “QLEQ” estimator (“quantile LEQ”) to distinguish it from the other LEQ factor as well as CCF or EADF.
4. Econometric Models
Various techniques have been employed in the finance and economics literature to classify data in models with constrained dependent variables, either qualitative or bounded in some region. However, much of the credit risk elated literature has focused upon qualitative dependent variables, which the case of PD estimation naturally falls into. Maddala (1981, 1983) introduces, discusses and formally compare the different models.^{11} We will discuss this framework here in order to motivate the discussion of special methods required in the case of EAD, in comparison and contrast to PD, and to highlight the similarities.
Let the i^{th} observation of dependent (or response) variable of interest, some measure of EAD risk be denoted (the LEQ, CCF or EADF), be observed independently. The vector of independent (or stimulus) variables corresponding to bounded random variable is denoted by . We assume that the conditional expectation of depends upon a linear function of the only through a smooth, invertible function :
