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Understanding and Predicting Ultimate Loss-Given-Default for Defaulted Bonds and Loans

Michael Jacobs, Jr.[1]

Office of the Comptroller of the Currency

Ahmet K. Karagozoglu

Hofstra University

Draft: June 2006

J.E.L. Classification Codes: G33, G34, C25, C15, C52.

Keywords: Recoveries, Default, Loss Given Default, Financial Distress, Bankruptcy, Restructuring, Credit Risk, Entropic Methods, Bootstrap Methods, Forecasting

Abstract

In this study we empirically investigate the determinants of and build a predictive econometric model for loss-given-default (LGD) using a sample of S&P and Moody’s rated defaulted firms. We extend prior work by incorporating contractual, firm specific, industry, macroeconomic debt/ equity market determinants of LGD (Acharya et al, 2003) in a Kullback-Leibler relative entropy framework (Friedman et al, 2003). We are also able to model to duration of time in default in an internally consistent manner, generating a predicted bivariate distribution of time to resolution and ultimate LGD. We confirm many of the stylized facts and findings of the literature in regard to the determinants of LGD and find in addition the independent significance of a macroeconomic factor, equity returns and the price of traded debt at default in explaining the LGD. The model is validated rigorously through resampling experiment in a rolling out-of-time and out-of-sample framework.

1. Introduction and Summary

Loss given default (LGD) [2], the loss severity on defaulted obligations, is a critical component of risk management, pricing and portfolio models of credit. This is among the three primary determinants of credit risk, the other two being the probability of default (PD) and exposure of default (EAD). However, LGD has not been extensively studied as, and is considered a much more daunting modeling challenge in comparison to, PD. Starting with the seminal work by Altman (1968), and after many years of actuarial tabulation by rating agencies, predictive modeling of default rates is currently in a mature stage. The focus on PD is understandable, as traditionally credit models have focused on systematic components of credit risk which attract risk premia, and unlike PD, determinants of LGD have been ascribed to idiosyncratic borrower specific factors. However, now there is an ongoing debate about whether the risk premium on defaulted debt should reflect systematic risk, in particular whether the intuition that LGDs should rise in worse states of the world is correct and how this could be refuted empirically given limited and noisy data (Carey and Gordy, 2004).

The recent heightened focus on LGD is evidenced the recent flurry of research into the relatively neglected area of LGD (Acharya et al (2005), Carey and Gordy [2004, 2005], Altman et al [1996, 2001, 2004], Gupton et al [2000, 2001], Araten et al [2003], Frye [2000 a,b,c], Jarrow [2001]). This has been motivated by the large number of defaults and near simultaneous decline in recovery values observed at the trough of the last credit cycle circa 2000-2002, regulatory developments (Basel) and the growth in credit markets. However, obstacles to better understanding and predicting LGD, including dearth of data and the lack of a coherent theoretical underpinning, have continued to challenge researchers. In this paper, we hope to contribute to this effort by synthesizing advances in financial theory, econometric methodology and data acquisition in order to build an empirical model of LGD that is consistent with a priori expectations and stylized facts, internally consistent and amenable to rigorous validation. In addition to answering the many questions that academics have, we further aim to provide a practical tool for risk managers, traders and regulators in the field of credit.

LGD may be defined variously depending upon the institutional setting or modeling context, or the type of instrument (traded bonds vs. bank loans) versus the credit risk model (pricing debt instruments subject to the risk of default vs. expected losses or credit risk capital. In the case of bonds, one may look at the price of traded debt at either the initial credit event[3] or the value of instruments received in settlement of a bankruptcy proceeding (Keisman et al, 2000; Altman et al, 1996). When looking at loans that may not be traded, the eventual loss per dollar of outstanding balance at default is relevant (Asarnow et al, 1995; Araten et al, 2003). There are two ways to measure the latter – the accounting LGD refers to nominal loss per dollar outstanding at default[4], while the economic LGD refers to the discounted cash flows to the time of default taking into consideration when cash was received.[5] The former is used in setting reserves or a loan loss allowance, while the latter is an input into a credit capital allocation model.

An aspect of LGD modeling deserving of special attention, until recently neglected altogether or grossly simplified, is the distributional characterization of this quantity. While the available theory and evidence suggests it to be stochastic and predictable with respect to other variables, LGD has been treated as either deterministic or as an exogenous stochastic process. Such assumptions are made for tractability and in practical application results in understated capital, mispricing and unrealistic dynamics of model outputs. We will contribute to resolving such deficiencies by attempting to model the ex ante distribution of LGD as a function of empirical determinants – contractual features, borrower characteristics and systematic factors. However, we will not directly model the interdependency between LGD and other parameters of interest, such as probabilities of default, by either estimating a structural or reduced form model in which they are determined simultaneously[6].

2. Review of the Literature

In this section we will examine the way in which different types of theoretical credit risk models have treated LGD – assumptions, implications for estimation and application. We will then turn to the empirical evidence, both on the estimation of LGD, and on the performance of these various models. Finally, we will look at some of the stat-of-the-art and vendor models of LGD, and how they have attempted to incorporate lessons learned.

2.1 Treatment of LGD in Theoretical Credit Risk Models

Credit risk modeling was revolutionized by the approach of Merton (1974), who built a predictive theoretical model in the option pricing paradigm of Black and Scholes (1973), which has come known to be the structural approach. Equity is modeled as a call option on the value of the firm, with the face value of zero coupon debt serving as the strike price, which is equivalent to shareholders buying a put option on the firm from creditors with this strike price. Given this capital structure, log-normal dynamics of the firm value and the absence of arbitrage, closed form solutions for the default probability and the spread on debt subject to default risk can be derived. The LGD can be shown to depend upon the parameters of the firm value process as is the PD, and moreover is inversely related to the latter, in that the expected residual value to claimants is increasing (decreasing) in firm value (asset volatility or the level of indebtedness). Therefore, LGD is not independently modeled in this framework; this was addressed in much more recent versions of the structural framework (Frye [2000], Dev et al [2002[, Pykhtin [2003]), to be discussed in more detail later.

Many extensions of Merton (1974) that relaxed many of the simplifying assumptions. Complexity to the capital structure was added by Black and Cox (1976) and Geske (1977), with subordinated and interest paying debt, respectively. The distinction between long- and short-term liabilities in Vasicek (1984) was the precursor to the KMV( model. However, these models had limited practical applicability, the standard example being evidence of Jones, Mason and Rosenfeld (1984) that these models were unable to price investment grade debt any better than a naïve model with no default risk. Further, empirical evidence in Franks and Touros showed that the adherence to absolute priority rules (APR) assumed by these models are often violated in practice, which implies that the mechanical negative relationship between expected asset value and LGD may not hold. Longstaff & Schwartz (1995) incorporate into this framework a stochastic term structure with a PD-interest rate correlation. Other extensions include Kim at al (1993) and Hull & White (1995), who examine the effect of coupons and the influence of options markets, respectively.

Partly in response to this, a series of extensions ensued, the so-called “second generation” of structural form credit risk models (Altman [2003]). The distinguishing characteristic of this class of models is the relaxation of the assumption that default can only occur at the maturity of debt – now default occurs at any point between debt issuance and maturity when the firm value process hits a threshold level. The implication is that LGD is exogenous relative to the asset value process, defined by a fixed (or exogenous stochastic) fraction of outstanding debt value. This approach can be traced to the barrier option framework as applied to risky debt of Black and Cox (1976).

All structural models suffer from several deficiencies which. First, reliance upon an unobservable asset value process makes calibration to market prices problematic and invites model risk. Second, the limitation of assuming a continuous diffusion for the state process implies that the time of default is perfectly predictable (Duffie and Lando [2000]). Finally, the inability to model spread or downgrade risk distorts the measurement of credit risk. This gave rise to the reduced form approach to credit risk modeling (Duffie and Singleton, 1999), which instead of conditioning on the dynamics of the firm, posit exogenous stochastic processes for PD and LGD. These models include Litterman & Iben (1991), Madan & Unal (1995), Jarrow & Turnbull (1995), Jarrow et al (1997), Lando (1998), Duffie (1998). The primitives determining the price of credit risk are the term structure of interest rates (or short rate), a default intensity and an LGD process. The latter may be correlated with PD, but this is exogenously specified, with the link of either of these to the asset value (or latent state process) not formally specified. However, the available empirical evidence (Duffie and Singleton [1999], Lando and Turnbull [1997]) has revealed these models deficient in generating realistic term structures of credit spreads for investment and speculative grade bonds simultaneously. A hybrid reduced – structural form approach of Zhou (2001), which models firm value as a jump diffusion process, has had more empirical success, especially in generating a realistic negative relationship between LGD and PD (Altman et al 2001, 2003).

The fundamental difference of reduced with structural form models is the unpredictability of defaults: PD is non-zero over any finite time interval, and the default intensity is typically a jump process (e.g., Poisson), so that default cannot be foretold given information available the instant prior. However, these models can differ in how LGD is treated. The recovery of treasury assumption of Jarrow & Turnbull (1995) assumes that an exogenous fraction of an otherwise equivalent default-free is recovered at default. Duffie and Singleton (1999) introduce the recovery of market value assumption, which replaces the default-free bond by a defaultable bond of identical characteristics to the bond that defaulted, so that LGD is a stochastically varying fraction of market value of such bond the instant before default. This model yields closed form expressions for defaultable bond prices and can accommodate the correlation between PD and LGD; in particular, these stochastic parameters can be made to depend on common systematic or firm specific factors. Finally, the recovery of face value assumption (Duffie [1998], Jarrow et al [1997]) assumes that LGD is a fixed (or seniority specific) fraction of par, which allows the use of rating agency estimates of LGD and transition matrices to price risky bonds.

It is worth mentioning the treatment of LGD in credit models that attempt to quantify unexpected losses analogously to the Value-at-Risk (VaR) market risk models, so-called credit VaR models (Creditmetrics™ [Gupton et al, 1997], KMV CreditPortfolioManager™ [KMV Corporation, 1984], CreditRisk+™ [Credit Suisse Financial Products, 1997], CreditPortfolioView™ [Wilson, 1998]). These models are widely employed by financial institutions to determined expected credit losses as well as economic capital (or unexpected losses) on credit portfolios. The main output of these models is a probability distribution function for future credit losses over some given horizon, typically generated by simulation of analytical approximations, as it is modeled as a highly non-normal (asymmetrical and fat-tailed). Characteristics of the credit portfolio serving as inputs are LGDs, PDs, EADs, default correlations and rating transition probabilities. Such models can incorporate credit migrations (mark-to-market mode - MTM), or consider the binary default vs. survival scenario (default mode - DM), that principle difference being that in addition an estimated transition matrix needs to be supplied in the former case. Similarly to the reduced form models of single name default, LGD is exogenous, but potentially stochastic. While the marketed vendor models may treat LGD as stochastic (e.g., a draw from a beta distribution that is parameterized by expected moments of LGD), there are some more elaborate proprietary models that can allow LGD to be correlated with PD.

We conclude our discussion of theoretical credit risk models and the treatment of LGD by considering recent approaches, which are capable of capturing more realistic dynamics, sometimes called “hybrid models”. These Frye (2000a, 2000b), Jarrow (2001), Jokivuolle et al (2003), Carey & Gordy (2003), Pykhtin (2003) and Bakshi et al (2001). These models share in common the feature that dependence upon one or a set of systematic factors can induce an endogenous correlation between PD & LGD. In the model of Frye (2000a, 2000b), the mechanism that induces this dependence is the influence of systematic factors upon the value of loan collateral, leading to a lower recoveries (and higher loss severity) in periods where default rates rise (since asset values of obligors also depend upon the same factors). In a reduced form setting, Jarrow (2001) introduced a model of co-dependent LGD and PD implicit in debt and equity prices.[7]

2.2 Empirical Evidence on LGD and Tests of Credit Models

In this section we focus on the application of credit models and estimation of LGD from data. This ranges from simple quantification of LGD, calibration of credit models embedding LGD assumptions, and finally to empirical or vendor models of LGD.

There is a long tradition of actuarially estimating loss severities from bond or loan data, independent of any credit modeling framework and allied parameters such as PD. These have been conducted by academics, banks and rating agencies. The earliest studies relied exclusively on secondary market prices of bonds or loans. Altman & Kishore (1996) estimate LGDs for 300 defaulted senior secured and senior unsecured bonds from 1978-1995, yielding estimates ranging from 10% to 70% that could be statistically distinguished among various industry groups. In the ZETA™ model of Altman, Haldeman and Narayanan (1977), a 2nd generation of the Altman (1967) Z-Score PD estimation model, loan LGD estimates were based on a workout department survey. Later studies looked at ultimate recoveries on defaulted loans or bonds, either the nominal or discounted price at emergence from bankruptcy. Altman and Eberhart (1994) and Fridson et al (Merrill Lynch 2001) provided evidence that more senior significantly outperformed more junior bonds in the post-default period. Altman and Kishore (1996) find statistically different LGDs across broad industrial sectors.

LGD in bank loans have been studied by banks, rating agencies and academics. Bank studies focusing on internal loan data include Citigroup (Asarnow & Edwards, 1995), Chase Manhattan Bank (1996) and JP Morgan Chase (Araten et al, 2003)[8]. Average LGDs of 35% (861 large corporate obligors 1979-1993), 36% (412 large corporate obligors 1986-1993) and 40% (3800 wholesale loans 1982-2000) for were found in the Citigroup, Chase and JPMC studies, respectively. Finally, among those conducting historical analysis, some recent rating agency studies are worthy of note. Moody’s (Hamilton et al, 2001) reports an implied mean LGD of 30.3% (47.9%) for 121(181) senior secured (senior unsecured) secondary market loan prices a month after default. Various consortia of banks have published composite loss severity statistics from member banks, including Loan Pricing Corporation (2001; LPC) and the Risk Management Association (2000; RMA)[9]; however, the degree of segmentation by borrower and instrument characteristics is limited in these, and there are issues associated with normalizing data across banks. Standard and Poors (Keisman et al, 2000) presents empirical results from the LossStatsTM database. Analysis of 264 (690) bank loans (senior unsecured bonds) from 1987-1996 yields an average LGD of 16% (34%). This study also documents the independent influence of position in the capital structure (i.e., the proportion of debt above or below a claimant in bankruptcy), apart from collateral and seniority, in determining loss severities. Emery (2003) and Altman and Fanjul (2004) compare LGDs, as inferred from the prices of the traded instruments at default in a Moody’s database, on bank loans and bonds, respectively. A comparison of results reveals that loans experienced lower loss severity when controlling for seniority.[10] Varma et al (2003) document similar findings for corporate bonds as Altman and Fanjul (2004). Additionally, Altman and Fanjul (2004) document a differential LGD by rating at origination, such that “fallen angels” of the same seniority have significantly lower LGDs.[11]

Several recent empirical studies of LGD by academics have put more structure around this exercise, either by building predictive econometric models, or by attempting to directly test models. Frye (2000b) examines the LGD-PD correlation in 1982-1997 using the Moody’s Default Risk Service [DRSTM], finding a significant negative relationship at various levels of aggregation, consistent with the recent market experience in the 2001-2002 turn in the credit cycle. Hu and Perraudin (2002) also examines this relationship with Moody’s DRSTM for the 1983-2000 period, standardizing the data by transforming all borrowers to senior unsecured, thereby isolating the influence of systematic factors. They find LGD-PD correlations on the order of 0.2. Carey and Gordy (2004) analyze the correlation of LGD and PD in a combined database of defaults in 1970-1999[12]. Examining aggregated quarterly data at the obligor level, while they find almost negligible correlation with PD over the entire sample, in 1988-1998 there is a significant relationship that is in line with Frye’s (2000b) results[13]. However, they document that LGDs tend to rise more in recessionary periods than they fall during expansions, suggesting that more is at play than a macroeconomic factor influencing the value of collateral.[14] In the Araten et al (2001) bank study, unsecured U.S. large corporate borrower level LGDs are regressed average Moody’s All-Corporate default rate for the period 1984-1999 on an annual basis, yielding an r-squared of 0.2, in line with Carey and Gordy (2004) for their restricted sample, yet significantly higher than Hu and Peraudin (2002). Altman et al (2001,2004) find an that LGDs increase as the credit cycle worsens, going from 75% in 2001-2003 to 55% in 2003 as default rates decreased beneath their long run average of 4.5%. This is verified by Keisman (2003), who finds that LGDs of all seniorities rise during this stress period, in the S&P LossStatsTM database. However Altman (2004) finds that a systematic variable has no effect on LGD when bond market conditions (e.g., supply-demand imbalances) are accounted for. However, Acharya et al (2005) examine the same data as Keisman (2003) for the period 1982-1999, and while they verify that seniority and security are key determinants of LGD, in addition they find industry specific factors influencing LGD independently of the macroeconomic state and bond market conditions seen in Altman (2004). In particular, they find elevated LGDs in distressed industries (less redeployable assets, greater leverage and lower liquidity), after controlling for firm / contract systematic factors. This constitutes a test of the Schleifer and Vishny (1992) “fire-sale” hypothesis – an industry equilibrium phenomenon in which macro and bond market variables are spuriously significant due to omitting an industry factor.

Finally, we may mention vendor models of LGD, which incorporate approaches taken in the academic and agency literature, in addition applying proprietary methodologies and data sources. S&P (Friedman & Sandow, 2003) applies a Kulback-Leibler maximum entropy non-linear regression model to the LossStatsTM database, which incorporates Bayesian style prior information (point masses at 0 and 100%) to produce predictive densities of LGD. Moody’s LoosCalc2™ (Gupton, 2004) applies an econometric based models (local regression) to KMV’s proprietary LGD database.

Econometric Models

Data and Summary Statistics

We have built a database of defaulted firms (bankruptcies and out-of-court settlements), all having rated instruments (S&P or Moody’s) at some point prior to default. It contains data on 2,732 defaulted instruments from 1986-2003 for 650 borrowers, or which there is information on all classes of debt. All instruments are detailed by type, seniority, collateral type, position in the capital structure, original and defaulted amount, resolution type, instrument price at emergence from as well as the value of securities received in settlement from bankruptcy.

Estimation Results

Summary and Conclusions

Appendix – Tables and Figures

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[1] Corresponding author: Senior Financial Economist, Credit Risk Modelling, Risk Analysis Division, Office of the Comptroller of the Currency, 250 E Street SW, 2nd Floor, Washington, DC 20024, 202-874-4728, michael.jacobs@occ.treas.gov. The views herein are those of the authors and do not necessarily represent the views of the Office of the Comptroller of the Currency.

[2] This is equivalent to one minus the recovery rate, or dollar recovery as a proportion of par. We will speak in terms of LGD as opposed to recoveries with a view toward credit risk management applications.

[3] By default we mean either bankruptcy (Chapter 11) or other financial distress (payment default). In a banking context, this defined as synonymous with respect to non-accrual on a discretionary or non-discretionary basis. This is akin to the regulatory definition of default (Basel).

[4] In the context of bank loans, this is the cumulative net charge-off as a percent of book balance at default (the net charge-off rate).

[5] There is debate surrounding the appropriate choice for a discount rate. Bank studies (Araten et al, 2003) have put forward arguments for a punitive rate as consistent with the “low end” of what buyers of distressed assets look for as overall return, the uncertainty of recoveries (with the standard deviation about equal to the average), the rates used by commercial loan pricing models, and consistency with “peer practice”. A competing argument among academics (Acharya et al, 2004) as well as practitioners (Friedman et al 2003) and that it is proper to discount ultimate recoveries using the coupon on the debt prior to default. Finally, some have argued in favor of discounting recoveries at the default risk-free Treasury term structure (Carey and Gordy, 2005). We do not address this issue directly in this paper – however, to the extent that one can jointly forecast time-to-resolution and the ultimate recovery, an implicit estimate of the proper actuarial discount rate can be formulated based upon this research.

[6] Altman et al (2003) offers an extensive review of the theory and empirical evidence regarding the relationship between LGD and PD, and further documents the influence of the economic state on recovery values and hence the LGD estimate.WbuvwíÄ Ç Ñ £

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[11] Jarrow (2001) also has the advantage of isolating the liquidity premium embedded in defaultable bond spreads.

[12] Average LGDs of 35% (861 large corporate obligors 1979-1993), 36% (412 large corporate obligors 1986-1993) and 40% (3800 wholesale loans 1982-2000) for were found in the Citigroup, Chase and JPMC studies, respectively.

[13] LPC (RMA) find an average accounting LGD of 30% (27%) for 2,534 (977) loans, both in the 1998-2001period.

[14] In a comparable period, both authors find median LGDs on senior secured loans, senior secured bonds, unsecured loans and senior unsecured bonds were found to be 27.0%, 45.5%, 49.5% and 57.7%.

[15] Median LGDs of 49.5% and 66.5% for defaulted issuers originally investment and speculative grade, respectively.

[16] Moody’s DRSTM, S&P’s LossStatsTM and S&P’s CreditprioTM , and the Society of Actuaries private placements database.

[17] Correlations of 0.45 (0.80) for senior (subordinated) debt.

[18] Carey ands Gordy (2004) argue for a 2 stage approach to measuring LGD, first estimating an “estate LGD” at the obligor level, and then treating instrument level LGDs according to a contingent claims approach, as under the Absolute Priority Rule (APR) such recoveries can be viewed as collar options on residual value of the firm. However, they argue that the endogeneity of the bankruptcy decision will result in a measurement problem in the 1st stage borrower level. Furthermore, an extensive literature on violations of APR suggests a similar problem in the 2nd stage instrument level (Hotchkiss [1993], Eberhart et al [1989], Weiss [1990]).

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