Lgd Model Jacobs 10 10 V2[1]

An Option Theoretic Model for Ultimate Loss-Given-Default

with Systematic Recovery Risk and Stochastic Returns on

Defaulted DebtMichael Jacobs, Ph.D., CFA

Senior Financial Economist

Credit Risk Analysis Division

Office of the Comptroller of the Currency

October, 2010The views expressed herein are those of the author and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury.

Outline

• Background and Motivation

• Introduction and Conclusions

• Review of the Literature

• Theoretical Framework

• Comparative Statics

• Econometric Model & Empirical Results

• Implications for Downturn LGD

• Model Validation

• Summary and Future Directions

Background and Motivation• Loss Given Default (LGD) – ultimate economic loss per dollar of

outstanding balance at default• A critical parameter in various facets of credit risk modeling –

expected loss, pricing, capital (economic & regulatory)• Basel II Internal Ratings Based (IRB) advanced approach to

regulatory credit capital requires banks to estimate LGD• May be measured either on a nominal (undiscounted) or

economic (discounted) basis – we care about the latter• Here we measure the market values of instruments received in

settlement of default (bankruptcy or restructuring) as a proxy• “Workout” approach (discount recovery cash flows) vs. market

for distressed debt (trading or settlement prices at default or ultimate)

• Many extant credit risk models assume LGD to be fixed despite evidence it is stochastic and predictable with respect to other variables

Introduction and Conclusions• Develop a theoretical model for ultimate loss-given default in the Merton (1974)

structural credit risk model framework• Derive compound option formulae to model differential seniority of instruments &

an optimal foreclosure threshold• Extension that allows for an independent recovery rate process representing

undiversifiable recovery risk, with stochastic drift• Calibrate models to observed LGDs on extensive dataset of bonds and loans

from Moody’s 1987-2008 having both trading prices at default & at resolution of default

• Parameter estimates vary significantly across models & segments (volatilities of recovery rates and of their drifts are increasing in seniority & for bank loans as compared to bonds)

• Implications for downturn LGD: declining in ELGD but uniformly higher for bonds vs. loans & greater PD-LGD correlation

• Validate the model in out-of-sample bootstrap exercise: compares well to a high-dimensional regression model & a non-parametric benchmark based upon the same data

Review of the Literature• Structural models: Merton (1974), Black and Cox (1976), Geske (1977), Vasicek

(1984), Kim at al (1993), Hull & White (1995), Longstaff & Schwartz (1995)• Reduced form models: Litterman & Iben (1991), Madan & Unal (1995), Jarrow &

Turnbull (1995), Jarrow et al (1997), Lando (1998), Duffie & Singleton (1999), Duffie (1998)

• Credit VaR models: Creditmetrics™ (Gupton et al, 1997), Moody’s KMV™• Hybrid approaches: Frye (2000), Jarrow (2001), Bakshi et al (2001), Jokivuolle et

al (2003) • Various recent academic studies have appeared on this topic

– Hu and Perraudin (2002) – LGD/ PD correlation– Renault and Scalliet (2003) – beta kernel density estimation– Acharya et al (2004) - industry distress ( “fire-sale” effect)– Altman (2005) – debt market supply/demand– Mason et al (2006) – option pricing & returns on defaulted debt – Carey & Gordy (2007) – estate level LGD and the role of bank debt

Theoretical Framework• We propose an extension of Black and Cox (1976) with perpetual corporate debt & a continuous, positive foreclosure boundary

– Former assumptions removes the time dependence of the value of debt, thereby simplifying the solution and comparative statics – The latter assumption allows us to study the endogenous determination of the foreclosure boundary by the bank, as in Carey and Gordy (2007)

• Extend this by allowing the coupon on the loan (or drift / return on the collateral) to follow a stochastic process• We assume no restriction on asset sales, so that we do not consider strategic bankruptcy, as in Leland (1994) and Leland and Toft (1996)• Assume a firm financed by equity & perpetual debt divided between a single loan (class of bonds) with face value λ (1-λ)

– Loan: senior & potentially has covenants which permit foreclosure, and entitled to a continuous coupon at a rate c which may evolve randomly– Equity: receives a continuous dividend with constant & variable components δ+ρVt , where Vt is the value of the firm’s assets at time t – Impose the restriction 0 ≤ ρ ≤ r ≤ c where r is the constant risk-free

Theoretical Framework (continued)

• The asset value of the firm, net of coupons γ and dividends δ, follows a Geometric Brownian Motion with constant volatility, where C denotes the total fixed cash outflows per unit time:

tt

t t

dV Cr dt dZ

V V

1C c

• Default occurs at time t (dividends cease) and is resolved after a fixed interval τ (the loan coupon continues to accrue)

• At emergence, loan holders receive minimum of the legal claim or the value of the firm, and the total legal claim is, respectively:

exp , tc V

exp 1D c

• Assume firm value fluctuates through bankruptcy & as in Carey and Gordy (2007), model value of the loan at the threshold as equal to the recovery value at default time & remove the time dependency in loan value process, giving a 2nd order ODE debt that we solve for the optimal foreclosure boundary κ* (if positive and fixed cash flows to claimants other than the bank)


• Model undiversifiable recovery risk by introducing a separate process for recovery on debt Rt, state of underlying collateral from default:

tt t t

t

dRdt dZ dW

R t t td dt dB

1 t tt

t t

dA dRCorr

dt A R

1

,t t tCorr dB dWdt

• Economic LGD on the loan is given by following expectation under physical measure with modified formulae:

2 2exp, | , , , , , , , 1 min exp , exp

2

expˆ1 , | exp , , ,

Pt t t t t t t

tt t t

cLGD R c E c R Z W

cB R c

' 'ˆ, | exp , , , expt t t t tB R c R d c d

2

22 2 2 2 2 22

2 1ˆ 2 1 1

2e e

• A well-known result (Bjerksund, 1991) is that the maturity-dependent volatility is given by::

' 21 1

ˆlogexp 2ˆ

tt

Rd

c


• Closed-form solution for the LGD on the bond follows well-known formula for a compound option: “outer option” is a put & “inner option” is a call with expiry dates equal. Let R* be the critical level of recovery such that the holder of the loan is just breaking even:

, , | , , , , , , , ,PB t t tLGD V R c

2 2exp ,

1 min ,max exp exp , ,02t t t t tE B R Z W c

B

• The recovery to the bondholders is the expectation of the minimum of the positive part of the difference in the recovery and face value of the loan and the face value of the bond B, which is structurally identical to a compound option valuation problem (Geske, 1977):

*exp 1 , | , , , , , , ,Ptc LGD R c

exp, | , , , , , , , , , 1 , | , , , , , , , , ,bP

B t t B t t BLGD R c R cB


• Φ2(X,Y|ρXY): bivariate normal distribution function, ρ XY =[τX/τY].5 for respective “expiry times” (note assumption that loan resolves before bond a matter of necessity here and seen on average in the data)

• Can extend this framework to arbitrary tranches: debt subordinated to the dth degree results in a pricing formula that is a linear combination of d+1 variate Gaussian distributions

• These formulae become cumbersome very quickly, so for the sake of brevity we refer the interested reader to Haug (2006) for further details

2 2

, | , , , , , , , , ,

exp , ; , ; exp

t t B

t tB B

R c

B a b R a b c a

2*

1 1ˆlog

2ˆt

t

Ra

R

21 1ˆlog

2ˆt

B t

B

Rb

B

Comparative Statics

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Figure 9: Ultimate Loss-Given-Default vs. Value of Firm at Default

Stochastic Collateral & Drift Merton ModelV_tL

GD

(V|c

or(

R,V

)={.

05

,.15

,.45

},a

lph

a=

.08

,lam

=.5

,c=

.06

,eta

=.3

,be

ta=

.5,n

u=

.5,k

ap

=.5

,kce

=.3

,tau

=1

)

Loan-cor(R,V)=0.15

Loan-cor(R,V)=0.05

Loan-cor(R,V)=0.45

Bond-cor(R,V)=0.15

Bond-cor(R,V)=0.05

Bond-cor(R,V)=0.45

0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 10: Ultimate Loss-Given-Default vs. Sensitivity of Recovery Process to LGD Side Systematic Factor

Stochastic Collateral & Drift Merton ModelnuL

GD

(nu

|co

r(R

,V)=

{.0

5,.1

5,.4

5},

V=

.5,a

lph

a=

.08

,lam

=.5

,c=

.06

,eta

=.3

,be

ta=

.5,k

ap

=.5

,kce

=.3

,tau

=1

)

Loan-cor(R,V)=0.15

Loan-cor(R,V)=0.05

Loan-cor(R,V)=0.45

Bond-cor(R,V)=0.15

Bond-cor(R,V)=0.05

Bond-cor(R,V)=0.45

• LGD is montonically decreasing at increasing rate in value of the firm at default, this increases in the correlation between PD and LGD side systematic factors, & also uniformly higher for bonds than for loans

• LGD increases at an increasing rate in LGD-side recovery volatility, for higher asset/recovery correlation at a faster rate, for bonds these curves lie above and increase at a faster rate

Comparative Statics (continued)• LGD increases at an

increasing rate in LGD volatility attributable to the PD-side systematic factor, for lower firm asset values LGD is higher but increases at a slower rate, for bonds these curves lie above & increase at a lower rate

• LGD decreases at a decreasing rate in recovery drift volatility, but sensitivity is not great (especially for loans), curves lie above for greater PD-LGD correlation & for bonds vs. loans

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Figure 11: Ultimate Loss-Given-Default vs. Sensitivity of Recovery Process to PD Side Systematic Factor

Stochastic Collateral & Drift Merton Modelbeta

LG

D(b

eta

|v=

{.3

,.5,.8

}.a

lph

a=

.08

,lam

=.5

,c=

.06

,eta

=.3

,ka

p=

.5,k

ce=

.3,ta

u=

1)

Loan-V=0.3

Loan-V=0.5

Loan-V=0.8

Bond-V=0.3

Bond-V=0.5

Bond-V=0.8

0.0 0.1 0.2 0.3 0.4

0.0

0.2

0.4

0.6

0.8

1.0

Figure 12: Ultimate Loss-Given-Default vs. Volatility in the Drtift of the Recovery Rate Process

Stochastic Collateral & Drift Merton ModeletaL

GD

(eta

|co

r(R

,V)=

{.0

5,.1

5,.4

5},

V=

.5,a

lph

a=

.08

,lam

=.5

,c=

.06

,be

ta=

.5,k

ap

=.5

,nu

=.5

,kce

=.3

,tau

=1

)

Loan-cor(R,V)=0.15

Loan-cor(R,V)=0.05

Loan-cor(R,V)=0.45

Bond-cor(R,V)=0.15

Bond-cor(R,V)=0.05

Bond-cor(R,V)=0.45

Empirical Methodology: Calibration of Models

• Estimate parameters of different models for LGD by full-information maximum likelihood (FIML) • Consider LGD implied in the market at time of default ti

D for the ith instrument in recovery segment s, denoted LGDi,s,tiD as expected, discounted (at ri,s

D) ultimate LGDi,s,tiE at time of emergence tiE given by a model m (w/parameters θs,m) :

• Since cannot observe expected recovery prices ex ante we invoke market rationality: i.e., for a homogenous recovery segment expect that the normalized forecast error should be “small”:

• A “unit-free” measure of recovery uncertainty normalized by the root of the time-to-resolution: idea being an economy with information revealed independently & uniformly over time at a rate of 1/sqrt(T)

, ,

, ,

, ,, ,

,1

Ei

D E Di i s i s

Pt i s t P

s m s mi s t t tDi s

E LGDLGD LGD

r

θ

, , , ,,

, ,, ,

Ei

Di

Ps m s m i s t

i s E Di s i si s t

LGD LGD

LGD t t

θ

Empirical Methodology: Calibration of Models (continued)

• Assuming that the errors are standard normal use full-information maximum likelihood: maximize the log-likelihood (LL) function:

• Equivalent to minimizing the squared normalized forecast errors:

• Derive a measure of uncertainty of our estimate by the ML standard errors from the Hessian matrix evaluated at the optimum:

, ,

,

, , ,1

, , , ,

1 , ,, ,

arg max arg max log

arg max log

Ds

s m s m

Ds E

i

s m Di

NP

s m i s s mi

PNs m s m i s t

E Di i s i si s t

LGD LL

LGD LGD

LGD t t

θ θ

θ

θ

θ

, ,

2

, , , , 2, , ,

1 1, , , ,

1arg min arg min

s sEi

Ds m s mi

PN NP s m s m i s ts m i s mE D

i ii s i s i s t

LGD LGDLGD

t t LGD

θ θ

θ

,

,

, ,

1

22

ˆ

,

ˆ

ˆs m

s m

s m s m

Ts m

LL

θ

θ θ

Σθ θ

Empirical Results: Data Description

• Moody’s Ultimate Recovery Database™ (“MURD”) August 2009 (4050 instruments 1985-2009 for 776 obligors & 811 defaults)

• Debt prices on bonds and loans of defaults (bankruptcies & out-of-court) of U.S. large corporates (mostly rated debt & traded equity)

• Merged with various other information sources: Compustat, CRSP, www.bankruptcydata.com, Edgar SEC filing, LPC DealScan

• Observations: instrument characteristics, complete capital structure, obligor demographics, key quantities & dates for LGD calculation

• Resolution types: emergence from bankruptcy, Chapter 7 liquidation, acquisition or out-of-court settlement

• Recovery / LGD measures: prices of pre-petition (or received in settlement) instruments at emergence or restructuring

• Sub-set: prices of traded debt at around default (30-45 day avg.) & equity prices prior to default to compute CARs

Empirical Results: Summary by Instrument & Default Event Type

• Overall mean LGD at default (ultimate) 54.7% (50.6%) & returns 28.6% • Loans have slightly lower LGDs default (ultimate) 52.5% (49.3%) than bonds

56.0% (51.3%) & higher returns 32.2% vs. 26.4%• Bankruptcies have higher LGDs default (ultimate) 55.7% (51.6%) than

restructurings 37.7% (33.8%) & lower returns 28.1% vs. 37.3%

Count Average

Standard Error of the Mean Count Average

Standard Error of the Mean Count Average

Standard Error of the Mean

Return on Defaulted Debt1 25.44% 3.75% 44.22% 21.90% 26.44% 3.74%

LGD at Default2 57.03% 1.97% 37.02% 5.40% 55.96% 1.88%

Discounted LGD3 52.44% 1.30% 30.96% 3.00% 51.30% 1.25%


LGD at Default2 53.31% 9.90% 38.86% 7.22% 52.50% 2.32%

Discounted LGD3 50.00% 1.68% 38.31% 5.79% 49.34% 1.62%


LGD at Default2 55.66% 0.86% 37.72% 3.12% 54.69% 0.84%

Discounted LGD3 51.55% 1.03% 33.76% 2.89% 50.58% 0.99%

Loan

s

485 29 514

Tot

al

1322 76 1398

Bon

ds

837 47 884

Table 1 - Characteristics of Loss-Given-Default and Return on Defaulted Debt Observations by Default and Instrument Type

(Moody's Ultimate Recovery Database 1987-2009)Bankruptcy Out-of-Court Total

Empirical Results: Summary by Seniority Class & Collateral Group

• For this data-set, rank ordering of mean LGD ( default or ultimate) not “intuitive” by collateral quality

• But rank ordering of mean LGD by seniority class in line with expectations

• Returns on defaulted debt generally increasing in seniority rank or collateral quality (more recovery risk?)

Seniority Class Collateral Type

Cash, Accounts

Receivables &

Guarantees

Inventory, Most

Assets & Equipment

All Assets & Real Estate

Non-Current

Assets & Capital Stock

PPE & Second

Lien

Unsecured & Other Illiquid

CollateralTotal

UnsecuredTotal

SecuredTotal

Collateral

Count 39 8 367 38 29 33 32 482 514

LGD at Default 66.81% 46.60% 51.95% 59.94% 55.02% 45.63% 46.25% 53.79% 53.31%

Ultimate LGD 64.38% 56.03% 48.58% 50.62% 56.53% 30.70% 31.78% 50.51% 49.34%

Returns 22.57% -5.80% 33.49% 35.68% 46.07% 22.39% 19.77% 33.03% 32.21%

Count 2 38 41 35 7 142 3 139 142

LGD at Default 61.50% 40.19% 36.02% 62.99% 61.24% 51.67% 50.73% 51.59% 51.57%

Ultimate LGD 76.81% 23.87% 36.67% 46.70% 60.32% 49.68% 50.15% 34.88% 35.04%

Returns 23.86% 47.53% 35.03% 55.99% 14.33% 17.44% -27.66% 38.02% 36.63%

Count 0 0 1 0 1 459 452 9 461

LGD at Default 0.00% 0.00% 85.00% N/A 80.00% 55.83% 55.94% 56.63% 55.96%

Ultimate LGD 0.00% 0.00% 78.76% N/A 74.25% 48.33% 38.14% 32.03% 38.00%

Returns 0.00% 0.00% 86.47% n 119.64% 23.40% 23.71% 25.62% 23.75%

Count 0 0 1 0 1 159 158 3 161

LGD at Default 0.00% N/A 85.00% N/A 90.50% 58.09% 57.98% 83.46% 58.48%

Ultimate LGD N/A N/A 74.72% N/A 97.74% 54.51% 36.50% 40.47% 36.46%

Returns 0.00% N/A 57.45% N/A -45.98% 33.57% 31.01% 150.30% 33.23%

Count 0 1 0 0 0 119 117 3 120

LGD at Default N/A 27.33% 0.00% N/A N/A 66.15% 66.58% 37.42% 65.81%

Ultimate LGD N/A 20.15% 0.00% N/A N/A 65.36% 33.62% 32.77% 33.54%

Returns N/A 72.13% 0.00% N/A N/A 15.11% 15.74% 9.49% 15.59%

Count 41 28 407 79 66 777 762 636 1398

LGD at Default 66.53% 41.57% 50.55% 61.56% 59.31% 57.41% 57.58% 53.40% 55.66%

Ultimate LGD 64.98% 32.93% 47.60% 48.58% 59.43% 51.46% 37.69% 36.13% 36.99%

Returns 22.63% 33.17% 33.82% 46.22% 28.96% 24.12% 34.46% 23.63% 28.56%

Table 2 - Losses-Given-Default and Returns on Defaulted Debt by Seniority Classes and Collateral Groups

(Moody's Ultimate Recovery Database 1987-2009)

Rev

olvi

ng

Cre

dit

/ T

erm

Lo

an

Sen

ior

Sub

ordi

nate

d B

onds

Juni

or

Sub

ordi

nate

d B

onds

Sen

ior

Sec

ured

B

onds

Sen

ior

Uns

ecur

ed

Bon

ds

Tot

al

Inst

rum

ents

Empirical Results: FIML Estimates• Overall recovery volatility

increasing in seniority 16.1 to 31.1%

• Recovery volatility to PD

- β (LGD - ν) side risk increasing in seniority class 10.0% to 18.2% (12.4% to 38.8%)

• But proportion of total variance attributable to LGD side increasing in seniority 59.4% to 20.0%

• Firm-value volatility σ increasing in seniority class 4.3% to 9.1%

• Mean-reversion speed in random drift of recovery κα humped in seniority class (3.3% sub, 5.5% sen-unsec.,4.0% for loans)

• Long-run mean of the random drift in the recovery α increasing in seniority 18.8% to 37.1% -> greater expected return of recovery lower ELGD paper

• Volatility of random drift in recovery ηα increasing in seniority 18.7% to 48.9% -> greater volatility in expected return lower ELGD instruments

Parameter β (6) ν (7) σR (8) πR

β (9) πR

ν (10) σ (βσ)0.5 κα

(11) α (12) ηα (13) ς (14)

Estimate 18.16% 36.83% 9.05% 12.82% 3.96% 37.08% 48.85% 20.88%Std Err 0.73% 1.37% 0.6192% 0.4190% 0.0755% 4.25% 3.21% 0.9215%

Estimate 16.54% 30.41% 8.19% 11.64% 4.40% 33.66% 44.43% 18.99%Std Err 0.60% 1.31% 0.6216% 0.7448% 0.0602% 3.51% 2.69% 0.8297%

Estimate 13.82% 24.38% 6.82% 9.71% 5.50% 28.07% 37.04% 15.83%Std Err 1.39% 1.99% 0.5993% 0.6165% 0.0281% 2.89% 2.24% 0.6504%

Estimate 12.02% 17.35% 5.47% 7.76% 4.42% 22.45% 29.68% 12.69%Std Err 1.05% 1.04% 0.5314% 0.9775% 0.0181% 2.01% 2.01% 1.00%

Estimate 10.24% 12.37% 4.32% 5.97% 3.34% 18.80% 18.69% 9.43%Std Err 1.0128% 1.08% 0.5474% 0.9142% 0.0106% 2.05% 2.00% 1.01%

Table 3 - Full Information Maximum Likelihood Estimation of Option Theoretic Two-Factor Structural Model of Ultimate Loss-Given-Default with Optimal Foreclosure Boundary, Systematic Recovery Risk and Random Drift in

the Recovery Process (Moody's Ultimate Recovery Database 1987-2009)

19.55% 80.45%

Senior Secured Bonds 34.62% 22.83% 77.17%

Recovery Segment

Se

nio

rity Cla

ss

Revolving Credit / Term Loan 41.06%

Senior Subordinated Bonds 21.11% 32.43% 67.57%

Senior Unsecured Bonds 28.02% 24.30% 75.70%

59.34%Subordinated Bonds

16.06% 40.66%

Value Log-Likelihood Function -371.09

P-Value of Likelihood Ratio Statistic - All Estimates 4.69E-03

Degrees of Freedon 1391

In-S

am

ple

/ T

ime

D

iag

nost

ic

Sta

tistic

s Area Under ROC Curve2 93.14%Komogorov-Smirnov Statistic (P-

Values)3 2.14E-08

McFadden Pseudo R-Squared4 72.11%Hoshmer-Lemeshow Chi-

Squared (P-Values)5 0.63

• Correlation of random drift & level of recovery ς increasing in seniority 9.4% to 20.9%

• Correlation of default & recovery process (βσ)1/2 increasing in seniority 6% to 12.8%

• Estimates statistically significant & magnitudes distinguishable across segments at conventional significance levels

• LR statistics indicate reject the null that all estimates are zero

• Model performs well-in sample & more elaborate models represent meaningful improvements over the simpler models

• AUROCs high by commonly accepted standards across models

• KS: very small p-values -> adequate separation in distributions low/high LGD

• MPR2 high by commonly accepted standards across models

• HL: high p-values -> high accuracy to forecast cardinal LGD

Empirical Results: FIML Estimates (continued)

Downturn LGD

• LGD mark-up declining in ELGD -> hirer recovery tail risk in lower ELGD paper

• Multiple higher for bonds vs. loans, higher PD-LGD correlation, collateral specific or volatility in the drift of the recovery rate drift process (differences narrow for higher ELGD

0.0 0.2 0.4 0.6 0.8 1.0

12

34

5

Figure 14: Ratio of Ultimate Downturn to Expected LGD vs. ELGD at 99.9th Percentile of PD-Side Systematioc Factor Z

Stochastic Collateral & Drift Merton Model (Parameters Set to MLE Estimates for Loans & Bonds)ELGD

DL

GD

/EL

GD

Loans/corr(R,V)=0.1

Loans/corr(R,V)=0.2

Loans/cor(R,V)=0.05

Bonds/corr(R,V)=0.1

Bonds/corr(R,V)=0.2

Bonds/cor(R,V)=0.05

0.0 0.2 0.4 0.6 0.8 1.0

12

34

5



DL

GD

/EL

GD

Loans/stdev(R|V)=0.1



Bonds/stdev(R|V)=0.1



0.0 0.2 0.4 0.6 0.8 1.0

12

34

5



DL

GD

/EL

GD

Loans/stdev(alpha)=0.1



Bonds/stdev(alpha)=0.1



Model Validation• Validate preferred 2FSM-SR&RD by out-of-sample and out-of-time rolling

annual cohort analysis • Augment this by resampling on both the training and prediction samples, a

non-parametric bootstrap (Efron [1979], Efron and Tibshirani [1986]) • Analyze distribution of key diagnostic statistics Spearman rank-order

correlation & Hoshmer-Lemeshow Chi-Squared (HLCQ) P-values• Alternative #1 for predicting ultimate LGD: full-information maximum

likelihood simultaneous equation regression model (FIMLE-SEM) built upon observations in URD at instrument & obligor level (Jacobs et al, 2010)

• 2nd alternative non-parametric estimation of a relationship with several independent variables & a bounded dependent variable• Boundary bias with standard non-parametric estimators using Gaussian kernel

(Hardle and Linton (1994) and Pagan and Ullah (1999)).

• Chen (1999): BKDE defined on [0,1], flexible functional form, simplicity of estimation, non-negativity & finite sample optimal rate of convergence

• Extend (Renault and Scalliet, 2004) to GBKDE: density is a function of several independent variables (->smoothing through dependency of beta parameters)

Model Validation (continued)

VariablePartial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -0.0903 2.55E-03Book Value -0.0814 0.0174Tobin's Q 0.0729 8.73E-03Intangibles Ratio (Industry Adjusted) 0.0978 7.02E-03Working Capital / Total Assets -0.1347 4.54E-03Operating Cash Flow -8.31E-03 0.0193Profit Margin (Industry) -0.0917 1.20E-03Senior Secured 0.0432 0.0482Senior Unsecured 0.0725 3.11E-03Senior Subordinated 0.2266 1.21E-03Junior Subordinated 0.1088 0.0303Collateral Rank 0.1504 4.26E-12Percent Debt Above 0.1241 3.84E-03Percent Debt Below -0.2930 7.65E-06Time Between Defaults -0.1853 7.40E-04Time-to-Maturity 0.0255 0.0084

Number of Creditor Classes -0.0975 1.20E-03

Percent Secured Debt -0.1403 7.56E-03Percent Bank Debt -0.2382 7.45E-03Investment Grade at Origination -0.0720 4.81E-03Principal at Default 8.99E-03 1.14E-03 1.35E-02 3.02E-02Industry - Utility -0.1506 8.18E-03Industry - Technology 0.0608 2.03E-03 0.0343 5.72E-03Cumulative Abnormal Returns -0.2753 1.76E-04Ultimate LGD - Obligor 0.5643 7.82E-06LGD at Default - Obligor 0.1906 4.05E-04LGD at Default - Instrument 0.2146 1.18E-14

Prepackaged Bankruptcy -0.0406 5.38E-03

Bankruprtcy Filing 0.1835 3.70E-13 0.1429 5.00E-03

1989-1991 Recession 0.0678 0.04742000-2002 Recession 0.0930 1.09E-02 0.1074 0.0103Moody's Speculative Default Rate 0.0726 1.72E-04S&P 500 Return -0.0139 2.88E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 1.72E-10 9.60E-08 1.72E-10 9.60E-08Pseudo R-Squared 0.6997 0.6119 0.5822 0.4744Hoshmer-Lemeshow 0.4115 0.3345 0.5204 0.3907Area under ROC Curve 0.8936 0.7653 0.8983 0.7860Kolmogorov-Smirnov 1.12E-07 4.89E-06 1.42E-07 6.87E-06

Con

trac

tual

Tim

e

Table 7 - FIML Simulataneous Equation Regression Analysis of Discounted Instrument and Obligor LGD

(Moody's Ultimate Recovery Database 1987-2009)

Cat

egor

y Instrument Obligor

Mac

roD

iagn

ostic

sC

apita

l S

truc

tur

e

Cre

dit Q

ualit

y /

Mar

ket

Lega

lF

inan

cial

• Model endogeneity of LGD at the firm & instrument levels

• Help understand LGD structural determinants & improve our forecasts

• 199 observations from the URD™ with variables• Financials: leverage ratio, book value

of assets (BVA), intangibles ratio, interest coverage ratio, free cash flow to BVA, profit margin

• Capital structure: number of major creditor classes, percent secured debt

• Credit: Altman Z-Score, debt vintage

• Macro: Moody's 12 month trailing speculative grade default rate

• Industry dummy

• Resolution: filing district and a pre-packaged bankruptcy dummies

Model Validation (continued)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Pearson Correlation

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Pro

babi

lity

Den

sity

Fig. 17 - Densities of Pearson Correlations for LGD Prediction100,000 Repetitions Out-of-Sample and Out-of-Time 1997-2008

Simulataneous Equation Regression Model2-Factor Merton Structural ModelNon-parametric Beta Kernel Density Model

0.1 0.3 0.5 0.7 0.9

P-Value of HL Statistic

0

2

4

6

8

Pro

babi

lity

Den

sity

Fig.18 - Densities of Hoshmer-Lemeshow P-Values for LGD Prediction100,000 Repetitions Out-of-Sample and Out-of-Time 1997-2008

Simulataneous Equation Regression Model2-Factor Merton Structural ModelNon-parametric Beta Kernel Density Model

Test Statistic Model GBKDE4 2FSM-SR&RD5 FIMLE-SEM6

Median 0.7198 0.7910 0.8316Standard Deviation 0.1995 0.1170 0.10545th Percentile 0.4206 0.5136 0.580395th Percentile 0.9095 0.9563 0.9987Median 0.1318 0.2385 0.2482Standard Deviation 0.0720 0.0428 0.03385th Percentile 0.0159 0.0386 0.040895th Percentile 0.2941 0.5547 0.5784

Table 8 - Bootstrapped1 Out-of-Sample and Out-of-Time Classification and Predictive Accuracy Model Comparison Analysis of Alternative Models for

Ultimate Loss-Given-Default (Moody's Ultimate Recovery Database 1987-2009)

Out

-of-

Sam

ple

/ T

ime

1 Y

ear

Ahe

ad

Pre

dict

ion Spearman Rank-

Order Correlation2

Hoshmer-Lemeshow Chi-Squared (P-

Values)3

Model Validation (concluded)• While all models perform decently out–of-sample in rank ordering, FIMLE-

SEM performs best, GBKDE worst & 2FSM-SR&RD in the middle (medians = 83.2, 72 & 79.1%, resp.)• It is also evident from the table and figures that the better performing models are

also less dispersed and exhibit less multi-modality

• However, the structural model is closer in performance to the regression model by the distribution of the Pearson correlation

• Out-of- sample predictive accuracy is not as encouraging for any of the models (in a sizable proportion of the runs we can reject adequacy of fit)

• Rank ordering of HL same as Pearson: FIMLE-SEM best (median = 24.8%), GBKDE worst (median = 13.2%), 2FSM-SR&RD middle (median = 23.9%)

• Structural model developed herein is comparable in out-of-sample predictive accuracy to the high-dimensional regression model

• While all models are challenged in predicting cardinal levels of ultimate LGD out-of-sample, remarkable that a parsimonious structural model of ultimate LGD can perform so closely to a highly parameterized econometric model

Summary of Contributions and Major Findings

• Developed a theoretical model for ultimate LGD with many intuitive & realistic features in the structural credit risk model framework

• Extension admits differential seniority, optimal foreclosure boundary, independent undiversifiable recovery risk process with stochastic drift

• Analysis of comparative statics: ultimate LGDs increasing in recovery volatilities, drifts on random return and PD-LGD correlation

• Empirical analysis: calibrated alternative models for ultimate LGD on instruments with trading prices & valkues at resolution of default using Moody’s URD™• 800 defaults are largely representative of the US large corporate loss

experience with complete capital structures & recoveries on all instruments to the time of default to the time of resolution

• Estimates vary significantly across models & recovery segments• Estimated volatilities of the recovery rate Z& their random drift are

increasing in seniority (bank loans vs. bonds)• Reflects the inherently greater risk in the ultimate recovery for higher

ranked instruments having lower expected loss severities

Summary of Contributions and Major Findings (continued)

• In an exercise relevant to advanced IRB under Basel II, analyzed quantification of a downturn LGD • Finding later to be declining for higher expected LGD & lower ranked

instruments• Increasing in the correlation between the process driving firm default and

recovery on collateral • Validated our leading model in an out-of-sample bootstrapping

exercise, comparing it to two alternatives• Benchmarks: high-dimensional regression model & non-parametric, both

based upon the same URD data • We found our model to compare favorably in this exercise

• Conclusion: our model is worthy of consideration to risk managers & supervisors concerned with advanced IRB under the Basel II• A benchmark for internally developed ultimate LGD models, as can be

calibrated to LGD observed at default (either market prices or model forecasts) & ultimate workout LGD

• Risk managers can use our model as an input into internal credit capital models

Lgd Model Jacobs 10 10 V2[1]

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