May 30, 2010

An Exposure at Default Model for Contingent Credit Lines

Pinaki BagUnion National Bank, United Arab Emirates

Michael Jacobs, Jr.Credit Risk Analysis Division

U.S. Office of the Comptroller of the Currency

The views expressed herein are those of the authors and do not necessarily represent the views of either Union National Bank, UAE or of the U.S. Office of the

Comptroller of the Currency.

EAD Modeling

• Vital building block for economic capital or regulatory (Basel II) capital

• Attempt to develop a parsimonious theoretical model with inputs from empirical study or expert opinion

Outline

1 Introduction - Motivation

2 Review of the Literature

3 The Model

4 Numerical Experiment

5 Conclusions

Introduction -Motivation Why is this important? What have been the challenges?

Probability of Default (PD)

Loss Given Default (LGD)

Exposure atDefault (EAD)

Basel II - 101

Why Modeling EAD?

Basel II regulatory capital is a function of PD, LGD and EAD – but EAD

and LGD has potentially larger impacts than PD

Contingent Credit Lines (CCL) are modeled using the Basel II suggested

Credit Conversion Factor (CCF) for capital calculation

AIRB allows banks to compute their own estimates of EAD for CCL,

provided these can be supported empirically

-

200,000

400,000

600,000

800,000

1,000,000

1,200,000

Probability of Default Loss Given Default Exposure at Default

Why Modeling EAD?The FDIC as of 9-09 reports close to 80% of all C&I loans are CCLs with outstanding close to $1.9 TrillionPopularity of CCLs attributed to financial flexibility (Avery & Berger, 1991), hedging (Kanatas, 1987), management of working capital (Hawkins, 1982)

Paucity of models Unsuitability of external dataInconsistency of internal data

Challenges (FSA,2007)

What We Did?

Portfolio

Segments

Segment Level

Usage

Unused Obligor

Limits

Each CCL as Portfolio of Put

Options

Basic CreditRisk+

Algorithm Fast Fourier

Transform

Moody's DRS

Database (Current Sample

Portfolio)

Moody's MURD Database & Compustat

(Reference Data for CCF

Estimates)

Review of the Literature What has been done? How have they been applied?

Review of the Literature

Thakor et al. (1981): option-theoretic CCL pricing as puts written by the bank & measure the sensitivity to interest ratesKaplan and Zingales (1997) & Gatev and Strahan (2003): empirical evidence that drawdowns on CCLs increase when firms more liquidity constrained or CP-Tbill rate spread rises

Jones and Wu (2009): model credit quality as a jump-diffusion process, draw-down & pricing functions of the difference between an opportunity rate & marginal cost of line borrowings

Review of the LiteratureEmpirical literature on additional partial draw-downs prior to default find

a decline as credit quality worsens

a.Asarnow & Marker (1995): Citibank agency rated firms 1987-1992

b.Jacobs & Araten (2001): JPMC internal rated firms 1995-2000

c.Agarwal et al (2005): HELC in the U.S. market

d.Jacobs (2009): Agency rated & marketable debt 1987-2008

e.Jiménez et al (2009): All C&I loans in Spain 1984-2005

Martin and Santomero (1997) study CCL pricing from the demand side

of firms & show CCL usage depends on firm's business growth

potential & uncertainty of such

Review of the LiteratureMoral (2006) examines modeling issues from a supervisory point of view &

analyzes different EAD risk measures

Sufi (2008) reports that firms with low cash flow or high cash flow volatility

rely more heavily on cash rather than credit lines

Jacobs (2009) finds that utilization is a stronger inverse driver than rating &

that EAD risk may be counter-cyclical

But Jiminez (2009) reports higher utilization for defaulting vs. non-

defaulting firms up to 3 years prior to default

Qi (2009) examines credit card usage in U.S. & finds borrowers are more

active than lenders in this game of “race to default”

Several of these studies report importance of macro factors, size of credit

line, borrower financials, collateralization, etc.

The Model What we did?

Model Overview

Individual obligors

Sub-segment

Segment

Segment Level Usage

Obligor Level Unused Limits

Portfolio

Each obligor’s CCL is modeled as portfolio of large number of put options to determine usage

Similar put size obligors are clubbed under each sub-segment

Each sub-segment having similarexpected usage are combined todetermine segment level usage

FFT used to convolute each segment to the overall portfolio usage distribution

Obligor Level Partial Draw-downsAssume obligor A, with a CCL having unused limit LA, has a very large number (n) of put options to exercise, which determines the level of partial draw-down. The size of each put can be given as:

(4)

• The amount of partial draw-down is r XQA ,where r is the number of puts exercised by A in the time horizon, from which it follows that the probability generating function (PGF) of r is defined as:

(5)

• Assume that expected usage of the CCL is αA LA, so the average number of puts used by A is:

(6)

Individual obligors

Sub-Segment Level Partial Draw-downsA Poisson process of exercise of each option makes the PGF:

(7)

Assuming independence of m≤N obligors in the portfolio having put size Q' the PGF for r number being exercised is:

(8)

Assume the overall expected additional usage on the unused in the segment is α & the unused limits of the m obligors to be LA:

(9)

Individual obligors

Sub-segment

Let and hence as this sub-segment has all put size equal to Q

(11)

The Poisson assumption implies the sub-segment PGF where each sub-segment i as

(15)

S=∑A=1

m

λA

Sub-Segment Level Partial Draw-downs

Individual obligors

Sub-segment

To find the overall segment usage distribution we convolute t sub-segments; assuming independence of each, the PGF is

(16)

Segment Level Partial Draw-downs

The segment exposure distribution follows from Taylor’s theorem

(17)

and Leibnitz’s nth order differentiation rule noting the fact that is constant.

∑i=1

t

S i

Individual obligors

Sub-segment

Segment

Hence, letting and after few algebraic manipulations we will have

(24)

Portfolio Level Partial Draw-downs

We can solve the above equation iteratively noting the fact that

Each Segment level usage distribution will than be convoluted using a standard Fast Fourier Transform to arrive at portfolio level usage distribution

W 0=e−∑

i=1

t

S i

Individual obligors

Sub-segment

Segment

Portfolio

Portfolio SegmentationThis is vital step required for apt implementation of the discussed algorithm

This may be done in various ways depending upon rating, product criterion, industry etc.Bank may segregate borrowers by keeping high commitment fees and low service fees in one contract, and low commitment fees and high service fees in another contract (Thakor and Udell,1987)Contract choice may not be always that simple, since it may also depend upon structure of the borrower’s industry (Maksimovic ,1990)

Numerical Experiment Experiment with Moody’s Data of

Contingent Credit Lines

Numerical Experiment with Moody's Data

For a typical CCL portfolio ∑Si may quite be large & we trying to

assign a probability to each dollar of usage, so calculation of a negative exponential of this leads to precision issues

I.e., the double-precision settings of common software applications

under default settings approximates W0 as zero, upon which

derivation of the usage distribution depends

Potentially many alternatives exist to circumvent the problem, such as use of libraries which can handle very high precision calculations, detailed discussion of which is beyond our scope

Herein we use Linux based Genius 1.0.7 as arbitrary precision calculator & Linux based Octave for FFT and final distribution evaluation

Numerical Experiment with Moody's Data

To illustrate we chose 2 sample segments of 13 obligors each with α = 65% & 40% for investment & junk grade, respectively

Taken randomly from Moody' Default Risk Service (DRSTM) database of CCLs rated as of 12/31/2009

Limits of each obligor varying from $25 MM to $235 MM

The values of α from Jacobs (2009) based upon estimated additional drawdowns on unused limits (or “LEQ” factors)

Moody's rated CCLs 1987-2009 defaulting withing a 1-year horizon in Moody's Ultimate Recovery Database (MURDTM)

Trace CCL usage prior to default in COMPUSTAT and Edgar SEC filings

Each of the obligor's limit is divided into 1,000 puts eachE.g., a CCL with limit $50mm is 1,000 puts with strike of $50,000 each

Numerical Experiment with Moody's Data

Investment Grade Number of Puts is set at 1000

Issuer Number Issuer Name Limit($ '000) Put Size ($ '000)

Moody's Senior Unsecured Credit

RatingMoody's Broad

Industry Category Debt Type Description153000 Central Maine Power Company 50,000 50 Baa1 PUBLIC UTILITY Revolving Credit Facility191670 Commercial Metals Company 235,000 235 Baa2 INDUSTRIAL Revolving Credit Facility

232000 Detroit Edison Company (The) 68,750 69 Baa1 PUBLIC UTILITY Revolving Credit Facility

252000 Duquesne Light Company 100,000 100 Baa2 PUBLIC UTILITY Revolving Credit Facility

404000 Indianapolis Power & Light Company 120,600 121 Baa2 PUBLIC UTILITY Revolving Credit Facility

490000 Michigan Consolidated Gas Company 81,250 82 A3 PUBLIC UTILITY Revolving Credit Facility576000 Orange and Rockland Utilities, Inc. 100,000 100 Baa1 PUBLIC UTILITY Revolving Credit Facility

687000 South Carolina Electric & Gas Company 75,000 75 Baa1 PUBLIC UTILITY Revolving Credit Facility

769000 Tucson Electric Power Company 120,000 120 Baa3 PUBLIC UTILITY Sr. Sec. Revolving Credit Facility

600045390 IDACORP, Inc. 250,000 250 Baa2 PUBLIC UTILITY Revolving Credit Facility

600050191 Rayonier Forest Resources, L.P. 50,000 50 Baa3 REAL ESTATE FINANCEGtd. Revolving Credit Facility600064222 Michigan Electric Transmission Company, LLC 25,000 25 Baa1 INDUSTRIAL Sr. Sec. Revolving Credit Facility

808653810 NASDAQ OMX Group, Inc. (The) 150,000 150 Baa3 SECURITIES Sr Sec 1st Lien Rev Credit Facility

Junk Grade Number of Puts is set at 1000

Issuer Number Issuer Name Unused Limit($ '000) Put Size ($ '000)

Moody's Senior Unsecured Credit

RatingMoody's Broad

Industry Category Debt Type Description

600040059 Accuride Corporation 212,000 212 C INDUSTRIALSr. Sec. Revolving Credit Facility/Gtd 1st Lien Sr Sec Revolver

809883143 Peach Holdings, Inc. 35,000 35 C FINANCE Gtd. Sr. Sec. Revolving Credit Facility820360433 Bravo Health, Inc. 25,000 25 B2 INSURANCE Sr. Sec. Revolving Credit Facility

199515 Conseco, Inc. 80,000 80 Caa3 INSURANCE Sr. Sec. Revolving Credit Facility600058771 GSCP (NJ ), L.P. 60,000 60 C OTHER NON-BANK Gtd. Sr. Sec. Revolving Credit Facility

566800 Covanta Energy Corporation 100,000 100 Ba3 PUBLIC UTILITY Gtd 1st Lien Sr Sec Revolver

807760066 Interstate Operating Company, L.P. 140,000 140 Caa3 REAL ESTATE FINANCEGtd. Sr. Sec. Revolving Credit Facility

820399560 TPG- Austin Portfolio Holdings LLC 100,000 100 Ca REAL ESTATE FINANCEGtd. Sr. Sec. Revolving Credit Facility

431200Kansas City Southern Railway Company (The) 100,000 100 B2 TRANSPORTATION Gtd. Sr. Sec. Revolving Credit Facility

809492974 Standard Steel, LLC 20,000 20 Caa1 TRANSPORTATION Gtd 1st Lien Sr Sec Revolver600038850 AEP Industries, Inc. 100,000 100 B2 INDUSTRIAL Gtd. Revolving Credit Facility600042238 Alliance Laundry Systems LLC 230,000 230 B3 INDUSTRIAL Gtd. Sr. Sec. Revolving Credit Facility

44000 American Greetings Corporation 75,000 75 B1 INDUSTRIAL Gtd. Sr. Sec. Revolving Credit Facility

Numerical Experiment with Moody's Data: Results

Investment Grade Junk Grade 65% 40%

Unused Limits ($ 000) 1,425,600 1,277,000 2,702,600Mean ($ 000) 926,640 510,800 1,437,400Standard Deviation( $ 000) 11,735 8,374 14,417Skewness 0.015698 0.020199 0.012426Kurtosis 3.0003 3.0005 3.0002

Convoluted Portfolio

The convoluted distribution has both higher mean and higher standard deviation than either the segments

However, the distributional statistics reveal these to be near Gaussian, which we would like to overcome in future extensions of the model

Numerical Experiment with Hypothetical Portfolio: Sensitivity Analysis

The standard deviation of the usage distribution decreases as we increase the number of puts used

May be explained that we assuming a known value of a in our model

Mean remains relatively stable but the extreme points converge

Additional usage rate also increases the volatility of the exposure distribution

To incorporate volatility in the model we can also use a mixed Poisson process

Commonly used distributions include Gamma, resulting in negative binomial

Argument against this is induces a second set of assumptions in our model

Variation of Usage Distribution ($) parameters with

Percentile n=700 n=800 n=900 n=1000 n=1100 n=1200 n=1300 n=1400 n=1500

50.00% 14,718 14,720 14,722 14,723 14,724 14,726 14,726 14,726 14,727

99.00% 17,272 17,100 16,970 16,849 16,741 16,662 16,583 16,520 16,460

99.50% 17,557 17,364 17,219 17,084 16,965 16,874 16,788 16,718 16,651

99.75% 17,823 17,612 17,452 17,304 17,173 17,074 16,978 16,903 16,829

99.90% 18,151 17,916 17,739 17,574 17,429 17,320 17,214 17,130 17,048

Standard Deviation 1,058 988 936 886 842 809 777 751 726

Variation of Usage Distribution ($) parameters with n

Percentile

50.00% 14,724 29,459 58,929 73,664 88,400 103,135 117,870

95.00% 16,214 31,552 61,876 76,955 92,001 107,023 122,024

99.00% 16,850 32,439 63,116 78,338 93,513 108,652 123,764

99.90% 17,575 33,445 64,519 79,901 95,220 110,492 125,728

99.97% 17,904 33,899 65,151 80,603 95,987 111,319 126,610

Standard Deviation 886 1,253 1,722 1,981 2,170 2,344 2,506

Conclusions So What? Where do we go from here?

Conclusions and Directions for Future Research

We formulated a parsimonious model for the estimation of portfolio level EAD in a typical CCL portfolio each as a portfolio of option instruments

Exercise of each has been modeled as a standard Poisson process where average additional usage α is assumed to be known.

Previous literature indicates α probably depends on obligor credit quality, “race to default”, pricing, utilization, etc.

Our algorithm accommodates different values of α to model this model this correlation, as the portfolio may be segmented by criterion of the bank

Various methods for estimating α have been outlined in the literature likely to work best for banks if this is based upon internal research

Further work may also be needed so that stable distribution parameters can be determined which will not be affected by choice of the number of puts used.

Most current credit risk models have a constant EAD as economic credit VaR input and stochastic exposures make a notable difference in capital

Conclusions and Directions for Future Research (continued)

Accurate EAD calculation is fundamental for liquidity risk management which poses a challenge to risk managers

E.g., HELC where all the accounts are undrawn but committed lines

This algorithm may prove helpful in providing insight into the problem

The other implication of the algorithm is EAD estimation for Basel II: as compared to PD, there has been limited research into this

Can provide a foundation for the banks under AIRB approach to Basel II.

This algorithm may also be used in stress testing for a worst case liquidity scenarios for the portfolio, as we have the complete distribution of usage

We can get a good estimate of our worst case scenarios from 99th or 99.9th percentile depending upon the risk appetite of the bank

We need further future work to improve the algorithm so as to use it in standard software applications with minimized hardware requirements

Finally, generalization of the Poisson assumption in order to model non-normality of the exposure distribution