a structural model of the term structure of credit spreads with stochastic recovery and contractual...

A Structural Model of the Term Structure of Credit Spreads with Stochastic Recovery and Contractual Design

1999

Prof. Hugues PirotteSolvay Business School

Université Libre de Bruxelles

Les Journées Bachelier – Journée Risque de CréditDépartement de Mathématiquesde l’Université d’EvryFebruary 7th, 2003

Special acknowledgments to Didier Cossin, Hélyette Geman, Rajna Gibson, Eduardo Schwartz, Suresh Sundaresan for their

contribution, insights, comments and review on the present paper

Friday, February 7th 2003

Les Journées Bachelier - Université d'Evry

2

Credit Risk Research Environment: Orientations

• Mitigation vs Pricing vs Behavioral studies• Structural vs Reduced-Form techniques• Determinants of Credit Risk

– Systematic vs specific risk sources

– Impact of Macro-variables

– Credit vs Liquidity vs Concentration risks

– Impact of legal rules/reputation effects on other sanctions

• The to components: P(def)+LGD• Credit risk term-structures• Reaction of models to changes in creditworthiness



3

Credit Risk Research Environment: Uses

• Risk in general in Asset Pricing• Corporate Credit Risk

– Structural default

– Cash-flow insolvency

• Sovereign Credit Risk+ Firm or Country growth linked to debt levels

- Impact of sanctions/Loss of reputation/Cuts in production or exports

• Integration of Market and Credit Risks Portfolio Management



4

Credit Risk Research Environment: Components

1. Default Boundary• Exogenous triggers : debt face value

• Endogenous triggers : debt service

2. Recovery-at-default• Ex-ante percentage depending on rating

• Variable with some-randomness

• Linked to default probability

• Endogenously derived

3. Strategic rules• Identification of a gain in defaulting on purpose!

UnwantedDefault

Default asan optimalsolution



5

Credit Risk Research Environment: Pricing Models

Security

MKTDATA

ECO.DATA

FIRMDATARATINGS

+Legal Treatment of Insolvency

Effective Triggers of Distress and Recovery



6

The Motivation: Ratings as Dynamic Indicators

• Priors:– Merton

– In the practice: KMV

The option principle applied to a « distance-to-default »

Mapping to ratings following empirical evidence.

« Implementing a Structural Valuation Model of Swap Credit-Sensitive Rates », 1999, H.Pirotte,

• Based on a previously published paper in European Financial Management.

V

DF



7

Four-Steps Iterative procedure

E=Call(V,...) v T

Infer V Compute v Compute T

r(T)

Compute r(T)



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Part II: What’s a Structural Model?

• The General Idea:

• The problems when defining...– The triggering of the probability of default

– The recovery rate

• Merton’s idea is, in some respect, less trivial than it appears!(see Crouhy and Galai)



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Assumptions of an extended model

• Assets’ value dynamics:

• Barrier:

• Passage time:

• Recovery rate definition:

• Two-factor interest-ratespecification:



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Derivations

• Bond price dynamicy (risk-neutral):

• With:

• Value preservation property:



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Derivations (cont’d)

• Value of bond at maturity is therefore of the form:

• Value of the bond at time 0:



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• With:



13


• With:

we have ...



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Results: debt pricing values



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Results: credit spreads

• Credit spreadbehavior:



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Results: credit spreads (cont’d)



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Results: Expected Cost of Default (ECD)

• ECD = EPD × ELGD



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Results: ECD (cont’d)



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Results: ECD (cont’d)



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Results: « Rates »



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Results: Distance-to-Default



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Results: credit spreads vs. maturity



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Endogenous choices at time 0!

• By absence of arbitrage:

• Required credit spread:

• Program to be solved:

• Equity valuation:



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Results



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Equity Delta



26



27

Iterating...



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Litterature and research fields...

• Rationing/Collateral/Mitigation

– The option to wait to invest and equilibrium credit rationing; Robert Lensink; Journal of Money, Credit, and Banking, Columbus; Feb 2002; Vol. 34, Iss. 1; pg. 221, 5 pgs

– Efficient frontier cutoff policies in credit portfolios; R M Oliver; The Journal of the Operational Research Society, Oxford; Sep 2001; Vol. 52, Iss. 9; pg. 1025

– Using project finance to mitigate telecom risk; Terry A Pratt; Journal of Project Finance, New York; Summer 2001; Vol. 7, Iss. 2; pg. 17

• Modellisation

– A pricing model for secondary market yield based floating rate notes subject to default risk; Manfred Fruhwirth; European Journal of Operational Research, Amsterdam; Dec 1, 2001; Vol. 135, Iss. 2; pg. 233

– The term structure of credit spreads with jump risk; Chunsheng Zhou; Journal of Banking & Finance, Amsterdam; Nov 2001; Vol. 25, Iss. 11; pg. 2015

– Counterparty risk and the pricing of defaultable securities; Robert A Jarrow; The Journal of Finance, Cambridge; Oct 2001; Vol. 56, Iss. 5; pg. 1765, 35 pgs

– An analytic approach to credit risk of large corporate bond and loan portfolios; Andre Lucas; Journal of Banking & Finance, Amsterdam; Sep 2001; Vol. 25, Iss. 9; pg. 1635

– A Markov chain model with stochastic default rate for valuation of credit spreads; Eiji Kodera; Journal of Derivatives, New York; Summer 2001; Vol. 8, Iss. 4; pg. 8

• Modellisation

– Floating-fixed credit spreads; Darrell Duffie; Financial Analysts Journal, Charlottesville; May/Jun 2001; Vol. 57, Iss. 3; pg. 76, 12 pgs



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• Basel– Market discipline of banks: The asset test; Donald P Morgan; Journal of Financial Services Research,

Dordrecht; Dec 2001; Vol. 20, Iss. 2,3; pg. 195– Basel Capital Accord implementation delayed; Anonymous; Journal of Taxation of Financial Institutions,

Kingston; Sep/Oct 2001; Vol. 15, Iss. 1; pg. 60 – Evaluating the basle guidelines for backtesting banks' internal risk management models; Andre Lucas;

Journal of Money, Credit, and Banking, Columbus; Aug 2001; Vol. 33, Iss. 3; pg. 826, 21– Basel Committee on banking surpervision proposes revision of Basel Capital Accord treatment of credit

risk and operational risk; Lawrence A Darby III; Journal of Taxation of Financial Institutions, Kingston; Jul/Aug 2001; Vol. 14, Iss. 6; pg. 76

– The impact of FDICIA and prompt corrective action on bank capital and risk: Estimates using a simultaneous equations model; Raj Aggarwal; Journal of Banking & Finance, Amsterdam; Jun 2001; Vol. 25, Iss. 6; pg. 1139

• Portfolio Management with Credit Risk + Risk Premiums + Products– Basis risk, credit risk and collateralization issues for insurance-linked derivatives and securities; Sylvie

Bouriaux; Journal of Insurance Regulation, Kansas City; Fall 2001; Vol. 20, Iss. 1; pg. 94, 27 pgs– Credit risk and the deposit insurance premium: A note; Jean Dermine; Journal of Economics and

Business, New York; Sep/Oct 2001; Vol. 53, Iss. 5; pg. 497• Credit Derivatives

– Credit derivatives in banking: Useful tools for managing risk?; Gregory R Duffee; Journal of Monetary Economics, Amsterdam; Aug 2001; Vol. 48, Iss. 1; pg. 25

• Sovereign Credit Risk– Crisis dynamics of implied default recovery ratios: Evidence from Russia and Argentina; John J Merrick Jr;

Journal of Banking & Finance, Amsterdam; Oct 2001; Vol. 25, Iss. 10; pg. 1921– Emerging market debt: Measuring credit quality and examining relative pricing; Robert E Cumby;

Journal of International Money and Finance, Kidlington; Oct 2001; Vol. 20, Iss. 5; pg. 591



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• Various empiricals:– Default probabilities in a corporate bank portfolio: A logistic model appraoch; Sjur Westgaard; European

Journal of Operational Research, Amsterdam; Dec 1, 2001; Vol. 135, Iss. 2; pg. 338– An analysis of default risk on residential mortgage loans; Shin-Ping Lee; International Journal of

Management, Poole; Dec 2001; Vol. 18, Iss. 4; pg. 421 – The determinants of credit spread changes; Pierre Collin-Dufresne; The Journal of Finance, Cambridge;

Dec 2001; Vol. 56, Iss. 6; pg. 2177, 31 pgs– The information content of bank exam ratings and subordinated debt prices; Robert Deyoung; Journal of

Money, Credit, and Banking, Columbus; Nov 2001; Vol. 33, Iss. 4; pg. 900, 26 pgs– Firm internationalization and the cost of debt financing: Evidence from non-provisional publicly traded

debt; David M Reeb; Journal of Financial and Quantitative Analysis, Seattle; Sep 2001; Vol. 36, Iss. 3; pg. 395, 20 pgs

– Estimating corporate yield curves; Antonio Diaz; The Journal of Fixed Income, New York; Sep 2001; Vol. 11, Iss. 2; pg. 95

– Explaining performance trends in leveraged loan investments; Albert J Friedman; The Journal of Fixed Income, New York; Sep 2001; Vol. 11, Iss. 2; pg. 83

– Testing for rating consistency in annual default rates; Richard Cantor; The Journal of Fixed Income, New York; Sep 2001; Vol. 11, Iss. 2; pg. 36

– Unique risk-return characteristics of high-yield bonds; Frank K Reilly; The Journal of Fixed Income, New York; Sep 2001; Vol. 11, Iss. 2; pg. 65

– A balanced approach to forecasting credit risk; Jeffrey S Morrison; The Journal of Business Forecasting Methods & Systems, Flushing; Summer 2001; Vol. 20, Iss. 2; pg. 16, 3 pgs

– On the term structure of default premia in the swap and LIBOR markets; Pierre Collin-Dufresne; The Journal of Finance, Cambridge; Jun 2001; Vol. 56, Iss. 3; pg. 1095, 21 pgs



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Default probabilities and « Recovery Rates »

S&P Moody's Bloomberg 1Y 2Y 3Y 4Y 5YAAA Aaa AAA 0.00% 0.00% 0.03% 0.06% 0.10%AA+ Aa1 AA1 0.00% 0.00% 0.00% 0.10% 0.21%AA Aa2 AA2 0.00% 0.00% 0.00% 0.03% 0.09%AA- Aa3 AA3 0.03% 0.09% 0.23% 0.35% 0.49%A+ A1 A1 0.02% 0.07% 0.15% 0.33% 0.46%A A2 A2 0.05% 0.11% 0.17% 0.22% 0.37%A- A3 A3 0.05% 0.17% 0.30% 0.48% 0.73%

BBB+ Baa1 BBB1 0.12% 0.29% 0.56% 0.87% 1.18%BBB Baa2 BBB2 0.22% 0.52% 0.74% 1.12% 1.50%BBB- Baa3 BBB3 0.35% 0.71% 1.12% 2.09% 3.02%BB+ Ba1 BB1 0.44% 1.21% 2.75% 4.08% 5.22%BB Ba2 BB2 0.94% 2.59% 4.62% 6.04% 7.34%BB- Ba3 BB3 1.33% 4.28% 7.42% 10.47% 13.00%B+ B1 B1 2.91% 7.74% 12.08% 15.44% 17.92%B B2 B2 8.38% 16.01% 21.00% 23.73% 25.73%B- B3 B3 10.32% 18.27% 23.32% 27.02% 29.40%

CCC Caa-C CCC 21.94% 29.25% 34.37% 38.24% 42.13%Investment Grade 0.08% 0.19% 0.31% 0.51% 0.72%Speculative Grade 4.14% 8.34% 11.93% 14.67% 16.84%

Class of Debt Recovery Rate Standard DeviationSenior Secured Bank 47.54% 21.33%

Equipment Trust 65.93% 28.55%Senior Secured Public 55.15% 24.31%

Senior Unsecured Public 51.31% 26.30%Senior Subordinated Public 39.05% 24.39%

Subordinated Public 31.66% 20.58%Junior Subordinated Public 20.39% 15.36%

All Subordinated Public 34.12% 20.35%All Public 45.02% 26.37%

a structural model of the term structure of credit spreads with stochastic recovery and contractual...

Documents

recovery slide

unwanted default default

luniversit devry

default probability

probability of default

creditworthiness slide

galai slide

default mapping