Actuarial Notes

Credit Risk: Pricing, Measurement, and Management

By Darrell Duffie and Kenneth J. Singleton · Princeton University Press · 2003 · INV101-108-26

Study guide: Exam INV-101

An integrated treatment of the conceptual, practical and empirical foundations of credit risk pricing and risk measurement, by two economists who model credit risk to measure portfolio risk and to price defaultable bonds, credit derivatives and other securities exposed to credit risk. It presents both the structural and the reduced-form approaches to pricing defaultable securities and assesses their fit to historical data, with a treatment of credit derivatives that includes credit swaps, collateralized debt obligations, credit guarantees, lines of credit and spread options. The SOA supplies Chapters 3 and 5 as study note INV101-108-26.

On the syllabus

  • Exam INV-101 — INV101-108-26, Chapters 3 and 5 of Credit Risk (outcome 3c): Ch. 3 Default Arrival: Historical Patterns and Statistical Models, and Ch. 5 Conceptual Approaches to Valuation of Default Risk

1 Introduction

  • 1.1 A Brief Zoology of Risks
  • 1.2 Organization of Topics

2 Economic Principles of Risk Management

  • 2.1 What Types of Risk Count Most?
  • 2.2 Economics of Market Risk
    • 2.2.1 Profit-Loss Asymmetries
    • 2.2.2 Minimum Capital Requirements
    • 2.2.3 Principal-Agent Effects
    • 2.2.4 Capital – A Scarce Resource
    • 2.2.5 Leverage and Risk for Financial Firms
    • 2.2.6 Allocation of Capital or Risk?
    • 2.2.7 Risk-Adjusted Return on Capital?
    • 2.2.8 Performance Incentives
  • 2.3 Economic Principles of Credit Risk
    • 2.3.1 Adverse Selection and Credit Exposure
    • 2.3.2 Credit Risk Concentrations
    • 2.3.3 Moral Hazard
  • 2.4 Risk Measurement
    • 2.4.1 Value at Risk
    • 2.4.2 Expected Tail Loss
    • 2.4.3 The Price of Market-Value Insurance
    • 2.4.4 Volatility as a Measure of Market Risk
    • 2.4.5 Coherency of Risk Measures
  • 2.5 Measuring Credit Risk
    • 2.5.1 Specialized Measures of Credit Risk
    • 2.5.2 Capital Guidelines for Credit Exposures

3 Default Arrival: Historical Patterns and Statistical Models

  • 3.1 Introduction
    • 3.1.1 The Changing Composition of Speculative Debt
    • 3.1.2 Forward Default Probabilities
  • 3.2 Structural Models of Default Probability
    • 3.2.1 The Black-Scholes-Merton Default Model
    • 3.2.2 First-Passage Models
  • 3.3 From Theory to Practice: Using Distance to Default to Predict Default
  • 3.4 Default Intensity
    • 3.4.1 Doubly-Stochastic Default
  • 3.5 Examples of Intensity Models
    • 3.5.1 Mean-Reverting Intensities with Jumps
    • 3.5.2 CIR Intensity Models
    • 3.5.3 Comparison of Jump and CIR Intensities
    • 3.5.4 Affine Intensity Models
    • 3.5.5 HJM Forward Default Rate Models
  • 3.6 Default Time Simulation
  • 3.7 Statistical Prediction of Bankruptcy
    • 3.7.1 Comparing Prediction Methods
    • 3.7.2 Default and Aging Effects

4 Ratings Transitions: Historical Patterns and Statistical Models

  • 4.1 Average Transitions Frequencies
  • 4.2 Ratings Risk and the Business Cycle
  • 4.3 Ratings Transitions and Aging
  • 4.4 Ordered Probits of Ratings
  • 4.5 Ratings as Markov Chains
    • 4.5.1 Time-Varying Transition Intensities
    • 4.5.2 Lando’s Stochastic Transition Intensity Model

5 Conceptual Approaches to Valuation of Default Risk

  • 5.1 Introduction
  • 5.2 Risk-Neutral Versus Actual Probabilities
  • 5.3 Reduced-Form Pricing
  • 5.4 Structural Models
    • 5.4.1 The Black-Scholes-Merton Debt Pricing MOdel
    • 5.4.2 First-Passage Debt Pricing
  • 5.5 Comparisons of Model-Implied Spreads
  • 5.6 From Actual to Risk-Neutral Intensities
    • 5.6.1 Reduced-Form Models
    • 5.6.2 Structural Models

6 Pricing Corporate and Sovereign Bonds

  • 6.1 Uncertain Recovery
  • 6.2 Reduced-Form Pricing with Recovery
    • 6.2.1 Fractional Recovery of Face Value
    • 6.2.2 Conditional Expected Recovery
    • 6.2.3 Fractional Recovery of Market Value
    • 6.2.4 Comparing Recovery Assumptions
  • 6.3 Ratings-Based Models of Credit Spreads
    • 6.3.1 General Pricing Framework
    • 6.3.2 Calibrating A Model to Historical Data
  • 6.4 Pricing Sovereign Bonds
    • 6.4.1 Credit Risk in Sovereign Bonds
    • 6.4.2 Parametric Models of Sovereign Spreads

7 Empirical Models of Defaultable Bond Spreads

  • 7.1 Credit Spreads and Economic Activity
  • 7.2 Reference Curves for Spreads
  • 7.3 Parametric Reduced-Form Models
    • 7.3.1 Square-root Diffusion Models of Spreads
    • 7.3.2 Jump-Diffusion Spreads
    • 7.3.3 Accommodating Observable Credit Factors
  • 7.4 Estimating Structural Models
  • 7.5 Parametric Models of Sovereign Spreads

8 Credit Swaps

  • 8.1 Other Credit Derivatives
    • 8.1.1 Total-Return Swaps
    • 8.1.2 Spread Options
  • 8.2 The Basic Credit Swap
    • 8.2.1 Settlement Issues
  • 8.3 Simple Credit-Swap Spreads
    • 8.3.1 Credit-Swap Spreads: Starter Case
    • 8.3.2 Repo Specials and Transactions Costs
    • 8.3.3 Payment of Accrued Credit-Swap Premium
    • 8.3.4 Accrued Interest on the Underlying Note
    • 8.3.5 If the Underlying is a Fixed-Rate Note?
  • 8.4 Model-Based CDS Rates
    • 8.4.1 The Case of Constant Intensity
    • 8.4.2 The Term Structure of Forward Default Rates
  • 8.5 The Role of Asset Swaps

9 Optional Credit Pricing

  • 9.1 Spread Options
    • 9.1.1 Pricing Framework
    • 9.1.2 Illustrative Numerical Example
    • 9.1.3 Numerical Option Valuation Algorithm
    • 9.1.4 Results
  • 9.2 Callable and Convertible Corporate Debt
    • 9.2.1 Capital Structure
    • 9.2.2 Default and Equity Derivative Pricing
    • 9.2.3 Convertible Debt as an Equity Derivative
    • 9.2.4 Call-Forcing Conversion
    • 9.2.5 Evidence of Delayed Calls
  • 9.3 A Simple Convertible Bond Pricing Model
    • 9.3.1 Background Modeling
    • 9.3.2 Convertible Bond Model Setup
    • 9.3.3 Pricing Algorithm
    • 9.3.4 Convertible Bond Hedging Strategies
    • 9.3.5 Exposure to Equity Volatility
    • 9.3.6 The Issuer’s Propensity to Call
    • 9.3.7 Duration and Convexity

10 Correlated Defaults

  • 10.1 Alternative Approaches to Correlation
  • 10.2 CreditMetrics Correlated Defaults
  • 10.3 Correlated Default Intensities
    • 10.3.1 Correlated Jump Intensity Processes
    • 10.3.2 Correlated Log-Normal Intensities
  • 10.4 Copula-Based Correlation Modeling
    • 10.4.1 Calibration of Copula to First-Passage Model
  • 10.5 Empirical Methods
  • 10.6 Default Time Simulation Algorithms
    • 10.6.1 Multi-Compensator Method
    • 10.6.2 First-Defaults Simulation
    • 10.6.3 Recursive Inverse-CDF Simulation
  • 10.7 Joint Default Events
    • 10.7.1 Multivariate Exponential Event Times

11 Collateralized Debt Obligations

  • 11.1 Introduction
  • 11.2 Some Economics of CDOs
  • 11.3 Default-Risk Model
    • 11.3.1 Obligor Default Intensities
    • 11.3.2 Multi-Issuer Default Model
    • 11.3.3 Sectoral, Regional, and Global Risk
    • 11.3.4 Recovery Risk
    • 11.3.5 Collateral Credit Spreads
    • 11.3.6 Diversity Scores
  • 11.4 Pricing Examples
    • 11.4.1 Collateral
    • 11.4.2 Sinking-Fund Tranches
    • 11.4.3 Prioritization Schemes
    • 11.4.4 Simulation Methodology
    • 11.4.5 Results for Par CDO Spreads
    • 11.4.6 Risky Reinvestment
  • 11.5 Default Loss Analytics
  • 11.6 Computation of Diversity Scores

12 OTC Default Risk and Valuation

  • 12.1 Exposure
    • 12.1.1 Potential Exposure
    • 12.1.2 BIS Capital Add-Ons for Exposure
    • 12.1.3 Interest-Rate Swaps
    • 12.1.4 Mid-Market Revaluation of Swaps
    • 12.1.5 Expected Swap Exposures
  • 12.2 OTC Credit-Risk Value Adjustments
    • 12.2.1 One-Sided Default Risk
    • 12.2.2 Adjustment with Netting and Collateral
    • 12.2.3 Two-Sided Default Risk
    • 12.2.4 Example: 10-Year Swap Adjustments
  • 12.3 Additional Swap Credit Adjustments
    • 12.3.1 Setup
    • 12.3.2 Base-Case Results
    • 12.3.3 Off-Market Swap Rate Credit Spreads
    • 12.3.4 Dependence of Default Risk on LIBOR Rate
    • 12.3.5 Asynchronous Swap Payments
    • 12.3.6 The Impact of Slope of the Yield Curve
    • 12.3.7 Term Structure of Credit Spreads
    • 12.3.8 When Both Counterparties are Risky
    • 12.3.9 The Impact of Netting
  • 12.4 Credit Spreads on Currency Swaps

13 Integrated Market and Credit Risk Measurement

  • 13.1 Market Risk Factors
    • 13.1.1 Preliminaries
    • 13.1.2 Modeling Return Distributions
    • 13.1.3 Shapes of Return Distributions
  • 13.2 Delta-Gamma for Derivatives, with Jumps
    • 13.2.1 Delta Measures of Derivatives Risk
    • 13.2.2 Beyond Delta to Gamma
  • 13.3 Integration of Market and Credit Risk
  • 13.4 Examples of VaR with Credit Risk
    • 13.4.1 Example: Loan Portfolio VaR with Credit Risk
    • 13.4.2 Example: Options Portfolio with Credit Risk

Appendix A Introduction to Affine Processes

  • 1 Introduction
  • 2 Analytical Solutions in Affine Settings
    • 2.1 Multivariate Example: Independent Coordinates
    • 2.2 Multivariate Example: The Heston Model
    • 2.3 The Riccati Equations
    • 2.4 Transforms of Affine Processes
  • 3 Some Applications
    • 3.1 Term-Structure Models
    • 3.2 Default Intensities and Probabilities
    • 3.3 Correlated Default
    • 3.4 Defaultable Term Structure Models
    • 3.5 Option Pricing
  • 4 Generalized Riccati Equations
  • 5 Solution for The Basic Affine Model
  • 6 Intensities for Stopping Times

Appendix B Econometrics of Affine Term-Structure Models

Appendix C HJM Spread Curve Models

Sources