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Risk management in banking - Joel Bessis

Year 2010


About the AuthorIntroduction1. BASIC POSTULATES OF THIS TEXT1.1 The Financial Crisis1.2 Scope and Goals of this Text2. POSITIONING OF THE TEXT2.1 The "Model Divide"2.2 Risk Management versus Risk Instruments2.3 Reverting to Better Risk Practices and Lessons of the Crisis3. BOOK STRUCTURE4. THE THIRD EDITIONSection 1. The Financial Crisis1. The 2007-2008 Financial Crisis1.1 THE SUB-PRIME CRISIS1.2 CLASSICAL CONTAGION MECHANISMS1.3 FINANCIAL RISKS1.4 REGULATIONS1.4.1 Risk Regulations1.4.2 Accounting Standards1.5 CONTAGION THROUGH SECURITIZATIONS1.6 LIQUIDITY CONTAGION1.7 CONTAGION AND PROCYCLICALITYTHROUGH FAIR VALUE RULES1.8 CONTAGION THROUGH RATING DOWNGRADES1.9 CONTAGION AND PROCYCLICALITY IN A LEVERAGED INDUSTRY1.10 SOME IMPLICATIONSSection 2. Business Lines, Risks, and Risk Management2. Banking Business Lines2.1 BUSINESS POLES IN THE BANKING INDUSTRY2.2 MANAGEMENT PRACTICES DIFFER ACROSS AND WITHIN BUSINESS LINES2.3 BANKING REGULATIONS AND ACCOUNTING STANDARDS3. Risks and Risk Management3.1 UNCERTAINTY, RISK,AND EXPOSURETO RISK3.2 TYPES OF RISKS3.3 CREDIT RISK3.3.1 Default Risk3.3.2 Migration Risk3.3.3 Exposure Risk3.3.4 Counterparty Risk of Derivatives3.3.5 Recovery Risk3.3.6 Correlation and Concentration Risks3.3.7 Credit Risk in the Trading Portfolio3.3.8 Spread Risk3.3.9 Country Risk3.4 LIQUIDITY RISK: FUNDING3.5 INTEREST RATE RISK3.6 MISMATCH RISK3.7 MARKET LIQUIDITY RISK3.8 MARKET RISK3.9 FOREIGN EXCHANGE RISK3.10 SOLVENCY RISK3.11 OPERATIONAL RISK4. Risk Management4.1 RISK OVERSIGHT CHALLENGES4.1.1 Motivations for Risk Oversight4.1.2 Risk Oversight: Banks' Challenges4.1.3 Risk Oversight: Regulators' Challenges4.2 RISK MANAGEMENT PROCESSES4.2.1 The "Three Lines of Defense" Principle4.2.2 Risk Management Organization and Central Functions4.2.3 Limits and Delegations4.3 SETTING-UP LIMITS4.3.1 Credit Risk and Lending Activities4.3.2 Market Risk and Trading Activities4.3.3 Asset-liability Management (ALM)4.4 ENTERPRISE-WIDE RISK MANAGEMENT (ERM)4.4.1 Top-down and Bottom-up Risk Management Processes and Systems4.4.2 Earnings Allocation and Fund Transfer Pricing (FTP) System4.4.3 Capital and Risk Allocation SystemSection 3. Financial Products5. Banking and Financial Products5.1 CLASSIFICATION OF LENDING PRODUCTS AND BASEL 2 CRITERIA5.2 SOVEREIGN, BANK, AND EQUITY EXPOSURES5.2.1 Bank Exposures5.2.2 Sovereign Exposures5.2.3 Equity Exposures5.3 RETAIL PORTFOLIO IN BASEL 25.3.1 Nature of Borrower or Low Value of Individual Exposures5.3.2 Large Number of Exposures5.3.3 Retail Portfolio: Further Differentiation for Capital Treatment5.4 TRANSACTION-SPECIFIC CREDIT RISK5.4.1 Credit Risk Mitigants5.4.2 Credit Risk Mitigation under Basel 25.5 SECURITY LENDING AND BORROWING5.6 SPECIALIZED LENDING5.6.1 Project Finance5.6.2 Object Finance5.6.3 Commodities Finance5.6.4 Income-producing Real Estate5.6.5 High-volatility Commercial Real Estate5.7 SECURITIZATIONS5.7.1 Securitizations: Traditional5.7.2 Securitizations: Synthetic5.8 OFF-BALANCE SHEET ITEMS6. Essentials on Derivative Products6.1 ASSETS AND POSITIONS6.2 DERIVATIVES: BASIC DEFINITIONS AND PRINCIPLES6.3 FORWARD CONTRACTS6.3.1 A Simple Example of a Forward Contract6.3.2 Spot and Forward Prices6.3.3 Hedging and Speculating with Forward Contracts6.3.4 Payoff of a Forward Contract6.4 FORWARD CONTRACTS VERSUS OPTIONS6.5 OPTIONS6.5.1 Options Allow Hedging Risks6.5.2 The Payoff of Option Contracts6.5.3 Sellers of Options6.5.4 Risk and Return of Option Contracts6.5.5 Option Pricing6.5.6 Underlying of Options7. Interest Rate Risk and Interest. Rate Derivatives7.1 INTEREST RATE RISK FOR BORROWERS AND LENDERS7.2 INTEREST RATES AND TERM STRUCTURE7.2.1 Lending, Borrowing and the Term Structure of Interest Rates7.2.2 Fixed and Variable Rates7.3 FORWARD RATES: DEFINITIONS7.3.1 Replicating a Forward Loan7.3.2 Forward and Spot Rates: No Arbitrage7.3.3 Example of Calculations of the Forward Yield Curves7.3.4 Usages of Forward Rate Contracts7.3.5 Valuation of a Forward Contract or FRA7.4 INTEREST RATE SWAPS (IRS)7.4.1 Definition of Interest Rate Swap7.4.2 Replicating an Interest Rate Swap7.4.3 Asset Swaps: Generalization7.4.4 Mark-to-market Value of an IRS7.4.5 Derivatives and P & L7.5 INTEREST RATE OPTIONS: CAPS AND FLOORS7.5.1 Payoff of Interest Rate Options7.5.2 Collar7.5.3 Options as Volatility Instruments7.6 HEDGING INTEREST RATE RISK BY CORPORATE BORROWER (CASE STUDY)7.7 WHERE DO INTEREST RATES COME FROM?8. Foreign Exchange Risk and Foreign Exchange Derivatives8.1 FOREIGN EXCHANGE RATES AND FORWARD CONTRACTS8.1.1 Replicating a Forward Exchange Rate8.1.2 Mark-to-market Valuation of Forward Contracts8.2 FOREIGN EXCHANGE OPTIONS8.3 HEDGING A CORPORATE LONG EXPOSURE IN FOREIGN CURRENCY (CASE STUDY)8.4 THE FOREIGN EXCHANGE MARKET AND RATES9. Credit Derivatives9.1 DEFINITIONS OF CREDIT DERIVATIVES9.2 CREDIT DEFAULT SWAPS (CDS)9.3 TOTAL RETURN SWAPS9.4 CREDIT SPREAD PRODUCTS9.5 BASKET SWAPS, FIRST-TO-DEFAULT, N-TO-DEFAULT9.6 SOVEREIGN RISK CREDIT DERIVATIVES9.7 MAIN SPECIFICS AND KEYTERMS9.7.1 The Underlying Assets9.7.2 Payment Terms9.7.3 Legal Issues9.7.4 Payments under a Credit Event9.7.5 Credit Events9.7.6 Materialization9.8 PRICING CREDIT DERIVATIVESSection 4. Valuation10. Distribution Functions10.1 RANDOM VARIABLES AND PROBABILITY DISTRIBUTION FUNCTIONS10.1.1 Discrete Variables10.1.2 Continuous Variables10.2 INVERSE FUNCTIONS10.3 THE MOMENTS OF A DISTRIBUTION10.3.1 Expectations10.3.2 Variance and Volatility10.4 UNIFORM DISTRIBUTION10.5 APPLICATION TO LOSS DISTRIBUTIONS AND LOSS PERCENTILES10.6 BERNOULLI VARIABLE10.7 INDICATOR FUNCTION10.8 BERNOULLI DISTRIBUTION10.9 BINOMIAL DISTRIBUTION OF SUM OF BERNOULLI VARIABLES10.10 NORMAL DISTRIBUTION10.11 LOGNORMAL DISTRIBUTION10.12 THE POISSON DISTRIBUTION10.12.1 Definition10.12.2 Financial Applications10.12.3 Number of Default Events over a Given Horizon10.13 THE EXPONENTIAL DISTRIBUTION AND TIME TO DEFAULT10.14 THE BETA DISTRIBUTION10.15 THE STUDENT DISTRIBUTION10.16 A SUMMARY10.17 APPENDIX: CALCULATION OF STANDARD DEVIATION FROM TIME SERIES11. Discrete and Continuous Returns11.1 DISCRETE AND CONTINUOUS RETURNS11.1.1 Single Period Discrete Return11.1.2 Compounding Discrete Returns over Multiple Periods11.1.3 From Discrete to Continuous Returns11.1.4 Logarithmic Returns11.1.5 Compounding Continuous Returns11.1.6 Comparing Continuous and Discrete Returns11.2 THE DISCOUNTED CASH FLOW MODEL11.2.1 Valuation in Discrete Time and the Rationale of Discounting11.2.2 The Case of Floaters11.2.3 Continuous Time11.2.4 Asset Value and Market Required Return11.2.5 Yield-to-Maturity and Zero-coupon Rates11.3 VALUATION UNDER UNCERTAINTY11.4 APPENDIX:THE TAYLOR EXPANSION FORMULA12. Stochastic Processes12.1 STOCHASTIC PROCESSES12.2 COMMON STOCHASTIC PROCESSES12.2.1 The Wiener Process12.2.2 Application: The Square Root of Time Rule for the Simple Wiener Process12.2.3 The Generalized Wiener Process12.2.4 The Ito Process12.2.5 The Stock Price Process12.2.6 The Mean Reverting Process12.2.7 The "Rare Event" Process12.3 APPLICATION: STOCK VALUE DISTRIBUTION12.4 INTEREST RATE PROCESSES12.5 SPECIFICS OF STOCHASTIC PROCESSES AND ITO LEMMA12.5.1 Deterministic Calculus12.5.2 Stochastic Calculus12.5.3 Ito Lemma12.5.4 Example: Stock Price Distribution13. Valuation and Pricing Risk13.1 CONSTRUCTING A RISK-FREE PORTFOLIO13.1.1 How to Form Risk-free Portfolios13.1.2 How to Derive the Pricing PDE13.1.3 How to Solve Numerically a PDE13.2 RISK-NEUTRAL VALUATION:THE CASE OF A STOCK PRICE13.2.1 Risk-neutral Probability13.2.2 Stock Price Dynamics under Risk-neutral Probabilities13.3 VALUATION: END RESULTS14. Some Applications of Valuation Techniques14.1 VALUATION OF RISKY DEBT FROM CREDIT SPREADS AND RISK-NEUTRAL PROBABILITIES14.2 VALUATION OF AN OPTION UNDER RISK-NEUTRAL PROBABILITIES14.3 THE VASICEK MODELSection 5. Risk Modeling15. Sensitivity15.1 SENSITIVITY DEFINITIONS15.2 COMMON SENSITIVITIES15.2.1 Stocks15.2.2 Bonds and Loans15.2.3 Options15.2.4 Forward Contracts and Interest Rate Swaps15.3 SENSITIVITY AND RISK FACTORS15.4 SENSITIVITIES AND RISK CONTROLLING16. Volatility16.1 VOLATILITY16.2 EQUALLY WEIGHTED HISTORICAL VOLATILITY16.2.1 Continuous Returns and Variance16.2.2 Historical Volatility with Equally Weighted Observations16.3 EXPONENTIALLY WEIGHTED MOVING AVERAGE (EWMA) MODEL16.4 GARCH MODELS16.5 MAXIMUM LIKELIHOOD METHODOLOGY16.6 ESTIMATING EWMA VOLATILITY17. The Value-at-Risk Measure17.1 MODELING POTENTIAL VARIATIONS AND PERCENTILES17.1.1 Modeling Potential Variations of Value17.1.2 Value Percentile17.2 THE VAR METHODOLOGY: MARKET RISK17.2.1 Sensitivity17.2.2 From Sensitivity to Volatility17.2.3 From Daily Volatility of the Position to Daily VaR17.2.4 VaR at Different Horizons17.2.5 Summary and Extensions17.3 CREDIT RISK VAR17.3.1 The Firm Value17.3.2 The Distribution of Firm Value18. VaR and Capital18.1 THE CONTRIBUTIONS OF VAR-BASED MEASURES18.2 LOSS DISTRIBUTIONS18.3 MEASURES OF POTENTIAL LOSSES18.3.1 Expected Loss18.3.2 Loss Percentiles18.3.3 Unexpected Loss and VaR18.3.4 Exceptional Losses18.3.5 Loss Distributions, Potential Losses and VaR18.4 VAR AND ECONOMIC CAPITAL18.5 EARNINGS-AT-RISK ("EAR")Section 6. Regulations19. Banking Regulations: Basel I and Market Risk19.1 REGULATORY ISSUES19.2 THE DILEMMAS OF THE REGULATOR19.2.1 Regulation and Competition19.2.2 Preemptive Risk Control versus Risk Insurance19.3 CAPITAL ADEQUACY19.3.1 Risk-based Capital Regulations19.3.2 Capital Requirements19.3.3 Risk-based Capital and Growth19.4 THE "BASEL I ACCORD" FOR CREDIT RISK19.4.1 The Cooke Ratio and Credit Risk19.4.2 Derivatives and Credit Risk19.4.3 Basel I Drawbacks19.5 THE ACCORD FOR MARKET RISK19.5.1 The Standardized Approach19.5.2 Proprietary Models of Market Risk VaR20. Banking Regulations: The Basel 2 Accord20.1 THE NEW BASEL ACCORD20.2 ASSET CLASSES20.3 CREDIT RISK COMPONENTS20.3.1 Probability of Default (DP) and Default Event20.3.2 Exposure at Default (EAD)20.3.3 Loss Given Default (LGD)20.3.4 Credit Conversion Factors (CCFs)20.3.5 Credit Components and Risk Weights20.4 THE STANDARDIZED APPROACH20.4.1 Regulatory Risk Weights for Corporates, Sovereigns and Banks20.4.2 The Retail Portfolio20.4.3 Effective Maturity (M)20.4.4 Off Balance Sheet Items20.5 INTERNAL RATINGS BASED FRAMEWORK20.6 CREDIT RISK MITIGATION20.6.1 Credit Risk Mitigation: Guarantees and Credit Derivatives20.6.2 Credit Risk Mitigation: Collateral Treatment20.6.3 Collateral: Haircut Calculations20.6.4 Effective LGD for Collateral-based Transactions20.7 COUNTERPARTY CREDIT RISK20.8 CAPITAL CALCULATION20.8.1 Risk-Weighted Assets for Corporate, Sovereign, and Bank Exposures20.8.2 Interpretation of Basel 2 Formulas for Risk Weights20.8.3 Retail Portfolio20.8.4 Equity Exposures20.9 SAMPLE COMPARISON BETWEEN BASEL I AND BASEL 2 CAPITAL FOR CORPORATE ASSET CLASS CREDIT RISK20.10 SPECIALIZED LENDING20.11 SECURITIZATIONS20.11.1 Standardized Approach20.11.2 IRB Approaches20.12 INTEREST RATE RISK20.13 OPERATIONAL RISK20.14 PILLAR 2: SUPERVISORY REVIEW PROCESS20.15 PILLAR 3: MARKET DISCIPLINE20.16 ADDITIONAL PROPOSALS21. Accounting Standards21.1 BANKS'FINANCIAL STATEMENTS21.2 INITIAL RECOGNITION OF FINANCIAL ASSETS AND LIABILITIES21.2.1 Financial Assets and Liabilities at Fair Value through Profit or Loss21.2.2 Loans and Receivables21.2.3 Held-to-maturity Financial Assets21.2.4 Available-for-sale Financial Assets21.3 VALUATION RULES21.3.1 Amortized Cost21.3.2 Method of Determining Fair Value21.3.3 Instruments Traded in Active Markets21.3.4 Instruments Traded in Inactive Markets21.4 IMPAIRMENT OF FINANCIAL ASSETS21.5 HEDGE ACCOUNTING21.5.1 Types of Hedges21.5.2 "Effectiveness" of Hedge21.6 COMPARISONS OF CLASSIFICATIONS: RISK REGULATIONS AND IFRSSection 7. Asset Liability Management (ALM)ALM GOALSALM SCOPE AND STRUCTURE OF THE SECTIONTHE ALM FUNCTION22. Liquidity Management and Liquidity Gaps22.1 LIQUIDITY DEFINITIONS22.2 LIQUIDITY GAP TIME PROFILES22.2.1 Three Basic Liquidity Positions22.2.2 Mismatch Risk22.3 LIQUIDITY GAP CALCULATIONS22.3.1 Static Liquidity Gaps22.3.2 Dynamic Liquidity Gaps22.3.3 How Gaps Change Through Time22.3.4 Fixed Assets and Equity22.4 LIQUIDITY MANAGEMENT22.4.1 Structuring Debt Maturities22.4.2 Numerical Example22.5 STRUCTURAL EXCESSES OF LIQUIDITY22.6 ISSUES FOR DETERMINING THE LIQUIDITY GAP TIME PROFILE22.6.1 Lines without Maturity22.6.2 Demand Deposits22.6.3 Contingencies Given (Off-balance Sheet)22.6.4 Amortizing Loans22.6.5 Multiple Scenarios22.7 LIQUIDITY SCENARIOS22.7.1 Liquid Assets22.7.2 Liquidity Crises and Stress Test Scenarios23. Interest Rate Gaps23.1 DEFINITION OF INTEREST RATE GAPS23.2 CALCULATIONS OF INTEREST RATE GAPS23.2.1 Interest Rate Gap and Liquidity Gaps23.2.2 Sample Gap Reports23.3 THE GAP MODEL23.4 NET INTEREST INCOME AND INTEREST RATE GAPS23.4.1 Projected Gaps23.4.2 Projected Net Interest Income23.4.3 Commercial Spreads23.4.4 The Sensitivity of Nil and Interest Rate Gap23.5 STATIC VERSUS DYNAMIC GAPS23.6 LIMITATIONS OF INTEREST RATE GAPS23.6.1 Embedded Options in Banking Products23.6.2 Lines without Maturity23.6.3 Mapping Interest Rates to Selected Risk Factors23.6.4 Regulated Rates23.6.5 Mark-ups and Mark-downs over Reference Rates23.6.6 Intermediate Flows and NIL Calculations23.7 FROM GAPS TO SIMULATIONS24. ALM and Hedging Policies24.1 MANAGING GAPS: SETTING UP LIMITS24.1.1 The Basic Mechanism of Limits24.1.2 Limits and "Nil at Risk"24.2 HEDGING: CLOSING INTEREST RATE GAPS24.2.1 Single Period24.2.2 Hedging Over Multiple Periods24.3 HEDGING THE VARIATIONS OF THE TERM STRUCTURE OF INTEREST RATES (CASE STUDY)24.4 HEDGING BUSINESS RISK AND INTEREST RATE RISK24.4.1 Multiple Business and Interest Rate Scenarios24.4.2 Matrix Methodology and The Risk-Return Profile of the Balance Sheet24.4.3 Implementing the Matrix Approach24.4.4 Hedging both Interest Rate and Business Risks24.4.5 The Matrices of Net Interest Income25. Implicit Options Risk25.1 OPTIONAL RISK25.1.1 Optional Risk is Always Adverse to the Bank25.1.2 Optional Risk and Hedging25.2 MODELING PREPAYMENTS25.3 PAYOFF OF PREPAYMENT25.4 THE VALUE OF IMPLICIT OPTIONS25.5 THE SIMPLE "BINOMIALTREE" TECHNIQUE APPLIED TO INTEREST RATES25.5.1 The Binomial Tree for Interest Rate25.5.2 Valuation of a Bond25.5.3 The Global Calibration of the Binomial Tree25.5.4 The Valuation of American Options25.5.5 The Current Value of the Prepayment Option25.5.6 The Original Loan25.5.7 Binomial Tree of Interest Rates25.5.8 The Market Value of Loan25.5.9 The Exercise Price and the Payoff of the Option25.5.10 Payoffs of the Option under Differed Exercise25.5.11 Option-adjusted Spread26. Economic Value of the Balance Sheet26.1 ECONOMIC VALUE (EV)26.2 ECONOMIC VALUE AND NET INTEREST INCOME FOR A BANK WITHOUT CAPITAL26.2.1 Sample Bank Balance Sheet26.2.2 EV and Projected Interest Income at Risk-free Rate26.3 NIL AND EV26.4 IDENTITY BETWEEN EV AND "ALL-IN" CASH FLOWS (CAPITAL AND INTEREST)26.4.1 Separating All-in Flows into Capital and Interest Flows26.4.2 Economie Value and NiL: General Example27. Economic Value and Convexity Risk27.1 DURATION PROPERTIES27.2 SENSITIVITY OF ECONOMIC VALUE AND DURATION GAPS27.2.1 Duration Gap27.2.2 Controlling Duration with Derivatives27.2.3 Duration Gap and Sensitivity of EV27.3 ECONOMIC VALUE, DURATION AND CONVEXITY27.3.1 Convexity27.3.2 Sources of Convexity27.3.3 Measure of Convexity27.3.4 Economic Value and Convexity Gaps27.4 CONVEXITY GAPS AND OPTIONS27.5 OPTIONAL RISK AND OPTIONAL GAPS27.5.1 Hedging Caps Sold to Clients27.5.2 Hedging Implicit Options to Renegotiate Fixed RatesSection 8. Funds Transfer Pricing Systems28. Funds Transfer Pricing Systems28.1 THE ORGANIZATION OF FTP SYSTEMS28.1.1 Exchanging Net Balances of Funds between Units28.1.2 Exchanging all Outstanding Balances28.2 CALCULATING INCOME WITHIN A FTP SYSTEM28.2.1 Allocating Income through the FTP System28.2.2 The Accounting N11 of the Bank28.2.3 Breaking Down the Bank Margin into Contributions of Business Units29. Economic Transfer Prices29.1 COMMERCIAL SPREADS AND MATURITY SPREAD29.2 ECONOMIC TRANSFER PRICES FOR LOANS29.2.1 The Cost of Existing Resources29.2.2 The "Notional" Funding of Loans29.2.3 The Cost of the Mirror Debt29.2.4 The Benefits from the Mirror Debt29.3 ECONOMIC TRANSFER PRICES FOR RESOURCES29.4 TRANSFERRING LIQUIDITY AND INTEREST RATE RISK TO ALM THROUGH THE FTP SYSTEM29.5 PRICING FOR LENDING29.5.1 The All-in Cost of Funds of the Mirror Debt29.5.2 The Cost of Credit Risk and the Risk Premium29.5.3 Transaction Versus Client Revenues and Pricing29.5.4 Economic and Commercial Transfer PricesSection 9. Dependencies and Portfolio RiskDEPENDENCIES MODELING ASA KEY BUILDING BLOCK FOR RISK MODELINGMEASURES OF DEPENDENCIESSIMULATIONSSECTION ORGANIZATION30. Correlations and Covariances30.1 CORRELATIONS AND COVARIANCES30.1.1 Correlations, Variances, and Covariances30.1.2 Correlation and Volatility of a Sum of Random Variables30.1.3 Visual Representation of the Diversification Effect30.1.4 Extension to any Number of Variables30.2 THE VARIANCE-COVARIANCE MATRIX30.2.1 Matrix Notations 30.2.2 Variance-Covariance Matrix of Portfolio Returns30.2.3 The Variance-Covariance Matrix of Portfolio Return30.2.4 Portfolio Return Volatility30.3 PORTFOLIO RETURN RISK AND CORRELATION30.4 APPENDIX: MATRIX NOTATIONS AND FORMULAS30.4.1 Transpose Matrix30.4.2 Variance-Covariance Matrix and Correlation Matrix30.4.3 Variance Formula and Correlation Matrix31. Conditional Probabilities31.1 DEFINITION OF CONDITIONAL AND JOINT PROBABILITIES31.1.1 Conditional Probabilities31.1.2 Conditional and Joint Probabilities31.2 DEFAULT PROBABILITY OF A FIRM DEPENDENT ON THE STATE OF THE ECONOMY31.3 THE TWO-OBLIGOR PORTFOLIO AND CONDITIONAL PROBABILITIES31.3.1 The Independence Case31.3.2 The Dependent Case3 1.4 APPENDIX: CONDITIONING, EXPECTATION AND VARIANCE32. Factor Models32.1 FACTOR MODELS32.2 THE GENERIC FORM OF FACTOR MODELS32.2.1 Example of a One-factor Model32.2.2 The Single-factor Model32.2.3 The Generic Form of Multiple-factor Models32.3 RISK OF A TWO-STOCK PORTFOLIO WITH THE ONE-FACTOR MODEL32.3.1 The One-factor Model32.3.2 The Vanance-Covanance Matrix32.3.3 Vanance-Covanance Matrices of Asset Returns and of Factors32.4 PORTFOLIO RISK WITH MULTIPLE-FACTOR MODELS32.5 RISK FOR A SINGLE ASSET AND TWO-FACTOR MODELS32.6 ATWO-ASSET PORTFOLIO WITH TWO-FACTOR MODELS32.6.1 The Two-factor Model Applied to a Two-asset Portfolio32.6.2 The Two-asset Portfolio Return32.6.3 Portfolio Systematic Risk: Two-factor Models32.6.4 Portfolio Systematic Risk Decomposition into Additive Items32.6.5 Two-asset Portfolio and Two-factor Model: Specific Risk32.6.6 Portfolio Total Risk and Two-factor Model32.7 ORTHOGONAL MULTIPLE FACTOR MODELS AND PCA32.7.1 Using PCA for Single Assets32.7.2 Two-asset Portfolio and Two Orthogonal Factors32.8 PORTFOLIO RISK: SUMMARY OF MATRIX FORMAT CALCULATIONS32.9 APPENDIX I:THE CONCEPTUAL FOUNDATIONS OF PCA32.10 APPENDIX 2: PORTFOLIO GENERAL RISK, ORTHOGONAL FACTORS33. Dependencies and Copula Functions33.1 KEY BENEFITS OF COPULA DEPENDENCY33.2 REMINDERS AND NOTATIONS33.3 DEFINITION OF COPULA FUNCTIONS33.3.1 Bivariate Copula33.3.2 Properties of Copula Functions33.3.3 Distribution Functions and Percentiles33.3.4 Various Forms of Bivariate Copula Functions33.3.5 The Bivariate Copula Function: Expanded Form33.3.6 The Copula Density Function33.3.7 Generalization to Several Variables33.4 BIVARIATE NORMAL STANDARD DISTRIBUTIONS33.4.1 Univariate and Bivariate Normal Distributions33.4.2 Bivariate Normal Standard Density33.4.3 The Bivariate Normal Copula33.4.4 Forms of the Gaussian Copula33.4.5 The Normal Standard Copula Density and the Joint Density of Two Variables33.5 CONDITIONAL PROBABILITY FROM THE COPULA DENSITY33.5.1 Conditional Density of Normal Bivariate33.5.2 Conditional Distributions and Simulations33.5.3 Application: Simulating Two Uniform Standard Variables with Conditional Copula33.6 APPENDIX: COPULA FUNCTION AND COPULA DENSITY33.6.1 Deriving the Copula Density from Copula Function33.6.2 Joint Probability Density Functions with Two Variables33.6.3 Derivation of the Normal Copula Density33.6.4 Joint Density of a Bivariate Normal Distribution and the Copula Density Function33.6.5 Conditional Distributions and Copula34. Simulations with Factor Models or the Copula Approach34.1 SIMULATIONS AND INVERSE FUNCTIONS34.2 SIMULATION OF CORRELATED NORMAL VARIABLES WITH FACTOR MODELS34.2.1 Standardizing the Single-factor Model34.2.2 The Simulation Algorithm for Standardized Normal Variables using Single-factor Models34.2.3 The Cholesky Decomposition Method: Two Variables34.2.4 Two Random Normal Variables34.2.5 Application of Cholesky Decomposition34.2.6 Classical Methodology: Summary34.3 SIMULATION OF DEPENDENT VARIABLES WITH THE COPULA APPROACH34.3.1 Main Principle34.3.2 Simulation Algorithm34.4 SIMULATION OF TWO DEPENDENT UNIFORM STANDARD VARIABLES34.5 SIMULATION OF TWO DEPENDENT NORMAL STANDARD VARIABLES34.6 SIMULATION OF TWO DEPENDENT TIMES TO DEFAULT34.6.1 Simulation of Time to Default for a Single Obligor34.6.2 Simulation of Two Exponentially Distributed Dependent Times to Default34.7 APPENDIX:THE CHOLESKY DECOMPOSITION METHODSection 10. Market Risk35. Delta-normal VaR35.1 PORTFOLIO DELTA-NORMAL VAR35.1.1 Steps for Determining VaR35.1.2 A Simple Portfolio of Two Zero-coupon Bonds35.2 MAPPING AN INSTRUMENTTO RISK FACTORS35.2.1 Interpolation of Interest Rate from Selected Risk Factors35.2.2 Value Preservation35.2.3 Volatility Preservation35.2.4 Mapping and Allocation35.3 THE EXAMPLE OF A FORWARD FOREIGN EXCHANGE RATE35.4 THE CONCEPTUAL FRAMEWORK OF DELTA-NORMAL VAR35.4.1 Deriving Sensitivities35.4.2 Mathematical Sensitivities35.4.3 Numerical Sensitivities35.4.4 Decomposition of the Forward as a Linear Function of Elementary Positions35.5 VOLATILITY AND DELTA-NORMAL VAR OF THE FORWARD VALUE35.6 THE DELTA-VAR OF THE FORWARD CONTRACT: SUMMARY36. Historical and Hypothetical Simulations36.1 HISTORICAL SIMULATIONS36.1.1 Principles36.1.2 Historical VaR: Forward Contract Example36.2 MONTE CARLO SIMULATIONS36.2.1 Grid Simulations36.2.2 Full Monte Carlo Simulations36.2.3 Example: Option Contract36.3 EXTENSIONS OF THE MARKET VAR METHODOLOGY36.3.1 E-VaR or Expected Shortfall36.3.2 Hypothetical Scenarios, Stress-tests and Extreme VaR37. Simulation of Interest Rates37.1 INTEREST RATES AND FACTOR MODELS37.2 PRINCIPAL COMPONENT ANALYSIS AND THE TERM STRUCTURE OF INTEREST RATES37.3 INTEREST RATE SIMULATIONS WITH PCA37.4 APPLICATION TO MARKET VAR37.5 ALM APPLICATIONS38. Back Tests, Benchmarks and Stress Tests 38.1 BACK TESTING38.2 BENCHMARKING38.3 STRESS TESTING, HYPOTHETICAL SCENARIOS AND SENSITIVITY ANALYSES38.3.1 Direct Effects or "Factor-push" Techniques38.3.2 Indirect Effects and Factor-push Scenarios38.4 VALIDATIONSection 11. Credit Risk: Standalone39. Credit Risk Data39.1 DEFAULT STATISTICS39.1.1 Annual Default Rates39.1.2 Cumulative Default Rates39.2 RECOVERY STATISTICS39.3 TRANSITION MATRICES40. Rating Systems40.1 CREDIT RATINGS40.2 CREDIT RATINGS AND LINKS BETWEEN COUNTERPARTIES40.3 INTERNAL CREDIT RATINGS AND BUSINESS RULES40.4 BUILDING BLOCKS OF THE INTERNAL CREDIT RATING SYSTEM40.5 RATING GRIDS40.5.1 Judgmental Ratings versus "Rating Models"40.5.2 Corporate Sample Rating Grid40.5.3 Banks: Sample Rating Grid40.6 MAPPING RATINGS TO DEFAULT PROBABILITIES40.7 APPENDIX: RATING SCALES OF RATING AGENCIES41. Statistical and Scoring Models41.1 SCORING41.2 THE ECONOMICS OF SCORING SYSTEMS41.3 LOGIT MODELS41.3.1 The "Basic" Linear Model Drawbacks41.3.2 Logit Model Family41.4 SCORING IN RETAIL BANKING: BEHAVIORAL VERSUS ORIGINATION MODELS41.5 IMPLEMENTATION OF SCORING IN RETAIL BANKING41.5.1 Discriminating Variables41.5.2 Identification of Potentially Significant Attributes41.5.3 Non-linear Relationship between Selected Attributes and Credit State41.5.4 Constructing Attributes from Observed Characteristics41.5.5 Fitting and Back Testing Scoring Models41.6 ACCURACY OF SCORING MODELS:THE "CAP"41.7 MAPPING SCORING MODELS TO A MASTER SCALE OF DEFAULT PROBABILITIES42. The Option Approach to Defaults and Migrations42.1 HOW FIRMS DEFAULT42.2 THE OPTION THEORETIC FRAMEWORK OF VALUATION OF EQUITY AND DEBT42.3 IMPLEMENTING THE STRUCTURAL MODEL OF DEFAULT42.4 MODELING DEFAULT PROBABILITY AND CREDIT STANDING AT HORIZON42.5 IMPLEMENTING THE EDF© MODEL42.6 THEORETICAL VALUES OF THE OPTION TO DEFAULT ANDTHE EDF©42.6.1 Valuation of the Put Option to Default42.6.2 Determination of the Default Probability42.6.3 Determining the Percentile of the Final Asset Value42.7 SAMPLE CALCULATIONS OF EDF© AND OF THE PUT AND CALL VALUES42.8 VARIATIONS ON THE MERTON'S MODEL42.9 MAPPING DEFAULT PROBABILITY TO THE STANDARDIZED NORMAL DISTANCE TO DEFAULT43. Default Probability and Default Intensity43.1 CUMULATIVE DEFAULT AND SURVIVAL PROBABILITIES43.2 FORWARD DEFAULT AND SURVIVAL PROBABILITIES43.2.1 Basic Formulas43.2.2 Calculating Marginal Default Probabilities43.3 MIGRATION MATRICES43.3.1 Migration Matrices and Cumulative Default Probabilities43.3.2 Direct Calculation of Default Probability over Two Periods43.3.3 General Calculation of Default Probability over Two Periods43.3.4 Generator Matrices43.4 CREDIT INTENSITY MODELS43.4.1 Default Intensity Models43.4.2 Forward Default Intensity43.4.3 Time to Default43.5 APPENDIX: MATRIX DIAGONALIZATION44. Credit Risk Potential Exposure44.1 BANKING PORTFOLIO EXPOSURES44.1.1 On Balance Sheet44.1.2 Off Balance Sheet Commitments44.2 MARKET INSTRUMENTS AND POTENTIAL FUTURE EXPOSURES (PFES)44.2.1 Interest Rate Swaps44.2.2 Foreign Exchange Swaps44.2.3 Options44.3 REGULATORY ADD-ONS FOR DERIVATIVES44.4 CREDIT RISK FOR DERIVATIVES: METHODOLOGY44.5 CALCULATING THE PFE FOR AN INTEREST RATE SWAP44.6 CREDIT RISK EXPOSURE FOR PORTFOLIOS OF DERIVATIVES45. Modeling Recoveries45.1 COLLATERALIZED SECURITIES LENDING/BORROWING45.1.1 Economic Derivation of Minimum Over-collateralization45.1.2 Basel 2 Treatment of Collateralized Transactions45.2 VALUATION OF CREDIT RISK GUARANTEES, INSURANCE OR CREDIT DERIVATIVES45.3 SUPPORT45.4 DISTRIBUTION OF RANDOM RECOVERIES45.5 COVENANTS46. Credit Risk Valuation and Credit Spreads46.1 CREDIT SPREAD, IMPLIED DEFAULT INTENSITY AND RECOVERY RATE46.2 CREDIT VAR AND MATRIX VALUATION46.3 CREDIT VAR AND MATRIX VALUATION: APPLICATION46.4 MIGRATIONS AND VAR UNDERTHE STRUCTURAL MODEL46.5 FORWARD VALUATION AND EXCESS SPREADSSection 12. Credit Portfolio Risk47. Credit Event Dependencies47.1 MODELING JOINT MIGRATIONS AND DEFAULTS WITH THE STRUCTURAL MODEL47.1.1 Dependency and Joint Default Probability47.1.2 Dependency and Joint Migration Probabilities47.2 JOINT DEFAULT PROBABILITY USING DISCRETE VARIABLES47.3 CONDITIONAL PROBABILITIES AND CORRELATION47.4 JOINT MIGRATION MATRICES47.5 SIMULATION OF JOINT DEFAULTS AND MIGRATIONS48. Example of Portfolio Loss Distribution 48.1 PORTFOLIO OF TWO OBLIGORS48.2 PORTFOLIO OF TWO INDEPENDENT OBLIGORS48.2.1 The Loss Distribution48.2.2 The Loss Statistics48.3 DEPENDENT DEFAULT EVENTS48.3.1 Calculation of Joint Default and Conditional Probabilities48.3.2 Loss Distribution48.3.3 Loss Statistics48.4 COMPARISON OF THE DEPENDENT AND THE INDEPENDENT CASES49. Analytical Loss Distributions49.1 INDEPENDENT DEFAULT EVENTS:THE BINOMIAL DISTRIBUTION49.1.1 Simulating Increased Diversification with Loss Independence49.1.2 The Effect of Diversification49.2 THE "STANDARDIZED" STANDALONE STRUCTURAL MODEL49.2.1 A Single Obligor Dependent on the State of the Economy49.2.2 The Asset Value and Default Probability Conditional on the State of the Economy49.2.3 The Default Probability Conditional on The State of the Economy49.2.4 Application: The Stressed Default Probability under Basel 249.3 MODELING DEFAULTS IN A UNIFORM PORTFOLIO:THE LIMIT DISTRIBUTION49.3.1 The Uniform Granular Portfolio49.3.2 Modeling Defaults under the Structural Model49.3.3 The Limit Distribution49.3.4 Application: Finding the Factor Value Matching a Given Portfolio Loss Percentile50. Simulation of Credit Portfolio Loss Distributions50.1 PRINCIPLES OF SAMPLE SIMULATIONS50.2 MONTE CARLO SIMULATIONS OF DEFAULT EVENTS BASED ON THE STRUCTURAL MODEL OF DEFAULT50.2.1 Asset Distribution and Default Probability50.2.2 The Multiple Simulations50.2.3 The Simulation Algorithm50.2.4 Simulations of Default Distributions50.2.5 Dealing with Different Default Probabilities and Discrepancies of Exposures50.3 SIMULATIONS OF TIMES TO DEFAULT50.3.1 The Simulation Algorithm50.3.2 Dealing with Different Default Probabilities and Discrepancies of Exposures51. Credit Portfolio Models51.1 CREDIT PORTFOLIO MODEL OVERVIEW51.2 MOODY'S-KMV CREDIT MONITOR AND MOODY'S-KMV PORTFOLIO MANAGER51.2.1 Conceptual Framework51.2.2 Credit Events and Credit State at Horizon51.2.3 Risk Factors and Dependence Structure51.2.4 Revaluation of Facilities at Horizon51.2.5 Generation of Portfolio Value Distribution51.2.6 Portfolio Optimization51.3 CREDIT METRICS51.4 CREDIT PORTFOLIO VIEW: ECONOMETRIC MODELS51.4.1 Credit Portfolio View Conceptual Framework51.4.2 Credit Events and Credit State at Horizon51.4.3 Risk Factors and Dependence Structure51.4.4 Revaluation of Facilities at Horizon51.4.5 Generation of Portfolio Value Distribution51.5 CREDITRISK+AND ANALYTICAL DISTRIBUTIONS51.5.1 Conceptual Framework51.5.2 Credit Events and Credit State at Horizon51.5.3 Risk Factors51.5.4 Dependency Structure51.5.5 Generation of Portfolio Value Distribution51.6 APPENDIX:THE GAMMA DISTRIBUTIONSection 13. Capital Allocation52. Economic Capital and Credit Risk VaR52.1 HORIZON FOR CREDIT CAPITAL52.2 FROM PORTFOLIO VALUE DISTRIBUTION TO CREDIT CAPITAL52.2.1 Capital Under Default Mode52.2.2 Capital Under Full Migration Mode52.2.3 Expected Loss52.2.4 Economic Capital and Loss Volatility52.2.5 From Future Values to Current Capital52.3 RAROC CALCULATIONS AT PORTFOLIO AND FACILITY LEVELS53. Capital Allocation and Risk Contributions53.1 NORMATIVE CAPITAL ALLOCATIONS AND CAPITAL EFFECTIVE UTILIZATIONS53.1.1 Normative Risk Allocations Versus Capital Utilizations53.1.2 Capital Utilizations53.2 DEFINITIONS OF RISK CONTRIBUTIONS53.3 STANDALONE INDIVIDUAL LOSSES53.4 RISK CONTRIBUTIONS AND MARGINAL RISK CONTRIBUTIONS TO PORTFOLIO LOSS VOLATILITY AND TO CAPITAL53.4.1 Risk Contributions, Portfolio Loss Volatility and Capital53.4.2 Marginal Risk Contributions53.4.3 Usages of Risk Contributions and of Marginal Risk Contributions53.5 BASIC PROPERTIES OF RISK CONTRIBUTIONS53.6 THE CAPITAL ALLOCATION MODEL AND RISK CONTRIBUTIONS53.6.1 Risk Contributions to Portfolio Loss Volatility53.6.2 Specific Cases53.6.3 From Risk Contributions to Capital Allocation53.7 SAMPLE CALCULATIONS OF RISK CONTRIBUTIONS53.7.1 Standalone Expected Loss and Portfolio Expected Loss53.7.2 Standalone Loss Volatilities and Portfolio Loss Volatility53.7.3 The Portfolio Loss Volatility53.7.4 Risk Contributions to Volatility53.7.5 Capital Allocation53.8 APPENDIX: CALCULATION OF ABSOLUTE RISK CONTRIBUTIONS FROM THE VARIANCE-COVARIANCE MATRIX54. Marginal Risk Contributions54.1 MARGINAL RISK CONTRIBUTIONS TO LOSS VOLATILITY54.2 MARGINAL RISK CONTRIBUTIONS TO CAPITAL54.3 GENERAL PROPERTIES OF RISK CONTRIBUTIONS AND MARGINAL RISK CONTRIBUTIONS54.4 MARGINAL RISK CONTRIBUTIONS TO VOLATILITY VERSUS RISK CONTRIBUTIONS54.5 MARGINAL RISK CONTRIBUTION AND SIZE OF AN EXISTING EXPOSURESection 14. Risk-adjusted Performance55. RaRoC and Shareholders' Value Added 55.1 RISK-ADJUSTED MEASURES OF PERFORMANCE55.1.1 Definitions55.1.2 The Hurdle Rate and the Cost of Capital55.1.3 The "Ex Ante" and the "Ex Post" Views55.1.4 The Reference Capital for Risk Contributions55.2 RISK-BASED PRICING AND MARGINAL RISK CONTRIBUTIONS55.2.1 Risk-based Pricing and Marginal Risk Contribution to Capital55.2.2 Risk-based Pricing and Risk Contribution to Capital55.2.3 The Pricing Paradox with Marginal Risk Contributions55.2.4 Risk-adjusted Performance versus Risk-based Pricing55.3 RAROC CALCULATIONS55.3.1 Revenues55.3.2 Expected Loss55.3.3 Other Costs55.3.4 Expanded RaRoC Formula55.4 THE RISK PREMIUM EMBEDDED IN RISK-BASED PRICING55.5 SVA MEASURES55.6 THE PRICE OF RISK AND ARBITRAGE56. Economic Income Statements56.1 THE CALCULATION OF RAROC AND SVA FOR CREDIT RISK56.1 Calculation of RaRoC and SVA56.2 RISK-BASED PRICING56.3 RISK-BASED PERFORMANCE, PRICING AND CAPITAL ALLOCATION56.3.1 Risk-based Pricing56.3.2 Risk-based Performance56.4 ORIGINATION AND POST-ORIGINATION FOLLOW UPSection 15. Credit Portfolio Management57. Portfolio Analysis57.1 THE SAMPLE PORTFOLIO AND THE SIMULATIONS57.1.1 Portfolio Data57.1.2 Portfolio Simulations57.2 PORTFOLIO LOSS DISTRIBUTION57.3 PORTFOLIO OVERVIEW57.3.1 Loss Statistics and Capital57.3.2 Portfolio Risk Return Profile57.3.3 Portfolio Concentration and Correlation Risk57.3.4 From Portfolio Risk to Individual Facilities57.4 REPORTING ALTERNATE METRICS OF RISK57.4.1 Exposures, Default Probability and Loss Given Default57.4.2 Exposure and Expected Loss57.4.3 Exposure and Capital Allocation or Loss Volatility57.5 RISK-ADJUSTED PERFORMANCE AND MISPRICING REPORTS57.5.1 Book Measures of Profitability versus Risk-adjusted Measures57.5.2 Mispricing Reports57.6 REPORTING RISK AND RETURN VERSUS BUSINESS DIMENSIONS57.7 APPENDIX I: SAMPLE PORTFOLIO INPUTS57.8 APPENDIX 2: SAMPLE PORTFOLIO OUTPUTS57.9 APPENDIX 3: PORTFOLIO ANALYSIS AND REPORTING ISSUES57.9.1 Traceability of Aggregated Measures and Risk Management57.9.2 Dealing with Multiple Dimensions57.10 PORTFOLIO ANALYSIS AND REPORTING TECHNICAL CHALLENGES57.10.1 Basic Specifications of Reporting Systems57.10.2 Stress Testing and What-if Analyses57.10.3 Interactive Implementation of Risk Systems58. Securitization and Capital Management58.1 ECONOMICS OF SECURITIZATIONS58.1.1 Expected Return on Assets and Compensations to Investors58.1.2 Variety of Securitizations58.1.3 Rationales For Securitizations58.1.4 The Securitization Organization58.2 STRUCTURING AND THE WATERFALL MECHANISM58.2.1 Structuring of Notes58.2.2 The Waterfalls of Cash Flows and Losses58.3 ECONOMICS OF SECURITIZATION FORTHE BANK58.3.1 Analysis of a Securitization Transaction (Case Study)58.3.2 The Costs of Funding On Balance Sheet and through Securitization58.3.3 The Effective Return on Capital for the Bank58.3.4 The Cost of Financing through Securitization58.3.5 The Annualized All-in Cost of Financing through Securitization58.3.6 The Pricing of Assets Sold to the SPE58.3.7 The All-in Cost of Funding through Securitization58.3.8 Securitization Economics and the Return on Equity58.3.9 Enhancing the Bank's Return on Capital through Securitization58.4 ASSESSING THE RISK OF ASSET-BACKED NOTES58.4.1 Rating Methodologies for Structured Notes58.4.2 Credit Portfolio Models and Securitizations58.4.3 Tranches are Subject to Correlation Risk59. Credit Portfolio Management59.1 THE RATIONALE FOR CREDIT PORTFOLIO MANAGEMENT59.2 TRADING CREDIT RISK59.3 APPLICATIONS OF CREDIT DERIVATIVES59.3.1 Hedging Credit Risk59.3.2 Trading Credit Risk59.3.3 Customizing Credit Risk59.3.4 Applications for Credit Portfolio Management59.3.5 Evolution of Credit Risk Models59.4 PORTFOLIO CREDIT RISK MANAGEMENT (CASE STUDY)59.4.1 Buying Protection for the Portfolio from Another Bank59.4.2 Buying Protection for the Portfolio and Selling Protection59.5 OTHER ARBITRAGE BETWEEN ECONOMIC PRICES AND RATING-BASED PRICESSection 16. Conclusion and Financial Reforms60. The Financial System and Reforms 60.1 THE FINANCIAL STABILITY FORUM RECOMMENDATIONS60.1.1 Capital60.1.2 Provisioning60.1.3 Valuation and Leverage60.2 THE WHITE PAPER FROM THE WHITE HOUSE60.2.1 Supervision60.2.2 Establish Comprehensive Supervision and Regulation of Financial Markets60.2.3 Protect Consumers and Investors from Financial Abuse60.2.4 Improve Tools for Managing Financial Crises60.2.5 Raise International Regulatory Standards and Improve International Cooperation60.3 AREAS UNDER SCRUTINY BY FINANCIAL AUTHORITIES: SUMMARY
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