Table of Contents Series page Title Page Copyright Preface About this book Whom is this book for? Outline of the book Acknowledgments Chapter 1: Introduction Banks and Risk Management Evolution of Bank Capital Regulation Creating Value from Risk Management Financial Risk Systems Model Risk Management Part One: Market Risk Chapter 2: Market Risk with the Normal Distribution Linear Portfolios Quadratic Portfolios Simulation-Based Valuation Chapter 3: Advanced Market Risk Analysis Risk Measures, Risk Contributions, and Risk Information Modeling the Stylized Facts of Financial Time Series Time Scaling VaR and VaR with Trading Market Liquidity Risk Scenario Analysis and Stress Testing Portfolio Optimization Developments in the Market Risk Internal Models Capital Regulation Part Two: Credit Risk Chapter 4: Portfolio Credit Risk Issuer Credit Risk in Wholesale Exposures and Trading Book Credit Models for the Banking Book Firmwide Portfolio Credit Risk and Credit Risk Dependence Credit Risk Stress Testing Features of New Generation Portfolio Credit Risk Models Hedging Credit Risk Regulatory Capital for Credit Risk Appendix Chapter 5: Counterparty Credit Risk Counterparty Pricing and Exposure CVA Risks Portfolios of Derivatives Recent Counterparty Credit Risk Developments Counterparty Credit Risk Regulation Part Three: Asset and Liability Management Chapter 6: Liquidity Risk Management with Cash Flow Models Measurement of Liquidity Risk Liquidity Exposure Hedging the Liquidity Exposure Structural Liquidity Planning Components of the Liquidity Hedging Program Cash Liquidity Risk and Liquidity Risk Measures Regulation for Liquidity Risk Chapter 7: Funds Transfer Pricing and Profitability of Cash Flows Basic Funds Transfer Pricing Concept Risk-Based Funds Transfer Pricing Funds Transfer Rate and Risk Adjusted Returns Profitability Measures and Decompositions Banking Book Fair Value with Funds Transfer Rates A Note on the Scope of Funds Transfer Pricing Regulation and Profitability Analysis Part Four: Firmwide Risk Chapter 8: Firmwide Risk Aggregation Correlated Aggregation and Firmwide Risk Levels Mixed Copula Aggregation Capital Allocation in Risk Aggregation Risk Aggregation and Regulation Chapter 9: Firmwide Scenario Analysis and Stress Testing Firmwide Scenario Model Approaches Firmwide Risk Capital Measures Regulatory Stress Scenario Approach The Future of Firmwide Stress Testing References Index End User License Agreement List of Illustrations Chapter 1: Introduction Figure 1.1 Risk Calculation Flow in a Financial Risk System Chapter 2: Market Risk with the Normal Distribution Figure 2.1 Profit-and-Loss Profiles for Barrier Options for a Grid of the Underlying Stock Value Figure 2.2 100 Normal Simulated Points from a Pseudo-Random Number Generator (Left) and a Quasi-Random Number Generator (Right) Figure 2.3 The Inverse Function Transformation for the Normal Distribution Figure 2.4 Eigenvalues and Variance Explained by Principal Components Analysis of the US Treasury Curve Figure 2.5 First Three Principal Components of the US Treasury Yield Curve Chapter 3: Advanced Market Risk Analysis Figure 3.1 VaR 99% Confidence Level Risk Measures, Incremental Risk, and Risk Contributions for the Portfolio Instruments Figure 3.2 Portfolio Trade Risk Profile for the European Call Option on Stock A as We Vary Its Holding from 1 to −1 Figure 3.3 Trade Risk Profile for Convex and Non-convex Risk Measures Figure 3.4 Empirical Distribution Function (CDF) and Block Maxima Distortion Function for n Equal to 5, 10, and 20 for a Loss Sample Figure 3.5 A Typical Financial Time Series of Returns that Displays the Well-Known Property of Volatility Clustering Figure 3.6 Autocorrelation of Returns and Squared Returns for the S&P 500 Index Figure 3.7 GARCH Filtered Lower Tail Residuals of the USD/SEK Exchange Rate. The Fitted Tail Distributions are the Empirical Distribution Function, Normal Density, t-Density, and the Generalized Pareto Distribution Figure 3.8 GARCH Filtered Lower Tail Residuals of the ABB Stock. The Fitted Tail Distributions are the Empirical Distribution Function, Normal Density, t-Density, and the Generalized Pareto Distribution Figure 3.9 Bivariate Frechet Bound and Independence Copulas Figure 3.10 Finite Tail Dependence for the Normal and t-Copula for Different Values of the Correlation Parameter and Threshold Figure 3.11 Bivariate Scatter Plots of Equity Returns Figure 3.12 Bivariate Scatter Plots of 10,000 Uniform Copula Simulations of Returns for Equity 1 and and Equity 2 from the Normal Copula, the t-Copula (Estimated with Kendall's Tau), and Archimedean Copulas Estimated with Kendall's Tau (Clayton and Frank) Figure 3.13 Bivariate Scatter Plots of 10,000 Empirical Density Copula Simulations of Returns for Equity 1 and Equity 2 from the Normal Copula, the t-Copula (Estimated with Kendall's Tau), and Archimedean Copulas Estimated with Kendall's Tau (Clayton and Frank) Figure 3.14 VaR at the 99% Confidence Level for Partial and Full Closeout at the Liquidity Horizons Figure 3.15 VaR at the 99% Confidence Level for Full Closeout for Different Stocks at the Liquidity Horizons and Full Closeout at the Liquidity Horizons Figure 3.16 Index Portfolio Returns Between May 3, 1993, and December 26, 2011 Figure 3.17 Conditional Mean, Most Likely Loss and 50% and 90% Confidence Intervals Around the Points for the Nikkei 225 and Hang Seng Index Risk Factors Figure 3.18 Efficient Frontiers for the Unconstrained and the Constrained Case Across CVaR 99% Confidence Risk Ranging Between 1 and 15% of Portfolio Mark to Market Figure 3.19 Asset 1 and Liability Cash Flow Scenarios for the 8 Time Buckets Chapter 4: Portfolio Credit Risk Figure 4.1 Merton Credit Spreads in Percent as the Debt Ratio, K/V, Varies Between 50, 65, and 75% and for 1, …, 10 Years Bond Maturity Figure 4.2 Example of Obtained Rating Grade Thresholds Using the Transition Probabilities of the Standard and Poor's Rating Grade Scale Figure 4.3 Corporate Bond Portfolio VaR at Different Confidence Levels Figure 4.4 Firm Portfolio VaR at Horizons 3, 6, 9, and 12 Months Figure 4.5 Credit Portfolio Loss Tail for the Normal and t(3) Copula for Confidence Levels 99% and Higher Figure 4.6 Mixed Firm and Pool Portfolio VaR and CVaR Risk Measures at the 12- Month Horizon Figure 4.7 99.9% VaR (Expressed in Percentage of Current Mark to Market) for the Firms Together with the 95% Lower and Upper Confidence Levels for the Different Simulation Sizes Figure 4.8 99.9% VaR (Expressed in Percentage of Current Mark to Market) for the Pools Together with the 95% Lower and Upper Confidence Levels for the Different Simulation Sizes Figure 4.9 Simulation Scatterplot of Default Times (Expressed in Years) from a Normal Copula with Correlation Parameter 0 Using 100,000 Simulations Figure 4.10 Simulation Scatterplot of Default Times (Expressed in Years) from a Normal Copula with Correlation Parameter 0.99 Using 100,000 Simulations Figure 4.11 Simulation Scatterplot of Default Times (Expressed in Years) from a Clayton Copula with Copula Parameter 30 Using 100,000 Simulations Figure 4.12 Historical Data for the Systematic Factor and the Price Indices Figure 4.13 Aggregate Portfolio Marginal Risk and Expected Loss at Horizons of 1, …, 10 Years Figure 4.14 99.9% VaR for the Credit Losses for the Large Pool Subportfolios 10, 11, and 14 Figure 4.15 99.9% VaR for the Credit Losses for the Large Pool Subportfolios 1, 3, and 6 Figure 4.16 99.9% VaR for the Net Credit Losses, Credit Losses, and the Expected Loss Adjusted Credit Losses for the Aggregate Portfolio Figure 4.17 Regulatory Capital 99.9% VaR and Expected Value over the Risk Horizon of 1, …, 10 Years for the Aggregate Portfolio Figure 4.18 Regulatory Capital 99.9% VaR and Expected Value over the Risk Horizon of 1, …, 10 Years for Subportfolio 11 in Grade 2 Figure 4.19 Aggregate Portfolio Percent Increase of Marginal Risk for 1, …, 10 Years Using Sampling for all the Portfolio Loans Figure 4.20 Subportfolio 8 Percent Increase of Marginal Risk for 1, …, 10 Years Using Sampling for all the Portfolio Loans Figure 4.21 Subportfolio 10 and 14 Percent Increase of Marginal Risk for 1, …, 10 Years Using Sampling for all the Portfolio Loans Figure 4.22 Loss Probability Density Obtained from the Adjusted Panjer Recursion When Average Loan Default Rates Are Reduced by a Factor of 10 and When the Average Deafult Rates Are Unadjusted Figure 4.23 Pools Portfolio Loss in Each Scenario Figure 4.24 Systematic Risk Factor Scenarios in the Extreme and Moderate Economic Distress Cases Figure 4.25 Aggregate Mortgage Portfolio Scenario Analysis of Net Earnings and Regulatory Capital Figure 4.26 Mortgage Portfolio Subportfolio 11 in Rating Class 2 Scenario Analysis of Net Earnings and Regulatory Capital Figure 4.27 Potential Recovery Process Figure 4.28 Credit Model with Cash Flows Figure 4.29 Sample Survival Curves A, B, and C Chapter 5: Counterparty Credit Risk Figure 5.1 Obtaining Default Probabilities of Counterparties Figure 5.2 Market Zero Rate and Instantaneous Forward Rate Term Structure Figure 5.3 Sample Simulation Paths for the Calibrated Hull-White Model Figure 5.4 10,000 Simulated Market Trajectories for the 1-Year Term Node Interest Rate on Days 1, …, 2600 Figure 5.5 10,000 Simulated Market Trajectories for the FX Forward Rate on Days 1, …, 2600 Figure 5.6 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 10-Year at-the-Money Payer Swap Figure 5.7 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 10-Year In-the-Money Payer Swap Figure 5.8 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 10-Year Out-of-the-Money Payer Swap Figure 5.9 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 10-Year at-the-Money Receiver Swap Figure 5.10 Comparison of Expected Exposure for 10-Year at-the-Money Receiver and Payer Swaps Figure 5.11 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 10-Year at-the-Money Cross Currency Payer Swap Figure 5.12 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 10-Year at-the-Money Payer Swap with Unequal Payment Frequencies—Pay Quarterly and Receive Yearly Figure 5.13 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 10-Year at-the-Money Payer Swap with Unequal Payment Frequencies—Pay Yearly and Receive Quarterly Figure 5.14 Sample Collateral Flows for a 1-Year at-the-Money Interest Rate Swap Exposure Path Figure 5.15 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 1-Year at-the-Money Payer Swap With No Collateral Figure 5.16 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 1-Year at-the-Money Payer Swap with Daily Margin and No Settlement Risk Figure 5.17 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 1-Year at-the-Money Payer Swap with Daily Margin and 5-Day Settlement Risk Figure 5.18 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 1-Year at-the-Money Payer Swap with Daily Margin and 10-Day Settlement Risk Figure 5.19 Expected Exposure for a 1-Year at-the-Money Payer Swap, Uncollateralized and with Different Settlement Risks Figure 5.20 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 1-Year at-the-Money Payer Swap with 10-Day Margin Frequency and No Settlement Risk Figure 5.21 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 1-Year at-the-Money Payer Swap with 10-Day Margin Frequency and 5-Day Settlement Risk Figure 5.22 Expected Exposure, the Running Mean of Expected Positive Exposures, and the Peak Exposure Profile for a 1-Year at-the-Money Payer Swap with 10-Day Margin Frequency and 10-Day Settlement Risk Figure 5.23 Expected Exposures for Portfolio Trades and the Netted Portfolio Figure 5.24 Peak Exposures for Portfolio Trades and the Netted Portfolio Figure 5.25 Marginal Expected Exposures for Portfolio Trades and the Netted Portfolio Figure 5.26 Marginal Worst-Case Exposures for Portfolio Trades and the Netted Portfolio Chapter 6: Liquidity Risk Management with Cash Flow Models Figure 6.1 Sample Cash Flows, Forward Liquidity Exposures, and Risk Reserve Process for Liquidity Scenario 1 Figure 6.2 Sample Cash Flows, Forward Liquidity Exposures, and Risk Reserve Process for Liquidity Scenario 2 Figure 6.3 Cash Liquidity Needs for the 3 Liquidity Scenarios Figure 6.4 Cash Liquidity Need for the 3 Liquidities, the Portfolio, and the Summed Liquidity Need Chapter 7: Funds Transfer Pricing and Profitability of Cash Flows Figure 7.1 Funding Curve Used in Calculation of Funds Transfer Rates Figure 7.2 FTP Breakdown of the Balance Sheet Figure 7.3 Sample Mortgage Net Interest Margin Relative to Synthetic FTP Legs Chapter 8: Firmwide Risk Aggregation Figure 8.1 Sample Risk Aggregation Hierarchies Chapter 9: Firmwide Scenario Analysis and Stress Testing Figure 9.1 Firmwide Risk-Based Capital Stress Testing Figure 9.2 Regulatory Stress Scenario Approach Figure 9.3 Firmwide Stress Testing Process List of Tables Chapter 2: Market Risk with the Normal Distribution Table 2.1 The VaR(0.99) and CVaR(0.99) Risk Measures for the Aggregate Portfolio and the Equity Positions, P , …, P . The Table also Displays the VaR(0.99) and 1 5 CVaR(0.99) Euler Risk Contributions with Respect to the Positions P , …, P 1 5 Table 2.2 Profit-and-Loss Profiles for Barrier Options for a Grid of the Underlying Stock Value Table 2.3 Mean and Standard Deviation of US Treasury Historical Yields Table 2.4 First Three Principal Components of the US Treasury Yield Curve Chapter 3: Advanced Market Risk Analysis Table 3.1 Sample Equity Portfolio Table 3.2 Sample Equity Portfolio Risk Measures for 1,000, 10,000, 100,000, and 1,000,000 Scenarios Table 3.3 Example Ordered Loss Matrix Table 3.4 VaR and CVaR 99% Confidence Level Risk Measures, Incremental Risk, and Risk Contributions for the Portfolio Instruments Table 3.5 Portfolio Trade Risk Profile for the European Call Option on Stock A as we Vary Its Holding from 1 to Table 3.6 VaR Risk Contributions and Smoothed Risk Contributions for the Portfolio Instruments Table 3.7 VaR and CVaR 99% Confidence Level Diversification Indices Table 3.8 Sample Distributions of X and Y (Bold Sample Numbers Indicate the Sample Belonging to the Left Part of the Distribution, i.e., the First Quintile for s = 2) Table 3.10 Risk Factor Information Measures and Correlation of Equity Risk Factors to Portfolio Profit and Loss Table 3.9 Risk Factor Information Measures and Correlation of Equity Risk Factors to Portfolio Profit and Loss When Holding in Equity A is Changed to 10 Units Instead of 1 Unit Table 3.11 Risk Distortion Measures Table 3.12 Unconditional Moments of Returns to the SP 500 Index Table 3.13 Hill-Based Exteme Value Portfolio Risk for Different Percentage Sample Points in the Tail Table 3.14 Sample Equity Returns for 10 Days Table 3.15 Estimation of Theta Parameter for Archimedean Copulas Clayton, Gumbel, and Frank Using Kendall's Tau and Maximum Likelihood Method Table 3.16 VaR Exceedances for the Models Table 3.17 Days Between VaR Exceedances for the Models Table 3.18 VaR Backtesting Comparison of the Models, Comparing Expected Number of Exceedances Versus Actual and p-Values from the Unconditional Coverage Test, the Independence and the Duration Test Table 3.19 Stock Allocation to Liquidity Groups Table 3.20 Percent Closeout VaR Profile and the No-Closeout VaR Profile on Either of Days 1, …, 15 Table 3.21 Stock Allocation to Full Closeout Horizons Table 3.22 Percent Tiered Full Closeout VaR Profile on Days 1, …, 15 Table 3.23 Delta Approximation of 3 Market Risk Books Table 3.24 Market Indices Volatility and Degrees of Freedom Table 3.25 VaR and CVaR 99% Confidence Level Risk Measures, Incremental Risk, and Risk Contributions for the Approximate Firmwide Market Risk Portfolio