POOYA FARAHVASH is vice president of Treasury Modeling and Analytics at American Express Company overseeing development of models used in ALM, liquidity risk management, stress testing, and deposit products. He previously worked at investment bank Jefferies in liquidity risk management and at CIT Group in asset-liability management. His experience in the banking industry is focused in treasury department activities, specifically in areas of interest rate risk, liquidity risk, asset-liability management, deposit modeling, and economic capital. Dr. Farahvash is also an adjunct instructor at New York University, teaching analytical courses. He received his PhD degree in Industrial and Systems Engineering and MS degree in Statistics both from Rutgers University, New Jersey. He currently lives in New York City.

About the Author xvii

Preface xix

Abbreviations xxiii

Introduction 1

Asset-Liability Management Metrics 5

ALM Risk Factors 7

Organization of This Book 8

Chapter 1 Interest Rate 17

Interest Rate, Future Value, and Compounding 18

Use of Time Notation versus Period Notation 22

Simple Interest 23

Accrual and Payment Periods 24

Present Value and Discount Factor 29

Present Value of Several Cash Flows 32

Present Value of Annuity and Perpetuity 33

Day Count and Business Day Conventions 34

Treasury Yield Curve and Zero-Coupon Rate 40

Bootstrapping 43

LIBOR 48

Forward Rates and Future Rates 49

Implied Forward Rates 50

Forward Rate Agreements 55

Interest Rate Futures 56

Swap Rate 58

Determination of the Swap Rate 61

Valuation of Interest Rate Swap Contracts 66

LIBOR-Swap Spot Curve 70

Interpolation Methods 75

Piecewise Linear Interpolation 76

Piecewise Cubic Spline Interpolation 78

Federal Funds and Prime Rates 84

Overnight Index Swap Rate 87

OIS Discounting 88

Secured Overnight Financing Rate 94

Components of Interest Rate 95

Risk Structure of Interest Rate 97

Term Structure of Interest Rate 98

Expectation Theory 100

Market Segmentation Theory 102

Liquidity Premium Theory 102

Inflation and Interest Rate 102

Negative Interest Rate 103

Interest Rate Shock 105

Parallel Shock 106

Non-Parallel Shock 107

Interest Rate Risk 109

Summary 110

Notes 112

Bibliography 114

Chapter 2 Valuation: Fundamentals of Fixed-Income and Non-Maturing Products 115

Principal Amortization 116

Bullet Payment at Maturity 116

Linear Amortization 117

Constant Payment Amortization 118

Sum-of-Digits Amortization 121

Custom Amortization Schedule 123

Fixed-Rate Instrument 124

Valuation 124

Yield 130

Duration and Convexity 133

Dollar Duration and Dollar Convexity 142

Portfolio Duration and Convexity 143

Effective Duration and Effective Convexity 144

Interest Rate Risk Immunization 145

Key Rate Duration 155

Fisher-Weil Duration 156

Key Rate Duration 160

Floating-Rate Instrument 165

Pre-Period-Initiation Rate Setting 166

Post-Period-Initiation Rate Setting 166

Valuation Using Estimated Interest Rates at Future Reset Dates 168

Using Implied Forward Rate 168

Using Forecasted Rate 171

Valuation Using Assumption of Par Value at Next Reset Date 177

Duration and Convexity 182

Valuation Using Simulated Interest Rate Paths 184

Non-Maturing Instrument 191

No New Business Treatment 192

No New Account Treatment 196

Constant Balance Treatment 197

Inclusion of Prepayment and Default: A Roll Forward Approach 198

Summary 207

Notes 210

Bibliography 210

Chapter 3 Equity Valuation 213

Dividend Discount Model 214

Discounted Free Cash Flow Method 217

Comparative Valuation Using Price Ratios 226

Summary 233

Note 234

Bibliography 235

Chapter 4 Option Valuation 237

Stock Option 238

Boundary Values 240

Call Option 241

Put Option 243

Put-Call Parity 247

Underlying Stock Does Not Pay Dividends 247

Underlying Stock Pays Dividends or Provides Yield 251

Binomial Tree 252

The Black-Scholes-Merton Model 267

Generalization of the Black-Scholes-Merton Model 272

Option Valuation Using Monte Carlo Simulation 273

Sensitivity of Option Value 282

Sensitivity to Underlying Price 282

Sensitivity to Volatility 288

Sensitivity to the Interest Rate 290

Sensitivity to the Passage of Time 291

Volatility 292

Historical Volatility 292

Implied Volatility 295

Non-Constant Volatility 297

ARCH and GARCH Models 298

Forecasting Volatility Using the GARCH Model 303

The GARCH-M Model 305

The Exponentially Weighted Moving Average Model 306

The EWMA Model for Covariance 310

Option Valuation Using a GARCH Model 312

Futures Options 319

Futures Contract 319

Option on Futures Contract 320

Put-Call Parity for Futures Options 323

Black Model 324

Using a Binomial Tree for Valuation of Futures Options 326

Summary 328

Annex 1: Derivation of Put-Call Parity When the Underlying Pays Dividends 331

Annex 2: Derivation of Delta, Gamma, Vega, Rho, and Theta 338

Notes 343

Bibliography 344

Chapter 5 Interest Rate Models 347

Instantaneous Forward Rate and Short Rate 347

Vasicek Model 354

Hull-White Model 358

Ho-Lee Model 366

Black-Karasinski Model 367

Interest Rate Options 368

Swaption 368

Interest Rate Cap and Floor 370

Analytical Valuation of Bonds and Options 373

Zero-Coupon Bond 373

Option on a Zero-Coupon Bond 374

Interest Rate Cap and Floor 375

Option on a Coupon-Bearing Bond 376

Swaption 376

Interest Rate Tree 377

The Hull-White Tree 382

The Black-Karasinski Tree 400

Calibration 405

Calibration Using the Analytical Method 408

Calibration Using the Interest Rate Tree 413

LIBOR Market Model 420

Summary 425

Annex: Derivation of Zero-Coupon Bond Price Using a ¿t-Period Rate from the Hull-White Tree 427

Notes 429

Bibliography 430

Chapter 6 Valuation of Bonds with Embedded Options 433

Callable Bond 433

Option-Adjusted Spread 441

Putable Bond 444

Summary 446

Note 447

Bibliography 447

Chapter 7 Valuation of Mortgage-Backed and Asset-Backed Securities 449

Mortgage-Backed Securities 450

Fixed-Rate Conventional Mortgage Loans 452

Prepayment 460

Impact of Prepayment on Mortgage-Backed Securities 463

Valuation of Mortgage-Backed Securities 476

Short Rate Model 476

Mortgage Refinancing Rate Model 480

Prepayment Model 483

Cash Flow Generator 483

Discounting and Aggregation Platform 484

Number of Simulated Paths and Convergence 486

Impact of Default on Mortgage-Backed Securities 488

Collateralized Mortgage Obligations 503

Valuation of Collateralized Mortgage Obligations 511

Asset-Backed Securities 513

Auto Loan ABSs 517

Collateral 517

Structure 520

Prepayment 521

Home Equity Loan ABSs 522

Collateral 522

Structure 523

Prepayment 524

Student Loan ABSs 524

Collateral 524

Structure 528

Prepayment 529

Credit Card Receivable ABSs 529

Collateral 529

Structure 530

Cash Flow Distribution Method 531

Prepayment 534

Early Amortization Event 534

Valuation of Asset-Backed Securities 535

Summary 550

Annex: Derivation of Survival Factor 552

Notes 553

Bibliography 554

Chapter 8 Economic Value of Equity 557

Economic Value of Equity: Basics 559

Duration Gap 562

Risk-Adjusted Yield Curve 567

Interest Rate Scenario Analysis 574

Product Type and Value Sensitivity 575

Impact of Interest Rate Shocks on EVE 584

Balance Sheet Type and EVE Sensitivity 593

Currency Exchange Rate Scenario Analysis 594

Economic Value of Equity Risk Limits 597

Balance Sheet Planning and EVE Forecasting 597

Basel Accord Guidance on EVE Analysis 600

Principles of Managing Interest Rate Risk in the Banking Book 601

Scenario Construction and EVE Analysis 604

Standardized Framework 607

Summary 608

Notes 610

Bibliography 611

Chapter 9 Net Interest Income 613

Interest Income and Expense: Basics 614

Interest Income and Expense for Floating-Rate Instruments 620

Using the Implied Forward Rate 621

Using the Forecasted Rate 631

Incorporating Balance Sheet Change in NII Analysis 638

Runoff View: No New Volume 638

Static View: Replacement of Matured Positions 642

Dynamic View: Incorporation of Business Plan 644

Earning Gap 648

Interest Rate Scenario Analysis 653

Parallel Shocks 654

Non-Parallel Shocks 664

Balance Sheet Type and NII Sensitivity 670

Impact of Interest Rate Options on NII 673

Currency Exchange Rate Scenario Analysis 683

Currency Forward and Interest Rate Parity 683

Exchange Rate Shock Scenarios 687

Net Interest Income Hedging 691

Net Interest Income Risk Limits 697

Required Data and Other Considerations in NII Analysis 699

Basel Accord Guidance on NII Analysis 701

Summary 702

Notes 704

Bibliography 704

Chapter 10 Equity and Earnings at Risk 705

Introduction to Value-at-Risk 706

Variance-Covariance Method 708

Historical Sampling Method 710

Monte Carlo Simulation Method 713

Conditional Value-at-Risk 717

Application of VaR Methodology in ALM 719

Scenario Generation 721

Historical Sampling 721

Monte Carlo Simulation 726

Standard and Generalized Brownian Motion 726

Multi-dimensional Brownian Motion 730

Geometric Brownian Motion 731

Mean-Reverting Brownian Motion 734

Geometric Mean-Reverting Brownian Motion 739

Calibration 743

Equity-at-Risk 743

Interest Rate Risk Factor 744

Component Contribution 748