Bond Pricing and Yield Curve Modeling: A Structural Approach

Description

In this book, well-known expert Riccardo Rebonato provides the theoretical foundations (no-arbitrage, convexity, expectations, risk premia) needed for the affine modeling of the government bond markets. He presents and critically discusses the wealth of empirical findings that have appeared in the literature of the last decade, and introduces the 'structural' models that are used by central banks, institutional investors, sovereign wealth funds, academics, and advanced practitioners to model the yield curve, to answer policy questions, to estimate the magnitude of the risk premium, to gauge market expectations, and to assess investment opportunities. Rebonato weaves precise theory with up-to-date empirical evidence to build, with the minimum mathematical sophistication required for the task, a critical understanding of what drives the government bond market. Frontmatter......Page 2 Dedication......Page 6 Contents......Page 8 Acknowledgements......Page 24 Symbols and Abbreviations......Page 26 Part I The Foundations......Page 30 1.1 My Goal in Writing This Book......Page 32 1.2 What My Account Leaves Out......Page 34 1.3 Affine Models......Page 35 1.4 A Simple Taxonomy......Page 37 1.5.1 Latent versus Observable Variables......Page 39 1.5.2 The Spanning Problem......Page 44 1.5.3 The Constraint Problem......Page 45 1.6 Why Do We Need No-Arbitrage Models After All?......Page 48 1.7 Stamp Collecting and Shallow versus Deep Explanations......Page 49 1.8 The Ideal Reader and Plan of the Book......Page 50 2.2.1 Arbitrage......Page 53 2.2.2 Pseudo-Arbitrage......Page 54 2.2.3 Sharpe Ratios......Page 56 2.2.4 Bond Prices and Yields......Page 57 2.2.5 Duration and Convexity......Page 60 2.2.6 Forward Rates......Page 61 2.3 Log Prices and Log Returns......Page 62 2.4 Dimensional Analysis......Page 63 2.5.1 Definition......Page 65 2.5.2 Transformations of Vectors......Page 66 2.5.3 Orthogonal Matrices......Page 67 2.5.4 Row Vectors......Page 68 2.5.5 Exponential of a Matrix......Page 69 2.6.1 The Ornstein–Uhlenbeck Process......Page 70 2.6.2 The AR(1) Process......Page 71 2.6.3 Parallels between AR(1) Processes and the Ornstein–Uhlenbeck Process......Page 72 2.7.1 Ito’s Lemma......Page 73 2.7.2 Stochastic-Calculus Rules for d pt dxt......Page 74 2.7.3 Expectations of Ito Integrals......Page 75 2.7.4 The Ito Isometry......Page 76 2.7.5 Risk-less Portfolios......Page 77 3.1 The Purpose of This Chapter......Page 78 3.2 The Monetary Channels......Page 79 3.3 A Modelling Framework......Page 81 3.4 The Monetary Actions: A Simple Model......Page 85 3.5.1 General Considerations......Page 87 3.5.2 Assessing the Quality of the Calibration Process......Page 89 3.5.3 State Variables versus Model Parameters......Page 90 4.1 The Purpose of This Chapter......Page 92 4.2.1 Inflation Risk......Page 93 4.2.2 Real-Rate Risk......Page 94 4.2.3 Putting the Pieces Together......Page 95 4.3 Real-World and Risk-Neutral Probabilities: The Market Price of Risk......Page 97 4.3.1 Introducing the P and Q Measures......Page 98 4.3.2 Introducing the Market Price of Risk......Page 101 4.4 An Important First Result: Bond Prices as Q-Expectations......Page 105 4.5.1 The General Case......Page 106 4.5.2 The Affine Case......Page 107 4.6 Nominal Rates, Inflation and Real Rates: Definitions......Page 108 5.2.1 An Account of What Happened......Page 110 5.2.2 Possible Explanations of What Happened......Page 114 5.3 How Can We Estimate Risk Premia?......Page 116 5.4 Different Types of Risk Premia......Page 117 5.5.1 Decomposition of the Risk Premium......Page 120 5.5.2 ‘Which’ Liquidity Are TIPS-Investors Compensated For?......Page 122 5.6 What Is and What Is Not a True Risk Premium......Page 123 5.7 Does It Matter if a Risk Premium Is ‘Really’ a Risk Premium?......Page 125 6.2.1 The Axis Rotation......Page 127 6.3 How Many Principal Components Do We Need for Yields?......Page 132 6.4 First Conclusions......Page 133 6.5 Some Mathematical Results......Page 134 7.2.1 Descriptive Features......Page 137 7.2.2 Mean-Reverting Properties – Each PC in Isolation......Page 141 7.2.3 The Joint Mean-Reverting Behaviour of Principal Components......Page 145 7.3 Real Rates and Break-Even Inflation......Page 151 7.4 Correlation between Nominal, Inflation and Real Principal Components......Page 157 Part II The Building Blocks: A First Look......Page 162 8.2 Linking Expectations with No-Arbitrage......Page 166 8.2.1 A One-Factor World......Page 167 8.2.2 Moving to Many Factors......Page 170 8.3 An Example: A Mean-Reverting Process for the Short Rate......Page 172 8.4 Expectations and Survey Data......Page 174 9.2 Where Does Convexity Come from?......Page 176 9.3 The Links between Convexity and Jensen’s Inequality......Page 178 9.3.1 A Special but Important Case: Gaussian Random Variables......Page 179 9.4 What Does Convexity Depend On?......Page 181 9.5 Why Convexity Is Different......Page 183 9.6 Why Isn’t Convexity ‘Always Good’?......Page 185 9.7 Who Sets the Price of Convexity? A Bit of Story-Telling......Page 186 10.1 The Purpose of This Chapter......Page 189 10.3.1 Distributional Properties of the Vasicek Model......Page 190 10.3.3 The Duration in the Vasicek Model......Page 194 10.4 Rate Expectations and the Shape of the Vasicek Yield Curve......Page 197 10.5.1 An Expression for Convexity......Page 199 10.5.2 Convexity and the Volatility of Yields......Page 200 10.5.3 How Big Should One Expect the Convexity Effect to Be?......Page 201 10.5.4 What Is the ‘Right’ Reversion Speed?......Page 202 10.6 The Risk Premium in the Vasicek Model......Page 204 10.7 The Functional Form of the Market Price of Risk......Page 205 10.8 The Link between the Market Price of Risk and the Sharpe Ratio......Page 207 10.9 Appendix 10A: Proof that rt = .........Page 210 Part III The Conditions of No-Arbitrage......Page 212 11.1 The Purpose of This Chapter......Page 214 11.2 Type-I Arbitrage......Page 215 11.3 Bounds to the Price-Correction Term: Type-I Arbitrage......Page 217 11.4 Bounds to the Price-Correction Term: Type-II Arbitrage......Page 218 11.5 A Useful Rewriting......Page 221 11.6 Extension to Many Factors......Page 222 11.7 The Task of the Long-Term Bond Investor......Page 224 12.2 Constructing a Risk-Less Portfolio: The Market Price of Risk Again......Page 225 12.3 Interpretations of the Market Price of Risk......Page 228 12.4 Excess Returns......Page 229 12.5 What the Market Price of Risk Can Depend On......Page 230 12.6 Appendix 12A: The Market Price of Risk and Excess Return with Many Factors......Page 231 13.1 The Purpose of This Chapter......Page 235 13.2 A Bird’s Eye View of the ‘Traditional’ and ‘Modern’ Approaches......Page 236 13.3 Pricing Assets: The Building-Blocks Approach......Page 237 13.4.1 Prices as Expectations in the Risk-Neutral Measure – Again......Page 240 13.4.2 The Equivalence of the State-Price Deflator and the Stochastic Discount Factor......Page 242 13.5 The Process for the State-Price Deflator......Page 243 13.7 Deriving the Drift of the State-Price Deflator......Page 245 13.8 The Short Rate Again......Page 247 13.9 Deriving the Volatility of the State-Price Deflator......Page 248 13.9.1 Evaluation of the Three Terms......Page 249 13.9.2 The Link between the Volatility of the State-Price Deflator and the Market Price of Risk......Page 250 13.9.3 Where Does the Volatility of Bonds Come from?......Page 251 13.9.4 Summary of Results......Page 252 14.2 The Expression for the Real State-Price Deflator......Page 253 14.3 The Process for the Real State-Price Deflator......Page 255 14.4 The Link between Break-Even Inflation and Inflation Expectations......Page 258 14.4.1 Inflation Expectation Under P......Page 259 14.4.2 Inflation Expectation under Q......Page 261 14.4.3 Inflation Expectation under T......Page 262 14.4.4 Inflation Expectations under Different Measures......Page 263 14.5 The Risk Premium as a Covariance......Page 264 14.6 Moving to an Affine World......Page 267 14.7 The Market Price of Inflation Risk – Affine Models......Page 268 15.1 The Purpose of This Chapter......Page 270 15.2 First Derivation of the SDF......Page 271 15.3 From the SDF to Risk Premia......Page 274 15.4 Real versus Nominal Prices......Page 277 15.5 Idiosyncratic Risk......Page 278 15.6 The Links between the SDF and the Risk-Less Rate......Page 279 15.6.1 The No-Uncertainty Case......Page 280 15.6.2 Reintroducing Uncertainty......Page 281 15.6.3 But Does It Work?......Page 282 15.7 SDFs in Continuous and Discrete Time......Page 285 15.8 A More General Result for the Sharpe Ratio......Page 286 Part IV Solving the Models......Page 290 16.1 Purpose of This Chapter......Page 292 16.2.1 The PDE Satisfied by Bond Prices......Page 293 16.3 A Special Case: Affine Term Structures......Page 295 16.4 The Vasicek Case......Page 298 16.5 Affinity of the Vasicek Model under P and under Q......Page 300 16.6.1 Yields......Page 301 16.6.3 Forward Rates......Page 302 16.6.4 Calibrating to the Volatility Structure: Factorization......Page 305 16.6.5 Fitting to the Yield Curve......Page 306 16.7 Why Do We Care about a Humped Volatility Curve?......Page 310 16.8 How to Lengthen the Short Blanket......Page 311 17.2 Affine Models with Many State Variables......Page 314 17.2.1 The N = 2 Case......Page 316 17.2.2 An Expression for the Variance for Generic N......Page 317 17.2.3 Stability......Page 319 17.2.4 Changing Variables......Page 320 17.3 Multivariable Exponentially Affine Models......Page 321 17.4.1 Yields and Forward Rates......Page 322 17.4.2 Distributional Properties......Page 323 17.5 Appendix 17A: Derivation of the Variance of a One-Dimensional Mean-Reverting Process......Page 324 17.6 Appendix 17B: Derivation of the Variance of a Multidimensional Mean-Reverting Process......Page 325 17.7 Appendix 17C: Stability of the Mean-Reverting System......Page 326 18.1 The Purpose of This Chapter......Page 328 18.2 What Is an Affine Model?......Page 329 18.3 The General Strategy......Page 330 18.4 Summary of the Equations Derived in Appendix 18A......Page 332 18.5.1 Expression for the Yield......Page 333 18.5.2 Expression for the Yield Covariance Matrix and the Yield Volatilities......Page 334 18.5.3 Expression for the Volatility of the Instantaneous Forward Rates......Page 335 18.6 Derivation of the Mean and Variance of the State Variables......Page 336 18.7.1 Simple Vasicek......Page 338 18.7.2 The Doubly-Mean-Reverting Vasicek Model......Page 339 18.7.3 The Trebly-Mean-Reverting Vasicek Model......Page 340 18.7.4 The Stochastic-Market-Price-of-Risk Model......Page 341 18.8 Appendix 18A: Solving for .........Page 342 18.8.1 Solving the ODE for .........Page 343 18.8.2 Solving the ODE for A(τ )......Page 345 18.9.1 The Meaning of eA......Page 352 18.10 Explicit Calculation of the Formal Solution .........Page 353 18.10.1 The Up-and-Down Theorem......Page 354 18.10.2 Commutation Relationships for A and f (A)......Page 355 18.10.4 Integral of eAt......Page 356 18.10.5 Evaluation of the Integral .........Page 357 19.1 The Purpose of This Chapter......Page 358 19.2 Motivation: Why the Shadow Rate Matters......Page 359 19.3 How the Shadow Rate Affects the Whole Yield Curve......Page 361 19.4.1 The Setting......Page 362 19.4.2 An Approximate Solution......Page 363 19.5.2 The Effect of the Shadow Rate on Long Yields......Page 368 19.6 A Broader View of the Topic......Page 372 Part V The Value of Convexity......Page 378 20.2 Break-Even Volatility – The Vasicek Setting......Page 380 20.3 Problems with the Vasicek Version of the Break-Even Volatility......Page 384 20.4 Generalizing to Many Factors......Page 386 20.4.1 Calculating the Terms.........Page 389 20.4.2 Expressing the Convexity in Terms of Yield Volatilities......Page 391 20.5 What to Do with This Expression for the Convexity......Page 392 20.5.1 An Important Aside......Page 393 20.6 An Intuitive Aside: Simplifying the Analysis......Page 394 20.7 A Graphical Interpretation......Page 397 20.8 Appendix 20A......Page 398 21.1 The Purpose of This Chapter......Page 400 21.2 Equivalent Affine Models......Page 402 21.3 The Expression for Convexity in an Affine Setting......Page 403 21.4 An Expression for the Theoretical Convexity of the Portfolio......Page 406 21.5.1 Theoretical Portfolio Convexity as a Function of Forward Rates......Page 409 21.5.2 The Portfolio Time Decay as a Function of ‘Carry’ and ‘Roll-Down’......Page 410 21.6 What These Results Imply......Page 412 21.7 Linking the Term 21 Tr[STDS] with Yield Volatilities......Page 413 21.8 Making the Weights (Almost) Model Independent......Page 415 21.9 How General Are the Results?......Page 417 21.10 Model-Based or Empirical?......Page 418 22.1 The Purpose of This Chapter......Page 420 22.2 The Strategy: A Reminder......Page 422 22.3.1 Determining the Optimal Weights......Page 424 22.3.2 Estimating the Yield Volatilities......Page 425 22.4.1 Is the Yield Curve Fairly Curved?......Page 427 22.4.2 Why Are the Strategies Not Always Profitable?......Page 431 22.4.3 Is the Strength of the Signal Correlated with the Money Made?......Page 434 22.4.4 Explaining the Residuals – Residual Exposure?......Page 435 22.4.5 Explaining the Residuals – Wrong Volatility Estimate?......Page 438 22.5 Conclusions......Page 440 Part VI Excess Returns......Page 442 23.2 The (Local) Expectation Hypothesis......Page 444 23.3 What One Really Tests for When One Tests the (L)EH......Page 446 23.4.1 General Exact Results......Page 448 23.4.2 Special Cases......Page 449 23.4.3 Approximate Results for the τ = n = 1 Case......Page 450 23.6 Excess Returns with Real Rates......Page 451 23.7 Excess Returns: Links with Carry, Roll-Down and Related Market Lore......Page 454 23.8 Why ‘Carry’ and ‘Roll-Down’ Matter......Page 458 24.1 The Purpose of This Chapter......Page 460 24.2.2 The ‘Forwards-Come-True’ Condition......Page 461 24.3.1 Market Yields versus Expected Yields......Page 462 24.4 The Link between Excess Returns and Term Premia......Page 465 24.5 The Link between Term Premia and Expected Excess Returns......Page 467 24.6 Reconciling Results......Page 469 24.7 Expected versus Realized Excess Returns......Page 470 24.8 Forwards-Come-True versus Yields-Don’t-Move: Roll-Down Again......Page 475 24.9 When to Invest......Page 477 25.1 The Purpose of This Chapter......Page 478 25.2.1 The Empirical Questions......Page 479 25.2.2 A Very Important Caveat on Spanning......Page 481 25.2.3 The Methodological Dilemma......Page 483 25.3 Unconditional Results: Nominal Bonds......Page 484 25.4.1 1- to 10-Year Returns......Page 486 25.4.2 Effectiveness of Various Regressors......Page 488 25.4.3 5-Year Returns: Comparison with Cochrane–Piazzesi (2005)......Page 490 25.5 Where Has the Volatility Gone?......Page 491 25.6 Regression Results: Real Bonds......Page 492 25.8 The Real Excess Returns......Page 493 25.9 Extracting the Real-Rate Risk Premium......Page 497 25.10 Estimating the Inflation Premium in Nominal Bonds......Page 499 25.10.1 Isolating the Liquidity Component......Page 500 26.1 The Purpose of This Chapter......Page 502 26.2 The Early Work......Page 503 26.3 Cochrane and Piazzesi (2005)......Page 504 26.5 Robustness of the Tent Shape: Tents versus Bats......Page 507 26.6 The Link Between the Tent and the Bat Factors: Constrained Regressions......Page 511 26.6.1 Constrained Regression: The Investigation Methodology......Page 512 26.6.2 Constrained Regression: Results......Page 514 26.6.3 Constrained Regression: First Conclusions......Page 516 26.7.1 Tent versus Slope Shape Similarity: The Investigation Methodology......Page 517 26.7.2 Tent versus Slope Shape Similarity: Results......Page 519 26.8.1 Tent versus Slope Robustness: Methodology......Page 520 26.8.2 Tent versus Slope Robustness: Results......Page 522 26.8.3 Tent versus Slope Robustness: Conclusions......Page 524 27.1 The Purpose of This Chapter......Page 526 27.2.1 Features and Highlights of the Radwanski Results......Page 527 27.2.2 The Methodology and Results......Page 528 27.2.3 Comments and Conclusions......Page 532 27.3.1 Main Results......Page 533 27.3.2 The Spanning of Yield Curve Factors Revisited: Implication for Affine Models......Page 536 27.4 Yield-Curve Spanning: Why One May Need Five Factors After All......Page 537 27.4.1 The Essentially Affine Description......Page 538 27.4.3 Augmenting the State Vector......Page 540 27.4.4 Switching Back to an ‘Augmented’ Set of Yields as State Variables......Page 541 27.4.6 Spanning in Principle versus Spanning in Practice......Page 542 27.5.1 The Set-Up and Main Features......Page 543 27.5.2 The Investigation Methodology and Results......Page 544 27.5.3 The Link with Forward-Rate-Based RPFs......Page 548 27.5.4 Intrinsic Limitations of Forward-Rate-Based Factors......Page 549 27.5.6 Re-Interpretation of the Cieslak–Povala RPF: Conditional Slope and Level......Page 551 27.6 Related Work......Page 554 28.1 The Purpose of This Chapter......Page 556 28.2 What Does Not Qualify as an Explanation......Page 557 28.3 Excess Returns, the Slope and the Real Economy......Page 558 28.4 The Data-Generating, Subjective and Risk-Neutral Measures......Page 560 28.5 Does It Matter?......Page 562 28.6 Why Is the Slope Significant? A Heterogeneous-Expectations Model......Page 563 28.7 Why Is the Slope Significant? An Over-reaction Model......Page 567 28.8.1 The Actions of the Central Bank......Page 568 28.8.3 The Bond Price Formation......Page 570 28.8.5 The Simulations......Page 571 28.8.6 Summary of Results......Page 575 29.2 What Is the Spanning Problem?......Page 576 29.3 The Empirical Spanning Problem......Page 577 29.4 The Theoretical Spanning Problem......Page 580 29.5 The Modelling Choices to Handle the Spanning Problem......Page 581 Part VII What the Models Tell Us......Page 586 30.1 The Purpose of This Chapter......Page 588 30.2 The Doubly Mean-Reverting Vasicek Model......Page 589 30.3 Bond Prices and Properties of the Solution......Page 590 30.4 The Volatility of the Instantaneous Forward Rate......Page 591 30.5 The Building Blocks......Page 593 30.6 Initial Conclusions......Page 597 30.7.1 Calibrating the Model to the Volatility Structure......Page 598 30.7.2 Calibrating the Model to the Yield Curve......Page 600 30.8 The Value of Convexity......Page 602 30.9 What Happened to the P-Measure?......Page 603 31.1 The Purpose of This Chapter......Page 604 31.2 Empirical Findings about Inflation......Page 605 31.3 The No-Arbitrage Relationships......Page 606 31.3.1 What the No-Arbitrage Relationships Really Imply......Page 608 31.4 The Assumptions About the Process for the State Variables......Page 610 31.5 Inflation Expectations and Risk Premia......Page 611 31.6 Adding Liquidity......Page 612 31.7 The Parameter Estimation Procedure and Results......Page 613 31.7.1 The Difficulty of Parameter Estimation......Page 615 31.8.1 Full-Sample Analysis......Page 617 31.8.2 Prediction of Nominal and Real Excess Returns......Page 618 31.8.3 Analysis of the May–September 2013 Events......Page 623 31.9 Conclusions......Page 625 31.10 Related Work......Page 628 32.1 The Purpose of This Chapter......Page 631 32.2 Turning a Snapshot Model into a Dynamic Model......Page 632 32.3 Turning a Dynamic Model into a No-Arbitrage Affine Model......Page 635 32.4 Are the Variables Really Principal Components?......Page 639 32.5.1 On-the-Run, Off-the-Run Bonds......Page 641 32.5.2 The Modelling Approach......Page 642 32.5.3 The Results......Page 644 32.5.4 Conclusions......Page 645 33.1 The Purpose of This Chapter......Page 647 33.2 Why PC-Based Models Are Special (Again)......Page 649 33.3 Specified-Variable Models Revisited......Page 651 33.3.1 Parameter Constraints for PCA Prespecified Models......Page 653 33.4 Our Strategy to Link the P- and Q-Measures......Page 654 33.5.2 The Geometry (Kinematics) of the Problem......Page 655 33.5.3 The Dynamics of the Problem......Page 656 33.5.4 Solution......Page 657 33.5.5 Necessary Conditions for Identifiability......Page 658 33.6.1 Impossibility of Identification When K Is Diagonal......Page 660 33.6.2 What Does It Mean to Require that the Factors −→xt Should Be Principal Components?......Page 661 33.6.3 Constraints on K for Identifiability......Page 662 33.6.4 What the Q-measure Reversion-Speed Matrix Affects......Page 664 33.7 Moving from the Q- to the P-Measure......Page 668 33.8 Estimating the Parameters of .........Page 670 33.9.1 Cross-Sectional Fit to Yields......Page 672 33.10 Calibration Results......Page 673 33.11 Generalizable Results on Term Premia from a PC-Based Affine Model......Page 678 33.12 The Existential Dilemma......Page 683 33.13.1 Preliminaries......Page 686 33.13.2 Some Ancillary Results......Page 687 33.13.3 The Derivation of the Main Result......Page 688 33.13.4 The Conditions on the Vector .........Page 689 33.14 Appendix 33B: Switching Regressors......Page 690 34.1 The Purpose of This Chapter......Page 692 34.2 The Strategy Behind the Adrian–Crump–Moench Model......Page 693 34.3 A High-Level Description of the Model......Page 694 34.4 State-Price Deflators: Generalizing the Results......Page 696 34.5 Establishing an Expression for the Excess Returns......Page 700 34.6 The Estimation Procedure......Page 705 34.7 Establishing a Link with the Affine Model: The Discount Factor......Page 706 34.8 Some Observations......Page 708 34.9.1 Full-Sample Analysis......Page 709 34.9.2 Analysis of the May–September 2013 Events......Page 712 34.10 Conclusions......Page 716 35.1 The Purpose of This Chapter......Page 717 35.2 Why Do We Need Another Affine Model?......Page 718 35.3 Another Justification for a Stochastic-Market-Price-of-Risk Model......Page 720 35.4 The Model......Page 722 35.5 In Which Measure(s) Are We Working?......Page 723 35.6 The Qualitative Behaviour of the Model......Page 725 35.7 Calibration of the Model......Page 727 35.8 Calibration Results......Page 729 35.9 Comments on the Solution......Page 734 35.10 Term Premia in the Stochastic-Market-Price-of-Risk Model......Page 737 36.1.1 The Road Followed......Page 743 36.1.2 The Case for the Prosecution: Models As Regurgitators......Page 744 36.1.3 The Case for the Defence: Models as Enforcers of Parsimony......Page 745 36.1.4 The Case for the Defence: Models as Enforcers of Cross-Sectional Restrictions......Page 747 36.1.5 The Case for the Defence: Models as Revealers of Forward-Looking Informations......Page 748 36.1.6 The Case for the Defence: Models as Integrators......Page 749 36.1.7 The Case for the Defence: Models as Enhancers of Understanding......Page 750 References......Page 754 Index......Page 766