FNCE_235_725_Fall2010_Ch1_6

FNCE_235_725_Fall2010_Ch1_6 - Important Announcements...

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Unformatted text preview: Important Announcements Adventures in Debentures 1.) Please bring pages 80 103 to the first day of class. 2.) In an effort to accommodate more students in upper level finance courses, the Finance Department has instituted a new schedule for dropping and adding elective courses. Please check with the Finance De partment concerning these dates. 3.) The chapter titled "Solutions to Questions" will be distributed over the course of the semester. Professor Michael R. Gibbons The Wharton School University of Pennsylvania Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures 17: Valuation by Monte Carlo Methods . . . . . . . . . . . . . . . 837 Contents at a Glance 1: Overview of Fixed Income Securities . . . . . . . . . . . . . . 16 2: The Grammar of Fixed Income Securities . . . . . . . . . . . . 45 3: Data for a Recurring Illustration . . . . . . . . . . . . . . . . . 76 4: Bond Valuation Using Synthetics . . . . . . . . . . . . . . . . 92 5: Interpreting Bond Yields . . . . . . . . . . . . . . . . . . . . . 168 6: Bond Values and the Passage of Time . . . . . . . . . . . . . . 232 7: Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . 288 8: Dollar Delta 1: Risk Measurement . . . . . . . . . . . . . . . . 376 9: Dollar Delta 2: Risk Management . . . . . . . . . . . . . . . . 430 10: Dollar Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . 490 18: Introduction to Bond Options . . . . . . . . . . . . . . . . . . 874 19: European Bond Options . . . . . . . . . . . . . . . . . . . . . 930 20: American Bond Options . . . . . . . . . . . . . . . . . . . . . 971 21: Deja Vu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 22: Bonds with Embedded Options 1 . . . . . . . . . . . . . . . 1042 23: Bonds with Embedded Options 2 . . . . . . . . . . . . . . . 1073 24: Floating Rate Notes . . . . . . . . . . . . . . . . . . . . . . 1138 25: Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . 1173 26: Options on Yields . . . . . . . . . . . . . . . . . . . . . . . . 1197 27: Floating Rate Notes with Embedded Options . . . . . . . . 1226 28: Home Mortgages . . . . . . . . . . . . . . . . . . . . . . . . 1269 29: Solutions To Questions . . . . . . . . . . . . . . . . . . . . . 1330 30: Errata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1451 11: Delta, Gamma, and Theta . . . . . . . . . . . . . . . . . . . . 538 12: Time-Adjusted Performance Profiles . . . . . . . . . . . . . . . 583 13: Vasicek 1: Properties of the Short-Term Rate . . . . . . . . . 647 14: Vasicek 2: The Term Structure . . . . . . . . . . . . . . . . . 697 15: Vasicek 3: More Term Structure . . . . . . . . . . . . . . . . . 737 16: Vasicek 4: The Greeks . . . . . . . . . . . . . . . . . . . . . . 791 Contents at a Glance Page iii Contents at a Glance Page iv Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 149 Table of Contents Part One: Introduction and Review 1: Overview of Fixed Income Securities . . . . . . . . . . . . . . . . . . 16 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . . 21 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2: The Grammar of Fixed Income Securities . . . . . . . . . . . . . . . 45 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 A: Interest Rate Compounding . . . . . . . . . . . . . . . . . . . . . . . . . 49 B: Interest Rate Quotes and Conventions . . . . . . . . . . . . . . . . . . . 59 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . . 68 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5: Interpreting Bond Yields . . . . . . . . . . . . . . . . . . . . . . . . . 168 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 175 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 A: The Problem with Redemption Yields . . . . . . . . . . . . . . . . . . 204 B: Par Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 219 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6: Bond Values and the Passage of Time . . . . . . . . . . . . . . . . 232 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 238 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 A: Thetas and Time Profiles for Coupon-Bearing Bonds . . . . . . . . . . 264 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 274 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3: Data for a Recurring Illustration . . . . . . . . . . . . . . . . . . . . . 76 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . . 82 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4: Bond Valuation Using Synthetics . . . . . . . . . . . . . . . . . . . . 92 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . . 99 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A: Creating Synthetics: Further Discussion . . . . . . . . . . . . . . . . . 119 B: Solving Simultaneous Equations with Electronic Spreadsheets . . . . . 130 C: Using Linear Programming to Search for Arbitrage . . . . . . . . . . . 133 Table of Contents Page v Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 7: Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 295 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 A: Forward Contracts: More Synthetics . . . . . . . . . . . . . . . . . . . 317 B: Alternative Interpretations of Forward Rates . . . . . . . . . . . . . . 323 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 349 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Part Two: Traditional Methods for Interest Rate Risk Table of Contents Page vi Adventures in Debentures Adventures in Debentures 8: Dollar Delta 1: Risk Measurement . . . . . . . . . . . . . . . . . . . 376 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 382 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 A: Omega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 417 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 594 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 A: Some Mathematical Derivations . . . . . . . . . . . . . . . . . . . . . 622 B: A Review of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 627 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 642 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 9: Dollar Delta 2: Risk Management . . . . . . . . . . . . . . . . . . . 430 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 436 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 467 Part Three: More Modern Methods for Interest Rate Risk 13: Vasicek 1: Properties of the Short-Term Rate . . . . . . . . . . . 647 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 654 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 A: Estimating Vasicek Parameters from Historical Data . . . . . . . . . . 682 B: More Details about the Vasicek Tree . . . . . . . . . . . . . . . . . . . 685 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 689 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 10: Dollar Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 495 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 523 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 14: Vasicek 2: The Term Structure . . . . . . . . . . . . . . . . . . . . . 697 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 702 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 A: Information about the Finance Museum . . . . . . . . . . . . . . . . . 722 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 725 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 11: Delta, Gamma, and Theta . . . . . . . . . . . . . . . . . . . . . . . . 538 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 543 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 574 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 15: Vasicek 3: More Term Structure . . . . . . . . . . . . . . . . . . . . 737 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 741 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 Table of Contents Page viii Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 12: Time-Adjusted Performance Profiles . . . . . . . . . . . . . . . . . 583 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 Table of Contents Page vii Adventures in Debentures Adventures in Debentures A: The Vasicek Model When h Goes to Zero . . . . . . . . . . . . . . . . 758 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 782 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938 A: Valuation of Put Options on Bonds: A Numerical Illustration . . . . . 949 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 959 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 16: Vasicek 4: The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . 791 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 796 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 824 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964 20: American Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . 971 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 976 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 993 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 17: Valuation by Monte Carlo Methods . . . . . . . . . . . . . . . . . . 837 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 841 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842 A: Use of Pseudo Probabilities for Valuation in a Binomial World . . . . 860 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015 21: Deja Vu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 1028 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 Part Four: Options on Bonds 18: Introduction to Bond Options . . . . . . . . . . . . . . . . . . . . . 874 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 880 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 A: Some Notation for Bond Options . . . . . . . . . . . . . . . . . . . . . 908 B: Exercise Policy for Calls and Puts . . . . . . . . . . . . . . . . . . . . 909 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 914 Part Five: Fixed Rate Bonds with Embedded Options 22: Bonds with Embedded Options 1 . . . . . . . . . . . . . . . . . . 1042 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 1047 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 1068 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 23: Bonds with Embedded Options 2 . . . . . . . . . . . . . . . . . . 1073 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 1080 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 A: A Useful Price Decomposition: A Numerical Illustration . . . . . . . . 1103 Table of Contents Page x Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924 19: European Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . 930 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 937 Table of Contents Page ix Adventures in Debentures Adventures in Debentures B: Sinking Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 1113 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 1253 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1260 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128 Part Seven: Mortgage Products Part Six: Fixed Income Securities with Adjustable Payments 28: Home Mortgages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269 24: Floating Rate Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 1143 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 1160 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 1275 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276 A: Mortgage Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 1305 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168 25: Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 1178 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 1188 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319 Part Eight: Miscellaneous 29: Solutions To Questions . . . . . . . . . . . . . . . . . . . . . . . . . 1330 The Grammar of Fixed Income Securities . . . . . . . . . . . . . . . . . . 1333 Bond Valuation Using Synthetics . . . . . . . . . . . . . . . . . . . . . . . 1337 Interpreting Bond Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193 26: Options on Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 1202 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 A: Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217 Practice Questions with Solutions . . . . . . . . . . . . . . . . . . . . . . 1221 Bond Values and the Passage of Time . . . . . . . . . . . . . . . . . . . . 1352 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355 Dollar Delta 1: Risk Measurement . . . . . . . . . . . . . . . . . . . . . . 1362 Dollar Delta 2: Risk Management . . . . . . . . . . . . . . . . . . . . . . 1367 Dollar Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378 Delta, Gamma, and Theta . . . . . . . . . . . . . . . . . . . . . . . . . . . 1381 Characterizing Uncertainty about Bond Returns . . . . . . . . . . . . . . 1384 Vasicek 1: Properties of the Short-Term Rate . . . . . . . . . . . . . . . . 1385 Vasicek 2: The Term Structure . . . . . . . . . . . . . . . . . . . . . . . . 1386 Vasicek 3: More Term Structure . . . . . . . . . . . . . . . . . . . . . . . 1389 Table of Contents Page xii 27: Floating Rate Notes with Embedded Options . . . . . . . . . . . 1226 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229 Announcements and Assignments . . . . . . . . . . . . . . . . . . . . . . . 1231 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232 Table of Contents Page xi Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Vasicek 4: The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392 Introduction to Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . 1403 European Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409 American Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413 Bonds with Embedded Options 1 . . . . . . . . . . . . . . . . . . . . . . . 1419 Bonds with Embedded Options 2 . . . . . . . . . . . . . . . . . . . . . . . 1421 Floating Rate Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428 Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1432 Floating Rate Notes with Embedded Options . . . . . . . . . . . . . . . . 1436 Home Mortgages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1440 30: Errata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1451 Part One Introduction and Review 1: Overview of Fixed Income Securities . . . . . . . . 16 2: The Grammar of Fixed Income Securities . . . . . . 45 3: Data for a Recurring Illustration . . . . . . . . . . 76 4: Bond Valuation Using Synthetics . . . . . . . . . . 92 5: Interpreting Bond Yields . . . . . . . . . . . . . 168 6: Bond Values and the Passage of Time . . . . . . . 232 7: Forward Contracts . . . . . . . . . . . . . . . . . 288 List of Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453 Table of Contents Page xiii Part One: Introduction and Review Page 14 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Chapter Question: What is the difference between a terrorist and a tenured finance professor? 1 Answer: You can negotiate with a terrorist. Overview of Fixed Income Securities Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Announcements and Assignments . . . . . . . . . . . . . . 21 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 22 Part One: Introduction and Review Page 15 Chapter 1: Overview of Fixed Income Securities Page 16 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Preface to Overview of Fixed Income Securities 1.) Summary of Chapter. The primary purpose of this chapter is to give you an outline of the topics that we will cover for the rest of the course. Hopefully, by reviewing this chapter as well as reading the Prospectus for the course, you will have a good idea of what we will cover during the semester, my approach to the subject matter, and why these topics are important. 2.) Road Map for Chapter. The topics in this chapter will be organized as follows: 1.) Course description. 2.) Administrative matters concerning the course. 3.) If time remains, some fundamental graphs. 3.) Required Reading. After attending the lecture, you should read: "Prospectus." Chapter 1: Overview of Fixed Income Securities Page 17 Chapter 1: Overview of Fixed Income Securities, Preface Page 18 Adventures in Debentures Adventures in Debentures 4.) Supplemental Reading. The following article provides a great preview of some of the topics that we will encounter during this course. First, the authors discuss a number of financial contracts (forwards, futures, options, and swaps) that we will study. Second, the authors describe a number of methods to measure financial risk,1 and we will also develop some measures of risk.2 Finally, the authors try to understand some securities by considering a complex security equivalent to some portfolio containing less complex securities -- this is sometimes called a "LegoTM " approach. Such decompositions are a tactic that we will use throughout the course. Clifford Smith, Jr., Charles Smithson, and D. Sykes Wilford. 1989. "Managing Financial Risk." Journal of Applied Corporate Finance. Winter. Pages 27 48. 5.) Other Assignments. After attending this lecture, you should do the following: Purchase bulk pack. Skim materials in bulk pack to get a better idea about course. Read the chapter titled "The Grammar of Fixed Income Securities." Be sure to answer the questions at the end of that chapter. (The questions begin on page 72.) 6.) New Vocabulary Used in this Chapter. The following buzz words will be used in this chapter: Discount function, performance profile, performance surface, term structure of interest rates, time profile, and yield curve. The following paper is a nice introduction to a variety of financial instruments -- some of which are quite complicated. Further, the article provides a very nice presentation as to why certain financial securities get created in the first place. Charles Smithson and Donald Chew. 1992. "The Uses of Hybrid Debt in Managing Corporate Risk." Journal of Applied Corporate Finance. Winter. Pages 79 89. 7.) Sources of Data for the Histograms. This chapter summarizes some of the historical data regarding the size of the U.S. market for fixed income securities as well as equities. The data were obtained from the following sources: Annual Statistical Digest, Banking and Monetary Statistics, Federal Reserve Bulletin, and NYSE Fact Book. The following article discusses many of the same issues that are contained in the Smithson and Chew (1992) article. While Finnerty's discussion provides many useful details, it does read like an encyclopedia. John Finnerty. 1988. "Financial Engineering in Corporate Finance: An Overview" by John Finnerty. Financial Management. Winter. Pages 14 -- 33. 8.) Acknowledgments. Throughout the course, I will make an analogy between LegoTM building blocks3 and constructing synthetics based on simple financial instruments and/or strategies. This connection to LegoTM building blocks was first made by Charles W. Smithson in a paper titled "A LegoTM Approach to Financial Engineering: An Introduction to Forwards, Futures, Swaps and Options." This article appeared in Midland Corporate Finance Journal in Winter, 1987 (pages 1628). 3 1 2 This course only examines interest rate risk. However, this article also incorporates risks associated with movements in commodity prices and foreign exchange. The authors do rely on yield duration, which is a traditional measure of interest rate risk. We will develop an alternative measure which is similar, but not identical to this risk measure. Further, the authors use a multiple regression to measure risk; such an approach could be viewed as a possible extension of some of the risk measures that we will develop during the course. LegoTM is an exclusive trademark of Interlego A.G. of Denmark. Chapter 1: Overview of Fixed Income Securities, Preface Page 19 Chapter 1: Overview of Fixed Income Securities, Preface Page 20 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Announcements and Assignments ecture notes L for Overview of Fixed Income Securities A Road Map 1.) Course description. 2.) Administrative matters concerning the course. 3.) If time remains, some fundamental graphs. Chapter 1, Announcements and Assignments Page 21 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 22 Adventures in Debentures Adventures in Debentures What is a Bond? "Bond : an interest-bearing certificate issued by a government or business, promising to pay the holder a specified sum on a specified date." Webster's New World Dictionary Objectives of this Course 1.) To describe important financial instruments which have market values that are sensitive to interest rate movements. 2.) To develop tools to analyze interest rate sensitivity and value fixed income securities. 3.) To define and explain the lexicon of the bond management business. Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 23 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 24 Adventures in Debentures Adventures in Debentures Objective #1: Course Surveys Fixed Income Assets and Related Securities Zero Coupon Government Bonds Coupon Bearing Government Bonds Forward Contracts 2001 Market Value (Trillions) $14.0 $12.0 $10.0 $8.0 $6.0 Bond Options $4.0 Bond Futures $2.0 Options on Bond Futures Bonds with Embedded Options Floating Rate Notes Caps, Collars, and Floors Floating Rate Notes with Embedded Options Interest Rate Swaps Home Mortgages $0.0 NYSE UST Bills, Notes, and Bonds Federal Agencies Mortgage Pools Total UST Debt Total Mortgage and Trusts Debt Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 25 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 26 Adventures in Debentures Adventures in Debentures Recent Market Values (in Billions) $14,000 2001 Daily Average Trading Volume (Billions) $350 $12,000 NYSE $10,000 UST Bills, Notes, and Bonds Federal Agencies Mortgage Pools and Trusts $8,000 Total UST Debt Total Mortgage Debt $300 $250 $200 $6,000 $150 $4,000 $100 $2,000 $50 $0 1980 1990 2000 $0 NYSE UST Bills, Notes, and Bonds Federal Agencies Mortgage Pools and Trusts Historic Market Values (in Billions) $700 Historic Daily Trading Volume (Billions) $600 $300.0 $500 $250.0 $400 $200.0 $300 NYSE $150.0 UST Bills, Notes, and Bonds $200 $100.0 $100 $50.0 $0 1950 1960 1970 $0.0 1970 1980 1990 2000 NYSE UST Bills, Notes, and Bonds Federal Agencies Total UST Debt Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 27 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 28 Adventures in Debentures Adventures in Debentures 2001 New United States Corporate Issues (Billions) $1,400 Objective #2: $1,200 $1,000 Course Surveys Tools for Bond Management Duration, Convexity, and Theta. Binomial Trees. Constructing Synthetics. $800 $600 $400 $200 $0 Stocks Bonds Historical US Corporate Issues $900.0 $800.0 $700.0 $600.0 (Billions) $500.0 $400.0 $300.0 $200.0 $100.0 $0.0 Stocks Bonds 1924 1950 1960 1970 1980 1990 2000 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 29 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 30 Adventures in Debentures Adventures in Debentures Decomposition of Complex Securities (or the LegoTM Approach1 to Valuation) Various types of complex fixed income securities will be decomposed into portfolios of simpler securities. These "analogies" are useful for developing intuition and even trading strategies (e.g., arbitrage in the extreme case). We can use these decompositions as a way to better understand the capital structure of firms and the balance sheet of a financial institutions. Objective #3: Course Surveys the Lexicon of Bond Management accreting swap accrual tranche accrued interest accrued interest rate option add-on yield adjustable rate mortgage American option amortizing swap amortization schedule annuity annuity yield curve arbitrage Asian option at-the-money balloon mortgage barbell barrier option binary option binomial tree break-even curve bullet butterfly trades callable annuity callable bond call option call option on yields caplet cap (or ceiling) caption cash settlement certainty equivalent cheapest-to-deliver option collar collateralized mortgage obligation compound option constant coupon yield curve conversion conversion factor convexity cost of carry (or carry) coupon yield current yield curve trading dedicated portfolio default (or credit) risk deferred swap delayed start swap delivery day delta () delta-based strategies delta profile digital option discount function dollar duration double-up option dynamic strategies 1 LegoTM is an exclusive trademark of Interlego A.G. of Denmark. Page 31 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 32 Adventures in Debentures Adventures in Debentures dynamic (structured) synthetics dynamic (unstructured) synthetics end-of-the-month option Eurodollar futures Eurodollars European option exercise (or strike) price first position day fixed/adjustable mortgage hybrid flat price floating rate note (or floater) floor forward contract forward rate forward rate agreement forward swap forward term structure full price futures gamma ( ) graduated payment mortgage holding period return horizon analysis immunization implied parameter values indexed (or inflation-indexed) bond initial margin intention day internal rates of return in-the-money intrinsic value inverse floating rate note invoice price KIKO (Knock In Knock Out) option ladder lambda () level pay fixed rate mortgage leverage LIBID LIBOR local expectations hypothesis long position lookback option maintenance margin marking to market mean reversion modified yield duration mortgage commitment mu () multifactor models omega () on-the-run securities option premium option writer origination fee out-of-the-money par swap par yield curve path dependent payment in arrears payment-in-kind (PIK) bond payoff (or revenue) diagram performance profile performance surface phi () plain vanilla bond points position day present value rule price limit putable bond put-call parity put option put option on yields quality option redemption yield replicating future payoffs replicating portfolio repurchase agreements (or repo's) riding the yield curve risk adjustment factor roller coaster floater or swap royal flush option sequential pay tranche settlement short position sigma () sinking fund slide effect spot rates of interest spot rate (yield) curve spread standard deviation static synthetics straddle strangle synthetics swap swaption switch option T-Bond futures term (or liquidity) premium term structure of foward deliveries term structure of interest rates theta () theta profile time-adjusted performance profile time profile time value timing option two-step mortgage variation margin when-issued security yield curve yield duration yield to call yield to maturity yield to worst Vasicek Term Structure Model wild card option Z bond tranche zero coupon bonds (or "zeros") Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 33 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 34 Adventures in Debentures Adventures in Debentures Tentative Outline of Topics Fundamentals in analyzing fixed income securities. Duration, immunization, and convexity. Binomial models of term structure. Bond options. Bonds with embedded options. Bond futures and options on bond futures. Floating rate notes, swaps, caps, collars, and floors. Home mortgages. Tracking the Milestones 1.) Course description. 2.) Administrative matters concerning the course. 3.) If time remains, some fundamental graphs. Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 35 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 36 Adventures in Debentures Adventures in Debentures Administrative Matters Tracking the Milestones 1.) Course description. 2.) Administrative matters concerning the course. 3.) If time remains, some fundamental graphs. Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 37 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 38 Adventures in Debentures Adventures in Debentures Some Basic Graphs for Studying Fixed Income Securities 1.) Performance surface. 2.) Performance profile. 3.) Time profile. 4.) Discount function. 5.) Term structure of interest rates. 0 1 0.75 0.5 0.25 0 5.0% 10.0% 15.0% term structure indicator (short term rate) Performance Surface: 20 Yr Zero. (Face=$1.00). time p passage g 15 10 5 20 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 39 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 40 Adventures in Debentures Adventures in Debentures Performance Profile: 20 Yr Zero. (Face=$1.00). 0.6 dollar value 0.5 0.4 0.3 0.2 0.1 0 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% term structure indicator (short-term rate) Dollar Value Discount Function 1 0.8 0.6 0.4 0.2 0 0 5 10 Maturity (in years) 15 20 Time Profile: 20 Yr Zero. (Face=$1.00). 1 0.8 dollar value 0.6 0.4 0.2 0 0 5 10 15 20 time passage (in years) Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 41 Term Structure of Interest Rates 12.0% Continuously Compounded Yield 11.0% 10.0% 9.0% 8.0% 7.0% 6.0% 0 5 10 Maturity (in years) 15 20 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 42 Adventures in Debentures Adventures in Debentures Worksheet Alternative Term Structures 12.0% Continuously Compounded Yield 11.0% 10.0% 9.0% 8.0% 7.0% 6.0% 0 5 10 Maturity (in years) 15 20 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 43 Chapter 1: Overview of Fixed Income Securities, Lecture Notes Page 44 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Chapter 2 The Grammar of Fixed Income Securities Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 A: Interest Rate Compounding . . . . . . . . . . . . . . . 49 B: Interest Rate Quotes and Conventions . . . . . . . . . 59 Practice Questions with Solutions . . . . . . . . . . . . . . 68 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 -- Wall Street Journal, June 26, 1992 Chapter 2: The Grammar of Fixed Income Securities Page 45 Chapter 2: The Grammar of Fixed Income Securities Page 46 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures 4.) New Vocabulary Used in this Chapter. Preface to The Grammar of Fixed Income Securities The following buzz words will be used in this chapter: Accrued interest, ask, bid, (bankers') discount rate, flat price, full price, handle, settlement, and simple interest. 1.) Summary of Chapter. This chapter interprets interest rates. The first appendix discusses interest rate compounding, and the second appendix explains various conventions for computing interest rates or bond yields. At the end of the chapter, there are problem set questions to allow you to confirm your understanding of the reading. Unlike most chapters, there are no lecture notes for this subject. You are expected to read the material and understand it. 2.) Required Reading. You should read: Appendix A, "Interest Rate Compounding." Appendix B, "Interest Rate Quotes and Conventions.." 3.) Assignments. You should do the following: Answer the questions at the end of this chapter. (The questions begin on page 72.) Chapter 2: The Grammar of Fixed Income Securities, Preface Page 47 Chapter 2: The Grammar of Fixed Income Securities, Preface Page 48 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures 2.) Interest Rates and Compounding Appendix A: Interest Rate Compounding* 1.) Introduction We call (default-free) bonds, annuities, and savings accounts "fixed income" instruments because they pay known or fixed cash flows at known points in time, as opposed to stocks which pay random amounts at random times. In addition, bond futures, floating rate notes, options on yields, and so forth have also come to be called fixed income securities because they are in some nominal way related to bonds or interest rates, although their payoffs may be no more "fixed" than those of a stock. Strictly speaking, only when an asset pays certain cash flows is it possible to speak of its yield or interest rate. Perhaps the simplest conceivable fixed income asset is one that requires an investment of $1 today and pays back a single cash flow in exactly one period of time. More generally, fixed income securities can pay a collection of known cash flows of different sizes at different times. It may be difficult to judge the relative value of such assets by simply looking at their prices. The idea of quoting an interest rate or yield on a fixed income security is to condense information about the size and timing of the cash flows associated with holding the asset into a single number that is in some rough sense comparable across securities. We will see in this course that yield is not in fact a reliable measure of economic value -- information about the timing of cash flows cannot be thrown away; nevertheless, yield is a concept that permeates the fixed income markets. 2.1.) Most Primitive Notion of an Interest Rate We said above that quoting an interest rate or yield is an attempt to distill information about the price and payoffs of a fixed income asset into a single number. Unfortunately, different ways of doing this have sprung up in different business environments. To be meaningful, an interest rate must come together with a list of qualifiers specifying the conventions (e.g, the extent of compounding) that underlie the basic quote. To explain those conventions, we'll strip away the bells and whistles, start with the simplest definition of an interest rate, and then describe everything else in terms of that simple idea. There's one whistle we can't strip away, even at the most primitive level. To make any sense of an interest rate, we need to agree on a particular period of time as a unit with which we will measure all other lengths of time. For example, we may declare one period to be one year or one month or one day, it doesn't matter which, we just need to pick one. Then we can give the following definition. A fixed income asset pays or yields an interest rate of r per period if each dollar invested in the asset at the beginning of each period earns a profit of r dollars at the end of the period. Example 1 (The Unit Investment). If you invest (lend) $1 today for one period at a rate of r per period, then you will receive $1 + r in exactly one period. Thus, if you invest $1 today at a rate of 10% per month for one month, you will pay $1 today and you will receive $1.10 after one month. If you borrow $1 today and pay back $1.06 after one year, then you are borrowing at a rate of 6% per year. * The first version of this appendix was prepared by Jennifer N. Carpenter in September, 1992, under the guidance of Michael Gibbons, Professor of Finance, the Wharton School of the University of Pennsylvania. This version reflects revisions from that original. Example 2 (Different Investment Sizes). If you lend $P today for one period at a rate of r per period then you will pay $P today and receive $P (1 + r) after one period has passed. Thus, if you invest $1/(1.10) $.91 today for one year at a rate of 10% per year, then you will receive $1 at the end of one year. If you lend $200 for one week at a rate of 1/4% per week then you will receive $200.50 at the end of one week. If your credit card charges 1.5% per month and your beginning of the month balance is $500 and you make no charges or payments during the month, then your month-end balance will be $507.50. If you buy $1,000 to be received in one year at a rate of 8% per year then you will pay a price of $1,000/1.08 $925.93 today and get $1,000 in one year. Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 50 Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 49 Adventures in Debentures Adventures in Debentures The examples above were extremely easy because in all cases, the investment lasted exactly one period. Handling an arbitrary initial investment size P just amounts to scaling the future value linearly, that is, just multiplying by the factor P . What if the investment endures for a length of time different than the period chosen as the basic time unit for the interest rate quote? Here is the basic rule: $P invested today at a rate of r per period grows to $P (1 + r)t at the end of t periods. We will use language about growth because it will help our intuition in more complicated examples. What the rule above means, of course, is that if you lend $P today at a rate of r per period for t periods, then at the end of the t periods you will receive $P (1 + r)t . Equivalently, if you lend $P today for a certain repayment of $F at the end of t periods, then your lending rate is r = (F/P )1/t - 1 per period. In all the examples above, the investment involves an initial payment today and a single receipt at the end of the investment horizon. How do we handle investments with multiple payoffs? We just remember the rule that an asset yielding a rate of r per period pays profit r per dollar per period and then scale linearly for investment size and scale exponentially for investment time. Example 5 (Multiple Cash Flows). Suppose you put $1,000 today in a savings account that pays 2% per quarter on all investments over the next 5 years. At the end of the first 6 months you withdraw $25. At the end of the seventh month, you withdraw $30. At the end of two years, you withdraw $500. At the end of five years, you withdraw what's left and close the account. What is the amount of the last withdrawal? We can figure this out in a variety of ways. One way involves keeping track of your account balance through time. At the end of six months (two quarters), your balance grows to $1,000(1.02)2 = $1,040.40 at which time you withdraw $25 leaving $1,015.40. From the end of the sixth month to the end of the seventh month, you earn one month's interest (a third of a quarter's worth) on $1,015.40 so your balance at the end of the seventh month is $1,015.40(1.02)1/3 $1,022.12. At this time you withdraw $30 leaving $992.12. From month 7 to month 24 you earn 17 months' interest on $992.12 so your balance grows to $992.12(1.02)17/3 $1,109.94 at the end of month 24 at which time you withdraw $500 leaving $609.94. From the end of the second year to the end of the fifth year, you earn twelve quarters' worth of interest so your ending balance is $609.94(1.02)12 $773.55 which is the size of your last withdrawal. Another way uses the interest rate to move all cash flows to the same point in time so that they may be added together directly. Since the problem is to determine the year 5 cash flow, let's move all cash flows to the end of year 5. First, the investment of $1,000 today is worth +$1,000(1.02)20 $1,485.94 in year 5. The withdrawals of $25 and $30 in months 6 and 7 are worth -$25(1.02)18 -$25.71 and -$30(1.02)53/3 -$42.57 in year 5, respectively. Similarly, the $500 withdrawal at the end of year 2 costs you $500(1.02)12 $634.12 in year 5 dollars. Thus you are left with $1,485.94-$35.71-$42.57- $634.12 = $773.54 in year 5, which matches ending balance we found above (up to a rounding error). Also you can check that by construction (of the quantity $773.55) the internal rate of return (or IRR) on an asset costing $1,000 today and paying $25 in six months, $30 in seven months, $500 in two years, and $773.55 in five years is exactly 2% per quarter (or (1.02)1/3 - 1 0.662% per month, or (1.02)4 - 1 8.24% per year). Indeed, to find the IRR per quarter on such an investment, Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 52 Example 3 (The t-Period Investment). If you invest or lend $100 today for 2 years at 12% per year, then you will receive $100(1.12)2 = $125.44 at the end of two years. If you borrow $100 today for 2 years at 1% per month, then you will owe $100(1.01)24 $126.97 at the end of two years. How can we reconcile this rule with the original definition? We reason as follows. Suppose we lend $1 for 2 years at 10% per year. Then at the end of the first year, according to our original definition, we have earned 10/ profit--the investment has c now grown in size to $1.10. At the end of the second year, according to our original definition, we earn another 10% profit, now on an investment of $1.10, so our second year profit is 11/ and our total profit is 21/, so we get a payoff of $1.21 = $(1.10)2 at c c the end of the two years. Notice that a rate of 10% per year is equivalent to a rate of 21% per two years--not 20% per two year! To account for the so-called interest on interest properly, time t enters the formula as an exponent, not as a factor. Finally, we point out that in the formula above the number of periods t need not be an integer. t may be any nonnegative number. Example 4 (Investing for Non-Integral Numbers of Periods). Suppose we borrow $100 today for six months at a rate of 12% per year. Then at the end of six months we will owe $100(1.12)1/2 $100(1.0583) = $105.83. If we lend $10 today for 29 months at a rate of 12% per year then we will receive $10(1.12)29/12 = $13.15 at the end of 29 months. Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 51 Adventures in Debentures Adventures in Debentures we find the r that satisfies the equation $1,000 = $30 $500 $773.55 $25 + + + . (1 + r)2 (1 + r)7/3 (1 + r)8 (1 + r)20 Here's general rule about compounding frequency: Let the positive integer m be the compounding frequency, and 1/m years be the compounding period. Then receiving an "annualized rate of r compounded every 1/m years" or an "annualized rate of r compounded m times a year" means receiving a rate of r/m every compounding period. Now let's use this rule to derive a formula to convert a rate from one compounding frequency to another. Let m = compounding frequency (e.g., m = 4 means compounded quarterly), let r(m) = the annualized rate compounded every 1/m years, let r be the annualized rate compounded annually that is mathematically equivalent to r(m) compounded every 1/m years. Multiplying through by (1 + r)20 , r must solve $1,000(1 + r)20 = $25(1 + r)18 + $30(1 + r)53/3 + $500(1 + r)12 + $773.55 , or $1,000(1 + r)20 - $25(1 + r)18 - $30(1 + r)53/3 - $500(1 + r)12 = $773.55 . We know that r = 2% solves this, because when r = 2%, this is exactly the equation we derived in our second solution above. Then, since getting an annualized rate of r(m) compounded m times a year means 2.2.) Compounding Frequency There is no single time period that is the natural unit in which to measure time for all business environments. For a mortgage banker, one month is natural because mortgage payments are typically monthly, and he typically recomputes principal balances on his outstanding loans after each payment. For a farmer, one year might be the natural time unit if that's his planting cycle. Corporations and the U.S. Treasury typically make interest payments on their loans every six months, so six months is the natural period. Credit card bankers usually keep track of interest on loans daily (although payments are supposed to be made monthly, they may be late). On the other hand, regardless of the choice of unit period, it is customary to quote all rates on an annualized basis--that is, to normalize the investment horizon to one year and adjust the periodic rate in a simple linear way. Thus, a mortgage banker who charges 1% per month would say he charges an annualized rate of 12%. A corporation paying 6% every six months would also say he pays an annualized rate of 12%. Eurobonds make annual payments, so a Eurobond trader would say 12% means each $1 invested earns 12/ at the end of each year. Clearly, these rates are different--on a dollar inc vestment, receiving 1/ at the end of each month is better than receiving 6/ at the end c c of every 6 months, which is in turn better than receiving 12/ at the end of every year. c To distinguish these rates, we say that the mortgage banker charges an annualized rate of 12% compounded monthly, the corporation pays an annualized rate of 12% compounded semi-annually, and the Eurobond trader quotes an annualized rate of 12% compounded annually. Often, we drop the term annualized, because the choice of one year as the standard investment horizon is so common that it is understood. Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 53 getting a rate of r(m) m every 1/m years, 1+r = 1+ r(m) m m . (2-1) We can invert this equation to solve for r(m) in terms of r: r(m) = m[(1 + r)1/m - 1] . (2-2) Therefore, we can convert from semi-annual compounding to monthly compounding and so on. To be specific, if we are given a rate of r(m1 ) compounded m1 times per year and we want to know the mathematically equivalent rate for a different compounding frequency, say m2 , we can always convert from m1 -compounding to annual compounding using formula (2-1) and then from annual compounding to m2 compounding using formula (2-2). Example 6. What is the semi-annually compounded rate that is equivalent to 10% compounded quarterly? According to equation (2-1), 10% compounded quarterly is 4 equivalent to 1 + .1 - 1 compounded annually. According to equation (2-2), the 4 semi-annually compounded equivalent to this annually compounded rate is 4 1/2 .1 - 1 = 10.125% . 2 1+ 4 Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 54 Adventures in Debentures Adventures in Debentures Continuous Compounding. To introduce the idea of continuous compounding, let's take a given annualized rate of r(m) compounded every 1/m years and consider the following as the notional asset underlying this quote. The asset costs $1 today and makes an interest payment once every compounding period (every 1/m years). In addition, at the end of a year, the asset returns the $1 principal. Since r(m) compounded every 1/m years means r(m)/m every period, the asset described pays r(m)/m every period for m periods, for a total nominal profit of r(m) (ignoring compounding, or "interest on interest"). For example, 12% compounded monthly means the notional underlying asset pays 1/ interest every month. c If we fix a value for r(m), then the annually compounded equivalent, r, increases as m increases, because, in terms of the notional asset, the same nominal profit, r(m) is paid out sooner. For a fixed value of r(m), it turns out that as m goes to infinity, the quantity on the right-hand side of equation (2-1) approaches the number er(m) where e is the so-called exponential number approximately equal to 2.718281828. (More pre1 m cisely, e = limm 1 + m .) For example, set r(m) = 12% for all m. The following table gives the annually compounded equivalent rates for different compounding frequencies. (There is no economic reason why r(m) should be the same for all m, x m ex as we have just fixed it to illustrate the mathematical result that 1 + m m .) We have computed all of the annually compounded equivalents in the last column according to equation (2-1) except for rate in the last row. To arrive at that number, we used the formula r = e.12 - 1. In general, the annually compounded rate, r, that is equivalent to a given continuously compounded rate, r(), is given by the formula Annually Compounded Rate Which Is Equivalent to 12% Compounded with Different Frequencies Compounding frequency m 1 2 4 12 52 365 8,760 525,600 31,536,000 Notional periodic payoff on $1 12/m/ c 12.0000000/ c 6.0000000/ c 3.0000000/ c 1.0000000/ c 0.2307692/ c 0.0328767/ c 0.0013699/ c 0.0000228/ c 0.0000004/ c flow at rate of 12//yr c Compounding periodicity every 1/m years annually semi-annually quarterly monthly weekly daily hourly every minute every second continuously Annually compounded equivalent r 12.00000000% 12.36000000% 12.55088100% 12.68250301% 12.73409872% 12.74746156% 12.74959249% 12.74968361% 12.74968522% 12.74968516% 1 + r = er() . (2-3) Note that, although this last formula tells us exactly the annually compounded rate r equivalent to r() compounded continuously, we can approximate r as closely as we wish with sufficiently frequent "discrete" compounding. In the case that r() = 12%, the approximation using 12% compounded hourly is correct to the nearest tenth of a basis point; 12% compounded every second is virtually indistinguishable from 12% compounded continuously. Of course we can invert formula (2-3) to compute the continuously compounded rate that is equivalent to a given annually compounded rate, r: r() = ln(1 + r) (2-4) where the function represented by "ln" -- the natural log, is the inverse of the exponential function. That is, if y = ex , then x = ln y. Equivalently, eln y = y = ln(ey ). Armed with formulas (2-2) and (2-4), we can fix a particular annually compounded rate, say r = 12%, and compute the equivalent rates for different compounding freChapter 2: The Grammar of Fixed Income Securities, Appendix A Page 55 Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 56 Adventures in Debentures Adventures in Debentures quencies. The results are in the table below. Rates with Different Compounding Frequencies Which Are Equivalent to 12% Compounded Annually Compounding frequency m 1 2 4 12 52 365 8,870 525,600 Notional periodic payoff on $1 12/m/ c 0.120000000/ c 0.058300524/ c 0.028737345/ c 0.009488793/ c 0.002181774/ c 0.000310538/ c 0.000012937/ c 0.000000216/ c flow at rate of 11.33//yr c Compounding periodicity every 1/m years annually semi-annually quarterly monthly weekly daily hourly every minute continuously Periodically compounded equivalent r(m) 12.00000000% 11.66010489% 11.49493789% 11.38655152% 11.34522692% 11.33462808% 11.33294184% 11.33286975% 11.33286853% Similarly, suppose we knew we could invest $1 today for one year at r0,1 compounded annually, and we could reinvest the end-of-year payoff at r1,2 compounded annually for another year. The value of the investment at the end of two years would be F = (1 + r0,1 )(1 + r1,2 ) . To compute the implied annually compounded two-year rate (or internal rate of return) on this investment we need to solve F = (1 + r0,2 )2 for r0,2 to obtain r0,2 = [(1 + r0,1 )(1 + r1,2 )]1/2 - 1 . However, if we work continuously compounded rates, we find that the two-year rate is the simple average of the two one-year rates. In terms of continuously compounded rates (r() = ln(1 + r)), we have F = er0,1 () er1,2 () = er0,1 ()+r1,2 () and also F = (er0,2 () )2 = e2r0,2 () , so r0,2 () = r0,1 () + r1,2 () . 2 Using the formulas above, we can convert an interest rate quote back and forth between continuous compounding and any other compounding frequency. Example 7. What is the semi-annually compounded rate that is equivalent to 10% compounded continuously? According to equation (2-3), 10% compounded continuously is equivalent to e.10 - 1 compounded annually. According to equation (2-2), the semi-annually compounded equivalent to this annually compounded rate is 2[(e.10 )1/2 - 1] = 10.254%. Why would anyone want to use continuously compounded interest rates? In a nutshell, discretely compounded rates are easy to understand but hard to use, while continuously compounded rates are hard to understand but easy to use. Remember how 10% per year for two years is equivalent to 21% per two years for two years, not 20%? ((1.10)2 = 1.21.) That's because time enters value formulas exponentially not linearly to account for interest on interest, as we said before. Well, since continuously compounded rates already appear as exponents, we can add them up or multiply them by time directly. In other words, earning a continuously compounded rate of r() per year for t years is equivalent to earning a continuously compounded rate of r() t over the whole investment horizon: (er() )t = er()t . Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 57 Notation. In this course we will almost always express interest rates as either annualized rates with annual compounding (notation: r) or as annualized rates with continuous compounding (notation: r() or r). Typically, we will reserve this no tation for zero-coupon bonds, and use something else such as y for yield or IRR in the case of bonds with multiple cash flows. In addition, to specify an interest rate completely, we will need to specify three dates. The first of these, u, is the trade date or quote date, the date on which the parties to the underlying loan negotiate the rate. The second is the settlement date, s, the date on which the parties exchange money for securities. The third is the maturity date, t, the date on which the borrower repays the lender. Thus, u rs,t is the annually compounded rate quoted at time u for a loan starting at time s and ending at time t. When u = s = 0, we will omit the subscripts u and s and just write rt . To summarize what we've done so far, the future value of $1 invested today for t years is: $F = $(1 + rt )t = $ert t . In future sessions we will review this notation and the proper interpretation of the subscripts. Chapter 2: The Grammar of Fixed Income Securities, Appendix A Page 58 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Appendix B: Interest Rate Quotes and Conventions* set so that the last coupon date is equal to the maturity date. For example, if the bond matures on August 15, 2022, then its coupon dates are February 15 and August 15 every year from issue to maturity. The par amount $F just tells the quantity of the bond under consideration, so prices are always quoted for $100 par. Usually, traders refer to a bond by its coupon and maturity in the following way. If a bond has c = 7.25% and M = May 15, 2016, they'll call it the "seven and a quarter's of May 15, 2016" or just the "seven and a quarter's of sixteen." Similarly, the "fourteen's of eleven" refer to the bond with c = 0.14 and M = November 15, 2011, and the "eight and seven-eighths' of seventeen" refer to the bond with c = 0.08875 and M = August 15, 2017. Settlement. Treasury bonds usually settle "next day," or one business day after the trade date. The trade date is the date on which the parties to the trade agree on a price and promise to transact, and the settlement date is the date on which the money and bonds actually change hands. While the conventional form of settlement in Treasury bonds is one business day, the actual time to settlement can be negotiated between the buyer and the seller. If a bond trade settles on a coupon date, the seller gets that day's coupon payment, and the buyer gets all of the remaining payments. If the trade does not settle on a coupon date, the seller does not receive the next coupon (or any of the other remaining payments) but receives a pro rata portion of that coupon as part of the total price. As we will describe more fully below, bond traders distinguish between the part of the total price that roughly compensates the seller for the partial coupon he has earned since the last coupon date (accrued interest) and the part of the total price that roughly counts towards the rest of the bond's cash flows (flat price). Yield. The "yield" or "bond-equivalent yield" of a bond is its semi-annually compounded internal rate of return. This is perhaps the easiest of all quoted rates to understand. Suppose a bond trade settles on a coupon date. Let $P be its price per $100 par and let y be its yield. Let T be the number of six month periods from settlement to maturity. Recall that a semi-annually compounded rate of y really means a rate of y/2 every six months. So P and y are related through the following formula: P = 100 = 100 t=1 1.) Introduction Bond yields provide an alternative way to quote the price of a given fixed income security. Unfortunately, different conventions are used in computing the yields, which make them difficult to interpret and compare across different markets. Here we will explain the conventions used for reporting interest rates in some bond markets. 2.) Interest Rate Quotes in U.S. Fixed Income Markets Dealers in U.S. fixed income markets often quote rates or yields rather than actual prices when trading securities, and they use different conventions for different securities. The following is a guide to yield calculations and other conventions such as accrued interest and settlement delay that are relevant to various types of securities. Some of this is taken from Marcia Stigum's Money Market Calculations: Yields, Break-Evens and Arbitrage or Lynch and Mayle's Standard Securities Calculation Methods Fixed Income Securities Formulas. 2.1.) Treasury Notes and Bonds A Treasury bond is a coupon-bearing instrument. The coupon rate, c, and maturity date, M , of a noncallable bond completely describe its cash flows. A given par amount $F of a Treasury with coupon c and maturity M makes a "coupon" payment equal to $cF/2 twice a year on coupon dates until maturity, and a "principal" payment of $F at maturity. The two coupon dates each year are spaced exactly six months apart and * The first version of this appendix was prepared by Jennifer N. Carpenter in September, 1992, under the guidance of Michael Gibbons, Professor of Finance, the Wharton School of the University of Pennsylvania. This version reflects revisions from that original. c/2 1 + c/2 c/2 c/2 + + ... + + (1 + y/2) (1 + y/2)2 (1 + y/2)3 (1 + y/2)T T c/2 (1 + y/2)t + 1 (1 + y/2)T . You can check that if c = y then P = 100 -- the bond is priced at par. This makes sense because if you invest $100 at a rate of y/2 per six-month period, then Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 60 Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 59 Adventures in Debentures Adventures in Debentures the interest accumulated at the end of each six-month period is $100y/2 which is exactly the amount of the coupon payment when c = y. Therefore, immediately after the coupon payment, the value of the investment is restored to the original principal amount of $100, so the par payment of $100 exactly pays off the loan, regardless of its maturity. Similar reasoning justifies the fact that when c < y, P < 100 -- the bond is priced at a discount, and when c > y, P > 100 -- the bond is priced at a premium. since the next coupon date is 2/15/93, TNM = 1, nSN = 160, and nLN = 184. Plugging these numbers into the formula above gives P = 105.5062627. If the size of the trade were $20 million par, the total market value traded would be $21,101,252.54. Accrued interest. The price P above is what some people call the "full price" of the bond. Even when c = y, this price will exceed 100 between coupon dates to account for the fact that some fractional period's worth of interest has accumulated on the bond since the last coupon date. Perhaps to adjust prices so that the bond is at a premium only when c > y and at a discount when c < y, bond traders quote "flat prices." The flat price of a bond is the full price minus what's called accrued interest. The accrued interest linearly approximates the economic interest that would have accrued on the bond since the last coupon date if the bond yield were always equal to its coupon rate. In particular, letting nLS = the actual number of days from the last coupon date to the settlement date, the accrued interest per $100 par is nLS . a = 100c/2 nLN Notice that accrued is computed purely by convention, it is not a market-based number. The flat price, p, that traders quote is p=P -a. It deserves emphasis that the "quoted" price is the flat price, not the full price. 32nds. Finally, bond traders quote flat prices in 32nds, not decimal. Thus a flat price of p = 92.47 would be quoted as 92 and 15/32nds or 92-15 and would appear in the Wall Street Journal as 92:15. Also, the 92 is sometimes called the "handle." Example 3 (Flat Price). What is the flat price of the bond in the example above? We have already computed the full price, so we just need to subtract the accrued interest. Plugging c = 0.08625, nLS = 24, and nLN = 184 into the formula for accrued we have a = 0.5625. Therefore p = 104.9438 or 104-30. Bonds with maturity less than one coupon period. The yield traders quote for bonds with less than one coupon period until maturity is what's sometimes called "simple interest." Generally, in a calculation using simple interest, time enters as a factor, not as an exponent. In this case, given a yield quote y, the full bond price P is 1 + c/2 P = 100 nSN . 1 + y/2 nLN As before, the price that traders quote is the flat price p, which is equal to the full price minus accrued interest. Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 62 Example 1 (Yield-to-Price on a Coupon Date). Price the 8 1/8's of August 15, 2021 at a yield of 7.34% for settlement August 15, 1992. Plugging c = 0.08125, y = 0.0734, and T = 58 into the formula above gives P = 109.37. (You can do it easily with a spreadsheet.) Actual/actual calendar. Now suppose that the trade settles on a day between coupon dates (and suppose also that the bond has more than one coupon period until maturity). Now we must discount the bond's cash flows back for a fractional period of time. Suppose, for example, the bond matures 6/30/93 and settles 11/30/92. What is T ? You might reason that it should be 1 1 since you're seven months from maturity. 6 That would in fact be correct for a corporate bond which uses what's called a 30/360 calendar, as we'll explain later, but Treasury bonds use what's called an actual/actual calendar for measuring fractional periods. To measure T , we count the number of six month periods from the next coupon date (12/31/92) to the maturity date and add to that the fraction of a period left from settlement to the next coupon date measured as the actual number of days from settlement to the next coupon date divided by the actual number of days from the last coupon date to the next coupon date. In our example, T would be 1 + 31/184. To get our yield-to-price formula, let's think about it in the following way. First, we'll discount all cash flows back to the next coupon date--that will give us an expression like the right-hand side of the formula above with one extra, undiscounted coupon payment--and then we'll discount that whole quantity back to the settlement date using the actual/actual calendar to measure the fractional period. Letting nSN = the actual number of days from settlement to next coupon, nLN = the actual number of days from last coupon to next coupon, and TNM = the number of six month periods from next coupon to maturity, we have P = 100 (1 + y/2)nSN /nLN TNM t=0 c/2 (1 + y/2)t + 1 (1 + y/2)TNM . Example 2 (Yield-to-Price not on a Coupon Date). Price the 8 5/8's of 8/15/93 at 3.21% for settlement 9/8/92. We have c = 0.08625, y = 0.0321, and Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 61 Adventures in Debentures Adventures in Debentures Example 4 (Bond in its Final Coupon Period). Price the 7 3/8's of 11/30/92 at 2.81% for settlement 9/17/92. The last coupon date of this bond is 5/31/92, rather than 5/30/92. (Bonds with maturity on the last day of the month typically pay coupons on the last day of the coupon month rather than on the day that matches the maturity day.) Thus we have nSN = 74, nLN = 183, c = .07375, and y = .0281. Plugging these numbers into the formula above, the full price of the bond is P = 103.101736. From this we need to subtract accrued interest. Since nLS = 109, our formula for accrued gives us a = 2.196380. The flat price, p, is the difference, 100.905356 or 100-29. Then the 30/360 number of days from date 1 to date 2 is n = 360(Y2 - Y1 ) + 30(M2 - M1 ) + (d2 - d1 ) . To compute the price of a corporate, agency, or municipal bond given its quoted yield, let nLS = 30/360 number of days from last coupon to settlement, let nSN = 30/360 days from settlement to next coupon, and let nLN = 180. Then use the same formulas for full price, flat price, and accrued interest that we gave for Treasuries with n replacing n. Example 5 (Agency Bond Price). Price the FNMA 5 1/2's of 12/19/94 at 4.28% for settlement 9/17/92.2 The last coupon date of this bond is 6/19/92 and the next coupon date is 12/19/92. Using the 30/360 calendar we have nSN = 92 and nLN = 180. In addition, c = .055, and y = .0428. Plugging these numbers into the formula for full price for a bond with more than one coupon period to maturity and replacing " n" in that formula with " n ," the full price of the bond is P = 103.933923. From this we need to subtract accrued interest. Since nLS = 88, our formula for accrued gives us a = 1.344444. The flat price, p, is the difference, 102.589479 or 102-19. 2.2.) Corporate, Municipal, and Federal Agency Bonds Corporates, munies and agencies follow many of the same conventions as treasuries; however, the calendar used to discount over fractional periods and to compute accrued interest is what's called a 30/360 calendar rather than an actual/actual calendar. Another difference is that corporates and munies (but not agencies) settle in three business days rather than one.1 Whenever you hear someone say "corporate settlement," they mean three business days until money changes hands between the buyer and the seller. 30/360 Calendar. Roughly speaking, the 30/360 calendar measures the number of days between two dates as if there were 360 days in each full year and 30 days in each full month. Here is the exact formula. Let date 1 be the earlier date and date 2 be the later date. Let Yi , Mi , and Di be the year, month, and day of date i, respectively, for i = 1, 2. Let d1 = 30 if D1 = 31, d1 = 30 if D1 = 28 or 29 and M1 = 2 and date 1 is a coupon date, d1 = D1 otherwise. Let d2 = 30 if D2 = 31 and D1 = 30 or 31, d2 = D2 otherwise. 1 2.3.) CD's and Simple Interest Securities Certificates of deposit or CD's are in the class of money market securities, which are instruments with less than one year from issue to maturity. There are two basic interest rate conventions for money markets: simple interest and banker's discount, or just discount, which we will discuss later. Actual/360 Calendar. Both simple interest and discount securities use an actual/360 calendar to measure periods of time that enter pricing calculations. In particular, letting 2 Prior to June, 1995, conventional settlement for corporates and munies was five business days, not three. FNMA stands for a bond issued by the Federal National Mortgage Association, a federally sponsored agency. Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 63 Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 64 Adventures in Debentures Adventures in Debentures nIM = actual number of days from issue to maturity, nIS = actual number of days from issue to settlement, and nSM = actual number of days from settlement to maturity, money market security calculations measure the time from issue to maturity as nIM /360, the time from issue to settlement as nIS /360, and so on, as though every year contained 360 days. CD's are issued at a price of par and pay a single cash flow at maturity consisting of par plus a predetermined interest payment. The interest payment on a CD is equal to a coupon quoted on an annual basis and scaled linearly down to the fraction of a year that the CD is outstanding. That fraction of a year is measured using the actual/360 calendar, so if the coupon rate is c and there are nIM actual days from issue to maturity, then the interest payment on $100 par amount of the CD that will be paid at maturity is nIM . interest payment = 100c 360 Simple Interest. The interest rate quoted for CD's is a simple interest rate like the one used for coupon bonds in their last coupon period that we described earlier. Given a simple interest quote, i, the full dollar price of $100 par amount of a CD with coupon rate c, nIM days from issue to maturity, and nSM days from settlement to maturity is 1 + c nIM 360 . PCD = 100 1 + i nSM 360 As we pointed out above, in a simple interest calculation, the investment time enters the growth factor (the denominator above) linearly rather than exponentially. The only difference between the use of the simple interest rate i for CD's and the bondequivalent yield y for coupon bonds in their last coupon period is the calendar. Accrued Interest. Finally, as with coupon bonds, the price traders quote is the flat price which is equal to the full price less accrued interest. The accrued interest on a CD is just like that on a coupon bond only it is computed using the actual/360 calendar. Thus, accrued for a $100 par amount of a CD with coupon rate c and nIS days from issue to settlement is nIS aCD = 100c , 360 and the flat price traders quote is pCD = PCD - aCD . Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 65 Example 6 (CD Price). Price a 6% CD issued 5/1/92 and maturing 11/1/92 at 3% for settlement 9/1/92. nIM = 184, so the interest payment on the CD is 3.066667. Since nSM = 61, the full CD price, PCD is 102.545394, using the formula above. Since nIS = 123, the accrued interest on this CD is 2.05, so the flat price, pCD , is 100.495394. 2.4.) Treasury Bills and Discount Securities Like CD's, Treasury bills, commercial paper, bankers' acceptances, and other discount securities are instruments with less than one year from issue to maturity that consist of a single cash flow at maturity. The difference is that at issue, bills are priced at a discount and pay par value at maturity, while CD's are issued at par and pay par plus a coupon at maturity. Economically, both simple interest and discount securities, as well as bonds in their final coupon period, are no different than zero-coupon bonds. T -bills in the secondary market usually settle one business day after the trade date. Sometimes, however, T -bills and commercial paper may settle "same day," that is, on the trade date. Discount Rate. The interest rate quoted for discount securities is something called a discount rate, or bankers' discount rate, which uses an actual/360 calendar. Given a discount rate, d, the price of $100 par amount of a discount security with nSM actual days from settlement to maturity is nSM . PD = 100 1 - d 360 Example 7 (Treasury Bill Price). Price the bill maturing 12/31/92 at 2.86% for settlement 9/17/92. Using the actual/360 calendar, we have nSM = 105. Since d = .0286, the price of the bill is 99.17. By the way, the asked yield that's quoted in the Wall Street Journal for T -bills is apparently the simple interest rate on an actual/365 calendar implied by the asked price for bills with 182 days or less to maturity, and the semi-annually compounded yield on an actual/365 calendar for bills with more than 182 days from settlement to maturity. 2.5.) Settlement The previous sections noted the standard form of settlement in each market. One day settlement is common for government bonds while three day settlement is the Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 66 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons traditional practice in the corporate and municipal bond market. Knowledge of the common practice for settlement is useful in interpreting bond quotes. However, two bond traders may agree to alternative time periods to settlement. Of course, as the settlement period changes, the price of the bond changes holding every thing else constant. The settlement period could (and sometimes does) extend for a period of several weeks as long as both parties to the transaction agree. Furthermore, settlement could be as short as "same day" settlement, which would require an immediate wire transfer of money from the buyer to the settlement. Practice Questions with Solutions* All interest rates quoted below are annualized; the compounding frequency associated with any given rate will depend on the question (remember, an annualized rate of r(m) compounded m times per year means a rate of r(m)/m every 1/m years.) The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." 3.) References Lynch, John J., Jr. and Jan H. Mayle, Standard Securities Calculation Methods: Fixed Income Securities Formulas. Security Industry Association, New York, NY, 1986. Stigum, Marcia, Money Market Calculations: Yields, Break-Evens, and Arbitrage. Dow Jones-Irwin, Homewood, IL, 1981. 1. ($) What is $100 invested today worth at the end of three years, assuming a rate of Part a. 20% compounded annually? Part b. 100% compounded annually? Part c. 0% compounded annually? Part d. 20% compounded semi-annually? Part e. 20% compounded quarterly? Part f. 20% compounded continuously? SOLUTION: Part Part Part Part Part Part a. b. c. d. e. f. $100(1.20)3 = $172.80. $100(2)3 = $800. $100(1)3 = $100. $100(1.10)6 = $177.16. $100(1.05)12 = $179.59. $100 e.203 = $182.21. 2. ($) An investment of $1000 today will return $2000 at the end of ten years with certainty. What is the annualized interest rate implied by this investment expressed with Part a. annual compounding? Part b. semi-annual compounding? Part c. quarterly compounding? * The first two questions are taken from a problem set developed by Wayne Ferson for a course on Investments at the University of Chicago. Chapter 2: The Grammar of Fixed Income Securities, Appendix B Page 67 Chapter 2: The Grammar of Fixed Income Securities, Practice Questions with Solutions Page 68 Adventures in Debentures Adventures in Debentures Part d. monthly compounding? Part e. daily compounding? Part f. continuous compounding? SOLUTION: so the flat price is the difference between the full price and the accrued interest, p = 99.77168365 - 0.65013587 = 99.1215478 . The quoted price in 32nd's is the flat price rounded to the nearest 32nd, for this bond the quoted price is 99:04. We are solving $1000(1 + r(m)/m)m10 = $2000 or r(m) = m(21/(10m) - 1). Part a. r(1) = 7.177%. Part b. r(2) = 7.053%. Part c. r(4) = 6.992%. Part d. r(12) = 6.952%. Part e. r(365) = 6.932%. Part f. To compute r we can either approximate it using the formula above with a very large m, or else compute it directly by solving $1000 er10 = $2000 for r. Doing 1 this we obtain r = 10 ln(2) = 6.931%. 4. ($$) On trade date 9/16/92 you buy $10 million par of FNMA 10.10's of 10/11/94 at 3.93%. When does the trade settle (assuming the standard convention)? What is the flat price of the bonds? How much money do the bonds cost you? What is the quoted price in 32nd's? SOLUTION: 3. ($$) On trade date 9/16/92 you buy $15 million par of long government bonds (the 7 1/4's of 8/15/22) at 7.322%. When does the trade settle (assuming the standard convention)? What is the flat price of the bonds? How much money do the bonds cost you? What is the quoted price in 32nd's? SOLUTION: The trade settles next day, 9/17/92. The next coupon date is 10/11/92 and the prior coupon date is 4/11/92. The agency bond uses a 30/360 calendar, so nSN = 24, nLS = 156, nLN = 180, and TNM = 4. Since c = .1010 and y = .0393, the full bond price (which represents the actual cost to purchase this bond) is: P = 100 1 (1 + .0393/2)24/180 4 t=0 1 .1010/2 + (1 + .0393/2)t (1 + .0393/2)4 . The trade settles next day, 9/17/92. The next coupon date is 2/15/93 and the prior coupon date is 8/15/92. The government bonds use an "actual/actual" calendar, so nSN = 151, nLS = 33, nLN = 184, and TNM = 59. Since c = .0725 and y = .07322, the full bond price (which represents the actual cost to purchase the bond) is P = 100 1 (1 + .07322/2)151/184 59 Using a spreadsheet to compute the quantity above, we find P = 116.5041539, so the cost of the bonds is $10,000,000P/100 = $11,650,415.39. The accrued interest per 100 par amount is 10.10 156 = 4.376667 a= 2 180 so the flat price is the difference, p = 112.127487 The quoted price in 32nd's is the flat price rounded to the nearest 32nd, for this bond the quoted price is 112:04. t=0 .0725/2 1 + (1 + .07322/2)t (1 + .07322/2)59 . Using a spreadsheet to compute the quantity above, we find P = 99.77168365, so the cost of the bonds is $15,000,000P/100 = $14,965,752.55. The accrued interest per 100 par amount is 7.25 33 = .65013587 a= 2 184 Chapter 2: The Grammar of Fixed Income Securities, Practice Questions with Solutions Page 69 5. ($) Would you rather borrow at an annualized rate of 10% compounded continuously, an annualized rate of 10.10% compounded monthly, or an annualized rate of 10.20% compounded semi-annually? Chapter 2: The Grammar of Fixed Income Securities, Practice Questions with Solutions Page 70 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons SOLUTION: Questions* All interest rates quoted below are annualized; the compounding frequency associated with any given rate will depend on the question (remember, an annualized rate of r(m) compounded m times per year means a rate of r(m)/m every 1/m years.) The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Consider borrowing $1 for 1 year. If the rate is 10% compounded continuously, then e.10 = $1.1052. If the rate is 10.10% compounded monthly, then (1 + .101/12)12 = $1.1058. If the rate is 10.20% compounded semi-annually, then (1 + .102/2)2 = $1.1046. Clearly, it is cheaper to borrow at 10.20% compounded semi-annually. 6. ($) The Treasury bill maturing November 12, 1993 traded at a discount rate of 2.83% for settlement on October 5, 1993. How much did $25,000 par amount of these bills cost? What is the continuously compounded yield (annualized) on this investment, assuming there are 365 days in a year? SOLUTION: Even if you decide not to submit your answer to this problem set, you should review my solution only after you attempt the questions to know if you could do them or not. Many times my solution is obvious AFTER you see it, but you need to know if the solution was obvious to you BEFORE you are told the solution. If you submit your answers to this problem set, please keep the following in mind: I sometimes fall behind where I expect to be in the lectures and an assignment is due before I get to some of the relevant material. If you believe the lectures have not yet covered the material necessary to answer a particular question and the required reading provides no guide, then you should indicate this on your solution and skip the question. Names of students along with their course and section numbers should be clear. The number of students in a study group should be less than or equal to five. Homework should be stapled. The final numerical answer should be "flagged" in some manner. Boxing, highlighting, and/or underlining the number are appropriate. Pages of spreadsheet printouts should be kept to a minimum. Only the essential information should be incorporated. You should not spend an excessive amount of time trying to solve any particular * The first two questions are taken from a problem set developed by Wayne Ferson for a course on Investments at the University of Chicago. There are 38 days from October 5 to November 12. The price of the bill is 100 1 - 38 .0283 = 99.7013 , 360 and the cost of $25,000 par is 25,000.997013 or $24,925.32. To find the continuously compounded interest rate, we solve .997013 = e-r(38/365) or r = - 365 ln(0.997013) = 38 2.8736%. Chapter 2: The Grammar of Fixed Income Securities, Practice Questions with Solutions Page 71 Chapter 2: The Grammar of Fixed Income Securities, Questions Page 72 Adventures in Debentures Adventures in Debentures question. If you cannot complete a question, just describe what you tried to do. purchased these STRIPs? 1. ($) What is $100 to be received in three years worth today, assuming the interest rate on a three year zero coupon bond is: Part a. 20% compounded annually? Part b. 100% compounded annually? Part c. 0% compounded annually? Part d. 20% compounded semi-annually? Part e. 20% compounded quarterly? Part f. 20% compounded continuously? 5. ($$) Today is the trade date. The trade date is Monday, October 10, 1994. The following table summarizes the relevant information for three securities: Issuer US Treasury Coupon Yield 10% 9% 10% Maturity Tuesday, 1/31/95 Monday, 10/2/95 Friday, 7/28/95 Quoted Yield 8.00% 7.00% 8.50% 2. ($) What continuously compounded interest rate is equivalent to Part a. 4% compounded annually? Part b. 20% compounded annually? Part c. 20% compounded quarterly? Part d. 100% compounded annually? Philadelphia (Muni) FNMA (agency) 3. ($$$) Consider the following: Part a. On trade date 9/16/92 the Wall Street Journal listed a bid price of 101:23 and an asked price of 101:25 for the 9 1/8's of 12/31/92 for settlement 9/17/92. What are the corresponding bid and asked yields for this government bond? Part b. On the same trade date, the Journal listed bid and asked discount rates of 2.88% and 2.86%, respectively, for a T -bill with exactly the same maturity, 12/31/92, for the same settlement, 9/17/92. (This is the bill of Example 14 in Appendix A titled "Interest Rate Quotes and Conventions.") Is there an arbitrage opportunity? (The terms "bid" and "asked" are from the point of view of the dealer; you sell at the bid price and buy at the asked price.) For all three bonds, the face value is $100, and the coupons are paid semi-annually. Note carefully that the last column in the above table reports the quoted yield; this reflects all the standard conventions for a particular type of bond given its maturity. In counting days, there are no leap years; you should also ignore holidays. Also keep in mind that a "business day" (for purposes of counting days till settlement) excludes Saturday and Sunday. Remember to quote prices to the nearest 32nd. It is very important for this question that you show your work! Do not try to use a calculator to compute the price. To receive any credit for this question, you must include all steps in the calculation! Part a. Calculate the quoted price for the bond issued by the US Treasury. Assume the usual settlement for this quoted yield. Part b. Calculate the quoted price for the muni bond issued by the City of Philadelphia. Assume the standard settlement for this quoted yield is three business days. Chapter 2: The Grammar of Fixed Income Securities, Questions Page 74 4. ($$) You buy $20 million par of the 11/15/21 STRIPs (zero-coupon bond) on trade date 9/16/92 at a price of 10-26. What is the yield quoted by the bond trader when you Chapter 2: The Grammar of Fixed Income Securities, Questions Page 73 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Part c. Calculate the quoted price for the agency bond issued by FNMA. Assume the usual settlement for this quoted yield. Chapter 3 Data for a Recurring Illustration Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Announcements and Assignments . . . . . . . . . . . . . . 82 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 83 Chapter 2: The Grammar of Fixed Income Securities, Questions Page 75 Chapter 3: Data for a Recurring Illustration Page 76 Adventures in Debentures Adventures in Debentures Day 3: The only way I can brush my teeth is by laying the Exercise Diary1 For my birthday this year my wife purchased me a week of private lessons at the local health club. Though still in great shape from when I was on the varsity chess team in high school, I decided it was a good idea to go ahead and try it. I called and made reservations with someone named Tanya, who said she is a 26-year-old aerobics instructor and athleticclothing model. tooth brush on the counter and moving my mouth back and forth over it. I am certain that I have developed a hernia in both pectorals. Driving was OK as long as I didn't try to steer. I parked on top of a Volkswagen. Tanya was a little impatient with me and said my screaming was bothering the other club members. The treadmill hurt my chest, so I did the stair monster. Why would anyone invent a machine to simulate an activity rendered obsolete by the invention of elevators? Tanya told me regular exercise would make me live longer. I can't imagine anything worse. My wife seemd very pleased with how enthusiastic I was to get started. They suggested I keep an "exercise diary" to chart my progress. Day 4: Tanya was waiting for me with her vampire teeth in full snarl. I can't help it if I was half an hour late; it took me that long just to tie my shoes. She wanted me to lift dumbbells. Not a chance, Tanya. The word "dumb" must be in there for a reason. I hid in the men's room until she sent Lars looking for me. As punishment she made me try the rowing machine. It sank! Day 1: Started the morning at 6:30 a.m. Tough to get up, but worth it when I arrived at the health club and Tanya was waiting for me. She's something of a goddess, with blond hair and a dazzling white smile. She showed me the machines and took my pulse after five minutes on the treadmill. She seemed a little alarmed that it was so high, but I think just standing next to her in that outfit of hers added about 10 points. Enjoyed watching the aerobics class. Tanya was very encouraging as I did my sit-ups, though my gut was already aching a little from holding it in the whole time I was talking to her. This is going to be GREAT! Day 5: I hate Tanya more than any human being has ever hated any other human being in the history of the world. If there were any part of my body not in extreme pain, I would hit her with it. She thought it would be a good idea to work on my triceps. Well, I have news for you, Tanya: I don't have triceps. And if you don't want dents in the floor, don't hand me any barbells. I refuse to accept responsibility for the damage. YOU went to sadist school. YOU are to blame. The treadmill flung me back into a science teacher, which hurt like crazy. Why couldn't it have been someone softer, like a music teacher, or a social studies teacher? Day 2: Took a whole pot of coffee to get me out the door, but I made it. Tanya had me lie on my back and push this heavy iron bar up into the air. Then she put weights on it, for heaven's sake! Legs were a little wobbly on the treadmill, but I made it the full mile. Her smile made it all worth while. Muscles ALL feel GREAT. Day 6: Got Tanya's message on my answering machine, wondering where I am. I lacked the strength to use the TV remote, so I watched 11 straight hours of the Weather Channel. 1 This story was written by W. Bruce Cameron and appeared in his 1997 "Exercise Diary". It was read on "Car Talk" on National Public Radio. Chapter 3: Data for a Recurring Illustration Page 77 Chapter 3: Data for a Recurring Illustration Page 78 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Day 7: Well, that's the week. Thank goodness that's over. Maybe next time my wife will give me something a little more fun, like a gift certificate for a root canal. Preface to Data for a Recurring Illustration 1.) Materials Needed for the Lecture. I plan to cover the lecture notes from the chapters titled "Data for a Recurring Illustration" and "Bond Valuation Using Synthetics." You should bring both chapters to the class. 2.) Summary of Chapter. This chapter has 2 goals. First, it reviews some basic concepts like the term structure of interest rates and the discount function. These concepts should have been covered in finance courses that you have already taken. Second, it presents some data that we will use many times in future chapters. I present the data here to keep all the relevant information in one central location. However, you may want to bring this data and graphs to future sessions for a convenient source of reference. 3.) Road Map for Chapter. The topics in this chapter will be organized as follows: 1.) Briefly explain interest rate compounding, the term structure, and the discount function. 2.) Discuss some data that will be used in future sessions. Chapter 3: Data for a Recurring Illustration Page 79 Chapter 3: Data for a Recurring Illustration, Preface Page 80 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons 4.) New Vocabulary Used in this Chapter. The following buzz words will be used in this chapter: Discount function, discount bonds, spot rates of interest, term structure of interest rates, and zero coupon bonds ("zero's"). Announcements and Assignments Chapter 3: Data for a Recurring Illustration, Preface Page 81 Chapter 3, Announcements and Assignments Page 82 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures ecture notes L for Data for a Recurring Illustration A Road Map 1.) Briefly explain interest rate compounding, the term structure, and the discount function. 2.) Discuss some data that will be used in future sessions. Zero-Coupon Bonds A zero-coupon bond maturing at time T is a bond that pays its face value at that time and no coupons prior. These are traded with names such as "STRIPS" and "zeros." These zero-coupon bonds sell at substantial discounts from their face value, with the discount representing interest earned on the investment through its life. Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 83 Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 84 Adventures in Debentures Adventures in Debentures Numeric Values for Yields and Discount Values Continuously Maturity (in years) 1/365 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Compounded Yield 8.0000% 8.0825% 8.1632% 8.2422% 8.3194% 8.3950% 8.4688% 8.5408% 8.6111% 8.6797% 8.7465% 8.8116% 8.8750% 8.9366% 8.9965% 9.0547% Annually Compounded Yield 8.3287% 8.4181% 8.5056% 8.5914% 8.6753% 8.7574% 8.8377% 8.9161% 8.9927% 9.0675% 9.1404% 9.2115% 9.2807% 9.3481% 9.4136% 9.4773% 0.999781 0.960393 0.921611 0.883704 0.846717 0.810686 0.775643 0.741613 0.708614 0.676660 0.645761 0.615919 0.587135 0.559405 0.532721 0.507072 Discount Continuously Maturity (in years) 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 Compounded Yield 9.1111% 9.1658% 9.2188% 9.2700% 9.3194% 9.3672% 9.4132% 9.4575% 9.5000% 9.5408% 9.5799% 9.6172% 9.6528% 9.6866% 9.7188% 9.7491% 9.7778% 9.8047% Annually Compounded Yield 9.5391% 9.5990% 9.6570% 9.7132% 9.7675% 9.8199% 9.8705% 9.9191% 9.9659% 10.0108% 10.0537% 10.0948% 10.1340% 10.1713% 10.2067% 10.2402% 10.2718% 10.3014% 0.482445 0.458822 0.436186 0.414515 0.393787 0.373979 0.355066 0.337022 0.319819 0.303431 0.287831 0.272990 0.258880 0.245475 0.232745 0.220664 0.209204 0.198340 Discount Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 85 Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 86 Adventures in Debentures Adventures in Debentures Continuously Maturity (in years) 17.0 17.5 18.0 18.5 19.0 19.5 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 Compounded Yield 9.8299% 9.8533% 9.8750% 9.8950% 9.9132% 9.9297% 9.9444% 9.9688% 9.9861% 9.9965% 10.0000% 9.9965% 9.9861% 9.9688% 9.9444% 9.9132% 9.8750% Annually Compounded Yield 10.3292% 10.3551% 10.3790% 10.4011% 10.4212% 10.4394% 10.4557% 10.4826% 10.5017% 10.5133% 10.5171% 10.5133% 10.5017% 10.4826% 10.4557% 10.4212% 10.3790% 0.188045 0.178293 0.169060 0.160322 0.152056 0.144238 0.136849 0.123263 0.111142 0.100339 0.090718 0.082156 0.074542 0.067775 0.061763 0.056426 0.051690 Discount The Discount Function Given the prices of zero-coupon bonds with various maturity dates, we know dT for various values of T . This set of dT constitutes the discount function. The discount function gives the price one must pay today to receive a dollar at various dates in the future. The following table provides a numerical illustration of a term structure and a discount function. Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 87 Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 88 Adventures in Debentures Adventures in Debentures Current Discount Function 1 0.8 Dollar Value Relation between Interest Rates and the Prices of Zero-Coupon Bonds dT is the current price of a zero-coupon bond which pays $1 in period T . For example, if a 12 year zero-coupon bond sells for 0.319819, then d12 = 0.319819. Let rT represent the current interest rate on a T period in vestment. For example, if d12 = 0.319819, then we can solve the following equation for rT : 1 = .319819(e12r12 ) , which implies that r12 = 9.50%. 0.6 0.4 0.2 0 0 5 10 15 20 25 30 Maturity (in years) Current Term Structure 0.12 Continuously Compounded Yield 0.11 0.1 0.09 0.08 0.07 0.06 More generally, 1 = dT (eT rT ), so we can solve for either dT or rT : dT = e-T rT rT = ln or 0 5 10 15 20 25 30 1 dT 1/T rT = ln(1/dT ) . T Maturity (in years) Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 89 Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 90 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Worksheet Chapter 4 Bond Valuation Using Synthetics Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Announcements and Assignments . . . . . . . . . . . . . . 99 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 100 A: Creating Synthetics: Further Discussion . . . . . . . . 119 B: Solving Simultaneous Equations with Electronic Spreadsheets . . . . . . . . . . . . . . . . . . . . . . 130 C: Using Linear Programming to Search for Arbitrage . . 133 Practice Questions with Solutions . . . . . . . . . . . . . . 149 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Chapter 3: Data for a Recurring Illustration, Lecture Notes Page 91 Chapter 4: Bond Valuation Using Synthetics Page 92 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Preface to Bond Valuation Using Synthetics 1.) Materials Needed for the Lecture. I plan to cover the lecture notes from the chapters titled "Data for a Recurring Illustration" and "Bond Valuation Using Synthetics." You should bring both chapters to the class. -- Oscar Wilde, Lady Windermere's Fan "What is a cynic? A man who knows the price of everything, and the value of nothing." 2.) Summary of Chapter. Some of this chapter is intended as a review of material that should have been covered in earlier finance courses -- for example, the present value rule. However, these ideas are so basic to fixed income securities that I need to discuss them here. We will extend the basic idea of the present value rule to three situations. First, we value a zero coupon bond from coupon-bearing bonds. Second, we specialize the present value rule to handle a constant coupon bond. Third, we analyze the impact of transaction cots on the applicability of the present value rule. In this chapter we will encounter our first illustration of a synthetic. The synthetics that we see in this chapter are very simple. In later chapters, we will use the concept of synthetics, but in a more complicated fashion.1 For all chapters, synthetics provide a useful technique for valuation, where the value is reasonable in order to preclude arbitrage. In the first part of the course, we focus on securities where the cash flows generated by some security are ultimately certain. For example, government bonds provide a stream of future coupons and final repayment of face value. If the investor sells such 1 The synthetics we encounter in this chapter will later be called "static (unstructured) synthetics." Chapter 4: Bond Valuation Using Synthetics Page 93 Chapter 4: Bond Valuation Using Synthetics, Preface Page 94 Adventures in Debentures Adventures in Debentures a bond prior to its maturity, then the future market value obtained at the time of the sale is uncertain. However, if the investor decides to hold the bond until it matures, then all cash flows are received with certainty. The fact that the cash flows could be received with certainty is sufficient to allow us to value such an instrument in a straightforward fashion. Later in the course we will encounter instruments (e.g., options) where the cash flows are ultimately uncertain -- that is, a option owner cannot guarantee a set of cash flows even if the option is held until maturity. Such instruments are more difficult to value. You should review the lecture notes in advance of the session. If you already understand the material, please feel free not to attend the lecture. 5.) Supplemental Reading. This would also be a good time to go back and review basic texts on corporate finance. In particular, you should focus on those chapters which discuss the present value rule. For example, one popular reference is: Richard A. Brealey and Stewart C. Myers. 2000. Principles of Corporate Finance. Sixth Edition. McGraw-Hill: New York. After attending this session, you may find the following reading useful: Appendix C, "Using Linear Programming to Search for Arbitrage." 3.) Road Map for Chapter. The topics in this chapter will be organized as follows: 1.) Illustrate the valuation process based on finding a replicating portfolio (or a synthetic alternative). 2.) Develop a general methodology for constructing a replicating portfolio which satisfies certain specifications for the quantity and timing of the cash flows. 3.) Find a synthetic alternative to a constant coupon bond. We generally think that financial markets are efficient. The following article provides some counter evidence to this viewpoint. On the surface the following article provides an interesting case study where the pricing of a security seems inconsistent with other prices in the market place. The article also provides some background information relating to "on-the-run" bonds as well as the "repo" market (and its connections to short positions). Bradford Cornell and Alan Shapiro. 1990. "The Mispricing of U.S. Treasury Bonds: A Case Study." The Review of Financial Studies. Volume 2. Number 3. Pages 297 310. The following article provides some useful details about the development of the market for zero coupon bonds. Deborah Gregory and Miles Livingston. 1992. "Development of the Market for U.S. Treasury STRIPS." Financial Analysts Journal. March/April. Pages 68 74. 4.) Required Reading. After attending this session, you should read: Appenidx A, "Constructing Synthetics: Further Details." Appendix B, "Solving Simultaneous Equations with Electronic Spreadsheets." Chapter 4: Bond Valuation Using Synthetics, Preface Page 95 6.) Other Assignments. After attending this session, you should do the following: Complete all questions which begin on page 160. Buy bulk pack from reprographics. Chapter 4: Bond Valuation Using Synthetics, Preface Page 96 Adventures in Debentures Adventures in Debentures 7.) New Vocabulary Used in this Chapter. The following buzz words will be used in the lecture notes, the readings, and/or the problem sets: Arbitrage, coupon-bearing bonds, dedicated portfolio, on-the-run securities, plain vanilla bonds, present value rule, relative value trades, replicating portfolio, repurchase agreements (repo's), synthetics, and trading universe. 9.) Acknowledgments. Some of the material in this chapter (including some of the appendices and problem set questions) is an outgrowth of a course that I co-taught with Charles Jacklin, Paul Pfleiderer, and William Sharpe at the Stanford Business School. 8.) Summary of Important Equations. The following three equations provide equivalent statements of the "present value rule." The rule allow one to aggregate cash flows that are received (or paid) at different points in time. Usually, the hardest part of the present value rule is properly stating the timing of the cash flows relative to the "present" as well as to the relevant discount rate. (Creating a time line should solve this timing problem.) Market Value = K1 e-r1 + K2 e-2r2 + + KT e-T rT . Market Value = K1 K2 KT + + + . (1 + r1 )1 (1 + r2 )2 (1 + rT )T Market Value = d1 K1 + d2 K2 + + dT KT . If AT is the price of a standard annuity paying one dollar per period until date T , then T AT = t=1 dt . If BT is the price of a constant coupon bond paying k dollars per period until date T with face value of $1, then BT = (kAT ) + dT . Chapter 4: Bond Valuation Using Synthetics, Preface Page 97 Chapter 4: Bond Valuation Using Synthetics, Preface Page 98 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Announcements and Assignments ecture notes L for Bond Valuation Using Synthetics A Road Map 1.) Illustrate the valuation process based on finding a replicating portfolio (or a synthetic alternative). 2.) Develop a general methodology for constructing a replicating portfolio which satisfies certain specifications for the quantity and timing of the cash flows. 3.) Find a synthetic alternative to a constant coupon bond. Chapter 4, Announcements and Assignments Page 99 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 100 Adventures in Debentures Adventures in Debentures Replicating Future Cash Flows Using Zero Coupon Prices Consider a twenty year bond with annual coupon payments of $1009.09 and a face value of $9363.03. This bond is viewed as default-free. The cash flows associated with this bond are: K1 = 1009.09, K2 = 1009.09, . . ., K19 = 1009.09, and K20 = 1009.09 + 9363.03 = 10,372.12. How much is this bond worth? Think of an alternative investment in a portfolio of zero coupon bonds maturing in years 1 to 20. Valuing Future Cash Flows Using Zero Coupon Prices (1) Maturity (in years) 1 2 . . . 19 20 (2) Number of Units 1,009.09 1,009.09 . . . 1,009.09 10,372.12 (3) Price per Unit 0.921611 0.846717 . . . 0.152056 0.136849 Sum (4) = (2) (3) Total Cost 929.99 854.41 . . . 153.44 1,419.41 10,000.03 Year 1 2 . . . 19 20 Cash Flow 1,009.09 1,009.09 . . . 1,009.09 10,372.12 Replicating Portfolio 1,009.09 one-year zeroes 1,009.09 two-year zeroes . . . 1,009.09 nineteen-year zeroes 10,372.12 twenty-year zeroes Thus, for a cost of 10,000.03, you can buy zero-coupon bonds that will give you the same payoffs at the same dates as the coupon bearing bond. Thus, the market value of the coupon bearing bond must also be 10,000.03.1 1 If the price of the coupon bearing bond differed from the cost of the portfolio containing the zero coupon bonds, this would generate arbitrage trades or relative value trades. Page 102 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 101 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Adventures in Debentures Adventures in Debentures Discounted Cash Flows What if you do not have zero coupon bond prices, but you do have interest rates? In this case, we know: dT = e -rT T Valuation Equation for Riskless Cash Flows Given riskless cash flows in the future (K1, K2, . . ., KT ), the potential for arbitrage requires (in the absence of transaction costs): Market Value = K1 e-r1 + K2 e-2r2 + + KT e-T rT , or in terms of annually compounded rates: Market Value = K1 K2 KT + + + . 1 2 (1 + r1) (1 + r2) (1 + rT )T . For example, assume the following continuously compounded rates from one to twenty years, r1 = 8.1632%, r2 = 8.3194%, . . ., r19 = 9.9132%, r20 = 9.9444%, then: Yr 1 2 . . . 19 Cash Flow 1,009.09 1,009.09 . . . Price of Zero (e-0.0816321) = 0.921611 (e-0.0831942) = 0.846717 . . . Current Value 929.99 854.41 . . . 153.44 1,419.41 or in terms of the discount function: Market Value = d1K1 + d2K2 + + dT KT . 1,009.09 (e-0.09913219) = 0.152056 20 10,372.12 (e-0.09944420) = 0.136849 Total Current Value: 10,000.03 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 103 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 104 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Illustrate the valuation process based on finding a replicating portfolio (or a synthetic alternative). 2.) Develop a general methodology for constructing a replicating portfolio which satisfies certain specifications for the quantity and timing of the cash flows. 3.) Find a synthetic alternative to a constant coupon bond. Example: Creating a 1 Year Zero Consider a trading universe of three bonds with the following cash flows: Bond A Yr 0 Yr 1 Yr 2 Yr 3 -B = K1 = K2 = -90.284055 5.00 5.00 Bond B -103.003910 10.00 10.00 110.00 Bond C -111.196735 15.00 115.00 0.00 K3 = 105.00 How can you generate a 1-year synthetic zero coupon bond with a par value of 100? That is, the cash flows that are wanted: Flow Yr 1 Yr 2 Yr 3 W1 = 100.00 W2 = W3 = 0.00 0.00 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 105 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 106 Adventures in Debentures Adventures in Debentures In other words, we want to find the number of units of bonds A, B, and C (that is, NA, NB , and NC ) such that this portfolio will give us the cash flows we want in years 1, 2, and 3. Mathematically, we need to solve three equations in three unknowns: 5NA + 10NB + 15NC = 100 5NA + 10NB + 115NC = 0 105NA + 110NB + 0NC = 0 Solving the above system of the three equations for NA, NB , and NC gives:2 Number How much will this synthetic zero coupon bond cost? We simply "price out" the portfolio, using the current prices of the bonds (shown below): Required Number Bond A Bond B Bond C -25.30 24.15 -1.00 Price per Bond Value 90.284055 -2,284.186592 103.003910 111.196735 Total 2,487.544427 -111.196735 92.161100 Bond A Bond B Bond C NA = -25.30 NB = NC = 24.15 -1.00 Thus, the price of a synthetic 1-year $100 zero coupon bond is $92.16. Clearly, the price of a synthetic 1-year $1 zero coupon bond would be $0.921611. Note that this involves short positions (negative holdings) and fractional numbers of bonds. The latter would not be a problem if we had wanted to create a very large par-value synthetic zero. The former may or may not be a problem for a large institutional investor. 2 The appendix titled "Creating Synthetics: Further Discussion" provides additional examples. You should review this appendix after this session . . . See the appendix titled "Solving Simultaneous Equations with Electronic Spreadsheets" for more details on how to do this calculation. Page 107 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 108 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Adventures in Debentures Adventures in Debentures A General Procedure for Creating A Synthetic Bond Let Kit be the cash paid by bond i in period t. Consider a trading universe with n bonds that provide different patterns of cash flows over n periods. Consider a portfolio with N1 of bond 1, N2 of bond 2, . . ., and Nn of bond n. The cash flows from this portfolio will be: N1K11 + N2K21 + . . . + NnKn1 in period 1 N1K12 + N2K22 + . . . + NnKn2 in period 2 . . . N1K1n + N2K2n + . . . + NnKnn in period n Now assume that you wish to receive W1 dollars in period 1, W2 dollars in period 2, . . ., and Wn dollars in period n. Can you construct a portfolio of bonds 1, 2, . . ., and n which will provide you with the cash flows that you want? We wish to find a portfolio (N1, N2, . . ., and Nn) that meets the following conditions: N1K11 + N2K21 + . . . + NnKn1 = W1 N1K12 + N2K22 + . . . + NnKn2 = W2 . . . N1K1n + N2K2n + . . . + NnKnn = Wn . This is a set of n simultaneous equations in n unknowns. If it can be solved, you can obtain the set of cash flows that you want. Keep in mind that you know values for all Kit and all Wt; you do not know N1, N2, . . ., and Nn. Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 109 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 110 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Illustrate the valuation process based on finding a replicating portfolio (or a synthetic alternative). 2.) Develop a general methodology for constructing a replicating portfolio which satisfies certain specifications for the quantity and timing of the cash flows. 3.) Find a synthetic alternative to a constant coupon bond. Comparing Constant Coupon Bonds: An Example Consider three 5 year bonds; each bond has a face value of $100. All bonds mature on the same date. All bonds pay annual coupons at the same point in time. The coupons, current prices, and yields (continuously compounded) for the three bonds are: BOND COUPON PRICE YIELD(%) A B C 12 7 2 111.36 ? 72.37 8.6863% ? 8.7333% Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 111 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 112 Adventures in Debentures Adventures in Debentures Worksheet Comparing Constant Coupon Bonds: Some Calculations A bond's value will be separated into two parts -- the value of the stream of coupons and the value of the principal. As an example, consider the 12% bond at 111.36 and the 2% bond at 72.37. Both bonds mature in 5 years. The valuation equations for the 12% and 2% bonds give: 5 111.36 = 12 t=1 5 dt + 100d5 dt + 100d5 . 72.37 = 2 t=1 Subtracting the second equation from the first gives: 5 38.99 = 10 t=1 dt . Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 113 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 114 Adventures in Debentures Adventures in Debentures The last equation implies that 5 3.899 = t=1 dt . Synthetic Alternatives to Constant Coupon Bonds: A Summary We established at the beginning of today's session that any fixed income security with known future cash flows (Kt) can be replicated with the appropriate combination of zero coupon bonds. (Again, this is an example of the LegoTM approach of modern finance.) However, in the special case of a constant coupon bond, there are two alternative synthetics to such a bond: 1.) Use a portfolio of zero coupon bonds (where the number of maturities equal the number of dates where the cash flow is not zero). This synthetic suggests the following valuation equation: B = kF d1 + kF d2 + + (1 + k)F dT , where B is the present value of a constant coupon bond, k is the coupon yield (annually compounded), F is the face value of the bond paid on date T. (The summation on the right hand side is the present value of an annuity of $1 per year for 5 years.) This suggests that the price of a zero coupon bond paying $100 in 5 years is $64.572; that is, 64.572 = 100d5 . Now lets value the 7% bond relative to the current prices for the 2% and 12% bonds. We know its value should be described by: 5 Value of 7% Bond = 7 t=1 dt + 100d5 = (7 3.899) + 64.572 = 91.865 . Thus, the appropriate value of Bond B relative to Bonds A and C is 91.86, which implies a yield to maturity (continuously compounded) of 8.7065% Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 115 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 116 Adventures in Debentures Adventures in Debentures 2.) Use a portfolio consisting of a single annuity and a single zero coupon bond (where the maturity of the annuity and the zero is the maturity of the constant coupon bond). This synthetic suggests the following valuation equation: B = kF AT + F dT , where AT is the present value of a standard annuity paying one dollar each period until period T . That is, AT = T dt. t=1 Worksheet Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 117 Chapter 4: Bond Valuation Using Synthetics, Lecture Notes Page 118 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Now we want to create a synthetic one year zero coupon bond with a face value of $100. That is, the cash flows that we want are: Appendix A: Creating Synthetics: Further Discussion This appendix provides further examples how to create synthetics involving plain vanilla bonds; at the end a general statement of the methodology is provided.1 Some of the examples in this appendix may have been discussed during the class session. However, most of this material was not presented in the lecture. Flow Yr 1 Yr 2 Yr 3 W1 = 100.00 W2 = W3 = 0.00 0.00 where Wt is the cash flow that is wanted in period t. This synthetic alternative should be based on Bonds A, B, and C. In other words, we want to find the number of units of bonds A, B, and C (that is, NA , NB , and NC ) such that this portfolio will give us the cash flows we want in years 1, 2, and 3. Mathematically, we need to solve three equations in three unknowns: 5NA 5NA 105NA + 10NB + 10NB + 110NB + 15NC + 115NC + 0NC = 100 = 0 = 0 1.) Creating Synthetic Zero Coupon Bonds. 1.1.) Example 1: Creating a 1 Year Zero. Consider bonds providing the following cash flows: Solving the above system of the three equations for NA , NB , and NC gives:3 Bond A Yr 0 Yr 1 Yr 2 Yr 3 -B = K1 = K2 = -90.284055 5.00 5.00 Bond B -103.003910 10.00 10.00 110.00 Bond C Number -111.196735 Bond A 15.00 Bond B 115.00 Bond C 0.00 We can verify that this particular composition is the correct synthetic alternative to the one year zero coupon bond by demonstrating that this composition generates net cash flows of $100, $0, and $0 in periods 1, 2, and 3, respectively. Note that this synthetic alternative involves short positions (negative holdings) and fractional numbers of bonds. The latter would not be a problem if we had wanted to create a very large par-value synthetic zero. The former may or may not be a problem for a large institutional investor. 3 NA = NB = NC = -25.30 24.15 -1.00 K3 = 105.00 where B is the current price of a bond and Kt is a cash flow at time t. These three bonds are available to buy and sell. We take the prices of these three bonds as given. We record the prices as negative values in the above table to reflect the price you pay is a cash outflow.2 1 2 Throughout the course we will encounter many types of synthetics. To distinguish among these different approaches to synthetics, we will label each methodology. Later in the course, we will call the methodology outlined in this appendix a "static (unstructured) synthetic." It seems a little silly to quote these prices to six decimal places. However, by doing so you can reproduce my numbers without rounding errors. I have carried the prices to six decimal places in order to reproduce the discount function used during the class session. See the appendix titled "Solving Simultaneous Equations with Electronic Spreadsheets" for more details on how to do this calculation in a convenient manner. Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 119 Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 120 Adventures in Debentures Adventures in Debentures How much will this synthetic zero coupon bond cost? We simply "price out" the portfolio, using the current prices of the bonds: Example 2: Desired Flows Example 1: Required Number Bond A Bond B Bond C -25.30 24.15 -1.00 Price per Bond Total Value 0 100 0 Bond A Bond B Bond C Required Number 3.30 -3.15 1.00 Price per Bond 90.284055 103.003910 111.196735 Sum 103.003910 111.196735 Sum 2,487.544427 -111.196735 92.161100 Thus, the price of the synthetic two year zero is $84.67, and the price (d2 ) of a two year zero with a face value of $1 is clearly 0.846718. The above portfolio composition for the synthetic is found by solving the following three equations in three unknowns: Thus, the price of a synthetic one year zero coupon bond with face value of $100 is $92.16. Clearly, the price (d1 ) of a synthetic one year zero coupon bond with face value of $1 is $0.921611. 5NA 5NA 105NA + 10NB + 10NB + 110NB + 15NC + 115NC + 0NC = 0 = 100 = 0 Total Value 297.937382 -324.462317 111.196735 84.671800 90.284055 -2,284.186592 Now we wish to find a synthetic three year zero coupon bond with face value of $100: Example 3: Desired Flows 1.2.) Examples 2 and 3: Creating Other Zeros. Now we find a synthetic two year zero with face value of $100 and then a three year zero with face value of $100. Since the appropriate solutions are very similar to the one used in the prior subsection, we will provide the appropriate answer with minimal detail. To construct a portfolio providing a synthetic two-year zero coupon bond, simply change the cash flows that you want and then repeat the process. This gives: Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 121 Required Number Bond A Bond B Bond C 2.00 -1.00 0.00 Price per Bond 90.284055 103.003910 111.196735 Sum Total Value 180.568110 -103.003910 0.000000 77.564200 0 0 100 Thus, the price of the synthetic three year zero is $77.56, and the price (d3 ) of a three year zero with a face value of $1 is clearly 0.775642. Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 122 Adventures in Debentures Adventures in Debentures The above portfolio composition for the synthetic is found by solving the following three equations in three unknowns: 5NA 5NA 105NA + 10NB + 10NB + 110NB + 15NC + 115NC + 0NC = 0 = 0 = 100 2.) Creating Synthetic Coupon Bonds. In the prior section, we found synthetic zero coupon bonds using coupon-bearing bonds as the components of the synthetic. Now we want to find synthetic couponbearing bonds. In the first subsection below, we will create this synthetic using coupon-bearing bonds as components; in the second subsection below, we will create this synthetic using zero coupon bonds as components. 1.3.) Summary of the Discount Function. 2.1.) Example 4: Creating Coupon Bonds from Coupon Bonds. In the prior 2 subsections, we have created synthetic zero coupon bonds with maturities of 1, 2, and 3 years. Based on those three synthetics, we have obtained part of the discount function.4 To summarize those numerical results: Discount Prices Yr 1 Yr 2 Yr 3 0.921611 0.846718 0.775642 Desired Flows 20 20 120 If we found the prices of the synthetic zero coupon bonds to be less than the prices of explicit zero coupon bonds, this suggests a profit opportunity. Apparently, one could create zero coupon bonds at a low price and sell them for a high price. In fact, in the late 1970's and early 1980's such a situation existed. As a result, investment banks started marketing zero coupon bonds using coupon-bearing government bonds as collateral. These zero coupon bonds were called "CATs," "LIONs," and "TIGRs." There have also been time periods when the synthetic zero coupon bonds were selling for a price greater than the explicit zero coupon bonds -- this provides an incentive to reconstitute the zero coupon bonds as coupon-bearing instruments. 4 What if someone asks you to value a new bond? For example, consider bond D which pays $20 in year 1, $20 in year 2, and $120 in year 3. To value bond D, you need to find a synthetic alternative to it using bonds A, B, and C. A straightforward way to do so is to simply use the cash flows that are wanted, then solve the simultaneous equations. This gives: Example 4: Required Number Bond A Bond B Bond C -2.00 3.00 0.00 Price per Bond 90.284055 103.003910 111.196735 Sum Thus, the price of the synthetic coupon-bearing bond is $128.44. The above portfolio composition for the synthetic is found by solving the following three equations in three unknowns: 5NA 5NA 105NA + 10NB + 10NB + 110NB + 15NC + 115NC + 0NC = 20 = 20 = 120 Total Value -180.568110 309.011730 0.000000 128.443620 Notice that the above discount prices match those used in the numerical examples in the class session. Thus, the three coupon-bearing bonds in this appendix are priced consistently with that discount function. This is not the only way to obtain a discount function based on the prices of coupon-bearing bonds. For example, one could use regression analysis . A regression would treat the prices of the coupon bonds as dependent variables and the coupons as the explanatory variables. The regression coefficients would then be estimates of the values for dt . The regression could be estimated subject to various restrictions on dt . Sometimes cubic splines are used to impose these restrictions. What if this particular coupon-bearing bond were already traded? If the price were not $128.44, there is a possible arbitrage. If the bond sells for more than $128.44, Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 124 Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 123 Adventures in Debentures Adventures in Debentures sell it and buy the portfolio of bonds A, B, and C (-2.00 of bond A, 3.00 of bond B, and none of bond C) providing the same cash flows. If the bond sells for less than $128.44, buy it and sell the portfolio of bonds A, B, and C providing the same cash flows. Notice that we do not know which of the four coupon-bearing bonds are mispriced (perhaps all of them are). All we know is that there is a relative mispricing among the four. Example 4 illustrates that a coupon-bearing bond can also be considered a portfolio of different coupon-bearing instruments. Usually, the synthetic based on zero coupon bonds is more intuitive. 3.) A General Procedure for Creating A Synthetic Bond. 2.2.) Example 5: Creating Coupon Bonds from Zeros. Of course, there are other synthetics (containing different components) that provide the same cash flows of $20, $20, and $120 in periods 1, 2, and 3, respectively. An alternative procedure would use the prices of the three synthetic zero coupon bonds obtainable using bonds A, B, and C: Example 5: Desired Flows 20 20 120 Synthetic Zero Price 0.921611 0.846718 0.775642 Sum Total Cost 18.43 16.93 93.08 128.44 Now assume that you wish to want W1 dollars in period 1, W2 dollars in period 2, . . ., and Wn dollars in period n. Can you construct a portfolio of bonds 1, 2, . . ., and n which will provide you with the cash flows that you want? Mathematically, we wish to find a portfolio (N1 , N2 , . . ., and Nn ) that meets the following conditions: N1 K11 N1 K12 N1 K1n + + N2 K21 N2 K22 + ... + + ... + Nn Kn1 Nn Kn2 = = . . . W1 W2 Here we formulate a general algebraic approach for creating synthetics. This algebraic framework is the obvious extension of the numerical examples that appear above. Let Kit be the cash paid by bond i in period t. Take n bonds that provide different patterns of cash flows over n periods. Consider a portfolio with N1 of bond 1, N2 of bond 2, . . ., and Nn of bond n. The cash flows from this portfolio will be: N1 K11 N1 K12 N1 K1n + + N2 K21 N2 K22 + ... + + ... + . . . Nn Kn1 in period 1 Nn Kn2 in period 2 + N2 K2n + . . . + Nn Knn in period n If desired, one could find the above portfolio composition by solving the following three equations in three unknowns: 1N1 0N1 0N1 + 0N2 + 1N2 + 0N2 + 0N3 + 0N3 + 1N3 = 20 = 20 = 120 + N2 K2n + . . . + Nn Knn = Wn . Of course, formally solving the above system of three equations is overkill, for the solution should be obvious by inspection. The great advantage of using zero coupon bonds as the components of the synthetic alternative is the computational convenience that results. Example 5 illustrates why we can always think of a coupon-bearing bond as a portfolio of zero coupon bonds. Of course, there is nothing unique about such a synthetic, for Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 125 This is a set of n simultaneous equations in n unknowns. If the n bonds are sufficiently different,5 one can obtain the desired set of cash flows. Keep in mind that you know numeric values for all Kbt and all Wt ; you do not know N1 , N2 , . . ., and Nn . The solution involves finding the values of N1 , N2 , . . ., and Nn consistent with the specified values for all Kbt and all Wt . 5 See the sections that follow for more detail about the phrase "sufficiently different." Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 126 Adventures in Debentures Adventures in Debentures If the cost of constructing a synthetic version of a bond with n other bonds does not equal the price of the actual bond, there is an arbitrage possibility. If there are no arbitrage possibilities, any set of n bonds (that are "different enough") will provide the same set of zero-coupon bond prices. We call this the discount function. It is only unique in a market with no arbitrage possibilities. For purposes of an examination, you should be able solve three equations in three unkowns.6 For problem set questions, you may be asked to work with a large number of equations and unknowns. If so, it is assumed that you will use an electronic spreadsheet to generate the solution. providing the following cash flows: Bond E Yr 0 Yr 1 Yr 2 Yr 3 -B = K1 = K2 = -77.56 0.00 0.00 Bond F -92.16 100.00 0.00 0.00 Bond G -8.49 5.00 0.00 5.00 K3 = 100.00 4.) Creating Synthetics: Some Caveats. If there are n periods, usually n bonds can be used to construct a synthetic version of any cash flow pattern over the n periods. This will work as long as the associated set of equations can be solved. If the equations cannot be solved, the bonds are not "sufficiently different."7 Consider two examples where the bonds are not "sufficiently different." Now we want to create a synthetic three year annuity where each annuity payment is $1000. That is, the cash flows that are wanted: Flow Yr 1 Yr 2 Yr 3 W1 =1000.00 W2 =1000.00 W3 =1000.00 4.1.) Example 6: Non-zero Desired Cash Flow, All Available Cash Flows Are Zero. This is perhaps the easiest example. Say you wish to create a synthetic with a desired cash flow different from zero in period t. The potential components of this synthetic all have cash flows of zero in period t. Obviously, any combination of these potential components will pay zero in period t, for anything times zero is still zero. Thus, this is a situation where the bonds are not "sufficiently different," and no synthetic can be constructed. For a specific set of numbers consider bonds E, F, and G below: Consider bonds 6 The synthetic alternative should be based on Bonds E, F, and G. We need to solve three equations in three unknowns: 0NE 0NE 100NE + 100NF + 0NF + 0NF + 5NG + 0NG + 5NG = 1000 = 1000 = 1000 If you try using Excel to find the solution, Excel will not be able to find an answer for the obvious reason. Excel will return "#NUM!" as the solution. 7 You could be given a potential solution to a problem where there are n equations in n unknowns (where n may be much larger than three). Then the question will ask you to verify whether or not the potential solution is correct. Even with large values for n, this is not a time intensive question. You need to just check if the net cash flows generated by the potential solution for each time period is in fact what was desired. In a linear algebra textbook, such a situation might be referred to as "linearly dependent" equations or that the matrix of future cash flows is "singular" or not of "full rank." 4.2.) Example 7: Synthetic Components Are Redundant. Now return to example 1. We still want to create a synthetic one year zero coupon bond with face value of $100. However, the synthetic components must be based on bonds A, B, and D (not C). Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 128 Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 127 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Mathematically, we need to solve three equations in three unknowns: 5NA 5NA 105NA + 10NB + 10NB + 110NB + 20ND + 20ND + 120ND = 100 = 0 = 0 An electronic spreadsheet will be unable to generate a solution to this problem. (Again, you will see "#NUM!" as the solution in Excel.) Why? Well recall that in example 4 that we generated a synthetic version of bond D using just 3 units of B and -2 units of A. Thus, using bond D is not "sufficiently different" from bonds A and B. It appears that we have 3 bonds when in fact one of the three bonds is redundant with the other two. Whenever your electronic spreadsheet fails to generate a solution, you can always check if some of the bonds are redundant with the other bonds. Appendix B: Solving Simultaneous Equations with Electronic Spreadsheets 1.) Introduction. When trying to find a synthetic alternative and the value of this synthetic, you may need to solve a system of simultaneous equations. Excel can be used to calculate such a solution. Perhaps the most straightforward way to use Excel is to rely on Solver, which is a tool in the Excel package. Solver is a very general algorithm which can be used to solve a system of linear equations. Another approach is to rely on Excel's matrix functions to solve a system of linear equations. This appendix discusses how to use these matrix functions. 5.) Extensions. This appendix implicitly assumes perfect markets (i.e., one can buy or sell any bond at the same price, in any amount). More realistic conditions alter the results, but the general relations will hold as approximations (i.e., values must be within ranges, rather than at precise levels). For situations where there are more than n assets to match up just n cash flows and/or where transaction costs are present, see the appendix titled "Using Linear Programming to Search for Arbitrage." 2.) Using Excel's Matrix Functions. Put the cash flows in a named array. For example, the one boxed below. Call it K. (To name an array use Define Names under the Insert menu.) Bond 1 Yr 1 Yr 2 Yr 3 5 5 105 Bond 2 10 10 110 Bond 3 15 115 0 In another location in the spreadsheet highlight a range of cells with the same number of rows and columns as in K above. Then in the upper left cell of this array, Chapter 4: Bond Valuation Using Synthetics, Appendix A Page 129 Chapter 4: Bond Valuation Using Synthetics, Appendix B Page 130 Adventures in Debentures Adventures in Debentures enter = minverse(K), followed by control+shift+enter if you are using Excel for Windows.1 The result will be a set of numbers that is called the inverse of K. In this case: -0.253 0.033 0.02 0.2415 -0.01 -0.0315 0.01 -0.01 0.00 formula. (When you are entering an array formula in Excel for Windows, press control+shift+enter; when you entering an array formula in Excel for the Macintosh, press command+enter.) Now required set of bonds will be in column 2 and the value of the "synthetic bond" will be in cell (5). To value another set of cash flows, simply change the numbers in column 1 (W) or change the cash flow array (K). Name the above array KInv. (Use the same procedure as we did above in naming the cash flow array K.) If KInv contains "#NUM!" in some or all of its cells, the bonds are not "different enough." If you get an inverse, they are "different enough." Next, set out ranges for (1) the set of desired flows, (2) the required numbers of the bonds, (3) the price of each bond, (4) the value of each set of bonds in the resulting portfolio, and (5) the total value of the portfolio. (1) Desired Flows 100 0 0 Bond 1 Bond 2 Bond 3 (2) Required Number -25.30 24.15 -1.00 (3) Price per Bond 85.50 97.75 107.87 (5) Total (4) Value -2163.15 2360.66 -107.87 89.64 Type the desired set of flows in column (1), and name this array W. (Again, use the same method as we used to name K and KInv.) Type the price per bond in column (3). Type formulas in column (4) so that each value is the product of the required number (column 2) times price per bond (column 3). Type a formula (=sum(..)) in cell (5) so that the value will be the sum of the numbers in column 4. Highlight the range where the required set of bonds are to be placed (i.e., column 2). Then in the top cell of this range, enter =mmult(KInv,W). This is an array 1 When using Excel for the Macintosh, press command+enter after you enter = minverse(K). Chapter 4: Bond Valuation Using Synthetics, Appendix B Page 131 Chapter 4: Bond Valuation Using Synthetics, Appendix B Page 132 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Appendix C: Using Linear Programming to Search for Arbitrage* Since the audience for this session is not homogeneous, I like to include some reading that will appeal to those who have prior knowledge of fixed income securities. This reading is intended for that group of individuals. Arbitrage is the ability to create positive profits today at no cost in the future. The purpose of this note is to illustrate two methods for solving for arbitrage -- the replication method and the minimum cost portfolio method -- and to demonstrate how linear programming can be used in the context of these two interrelated methods. The intent is to explain everything using examples which can be solved using Excel's Solver function.1 This note begins with a tutorial on linear programming and the Excel Solver function. The application to the arbitrage problems begins in Section 3. A linear program is simply a means of solving an optimization problem. Optimization problems (and the LP's used to solve them) exist in virtually any context, from engineering to biology to finance. The basic idea is that you have an objective function which needs to be maximized (or minimized) under certain constraints. Consider the following example:3 Example #1 Suppose we face a universe of bonds in which there are only three zeros and two coupon bonds: Security 0 d1 0 d2 0 d3 Type 1-year zero 2-year zero 3-year zero 3-year, 8% coupon 2-year, 12% coupon Face Value 100 100 100 100 100 Ask 94.32 91.41 87.65 109.02 113.70 Bid 92.81 89.97 85.14 106.07 111.90 B3 B2 1.) Linear Programming -- The Basics Numerous textbooks have been written on the subject of linear programs (or LP's as they are commonly called) and semester-long courses have been formed to teach students the art of constructing and solving LP's.2 This section attempts the seemingly impossible task of describing the basics of LP's in what amounts to only five paragraphs. The intent is to lay the groundwork for those of you who have never seen LP's before. Section 2 outlines how to solve LP's using Excel's Solver function. LP experts should find these sections to be a review and are welcome to skip to Section 3. * This appendix was prepared by John J. Kraska, III in March 1993 under the supervision of Michael Gibbons, Professor of Finance, the Wharton School of the University of Pennsylvania. 1 There are numerous software packages that can be used to solve linear programs (e.g., Lindo, Lotus, etc.). Excel appears to be the simplest package to use and is therefore used in this note. 2 One possible textbook is Roger Hartley's Linear and Nonlinear Programming, Ellis Horwood Limited, New York, 1985. In this example and throughout this note, the ask is the price at which we can buy the security, and the bid is the price at which we can sell the security.4 Now, suppose we want to construct a portfolio that generates cash flows of at least $20 in year 1, $120 in year 2, and $108 in year 3. Of course, there are infinitely many combinations of buying and selling the above securities that would meet our cash flow requirements. But, what if we want to find the cheapest portfolio that fulfills our needs? In essence, what we have done is to create an optimization problem. We can solve this problem using an LP. Our objective function, in this case, would be the cost of purchasing a portfolio of 3 4 For a more general version of the following problem, see Schaefer, "A Model for Bond Portfolio Improvement," in Journal of Financial and Quantitative Analysis, June 1977. The bid is the price at which you can sell a security if you already own that security. The price for which you can short a bond would differ from the bid price. Usually, the revenue you generate from shorting a security is less than the revenue you generate when you sell a security in your portfolio. In this note, however, we will assume that you can sell or short securities at the same price. Of course, we could extend this appendix to accomodate more general assumptions about the cost of short positions. Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 133 Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 134 Adventures in Debentures Adventures in Debentures these securities. Since we want to minimize this cost, we write the following:5 Minimize: L = -92.81N 1b - 89.97N 2b - 85.14N 3b - 106.07B3b - 111.90B2b + 94.32N 1a + 91.41N 2a + 87.65N 3a + 109.02B3a + 113.70B2a . Note when L is greater than zero, this implies a porfolio position that requires a cash outflow today. When L is less than zero, this implies a portfolio position that generates a cash inflow today. The first constraint of our problem is that the cash flows of this portfolio in year 1 must be greater than or equal to $20. This translates into the following equation: -100N 1b - 8B3b - 12B2b + 100N 1a + 8B3a + 12B2a 20 . 2.) Solving LP's Using Excel Solver Depending on the complexity of the LP, there are several different techniques that can be used to solve it. This section explores the use of Excel Solver as one such technique. In particular, we will focus on solving the problem introduced in Section 1. We can set our spreadsheet up in the following manner. Add two columns next to the prices of the securities that represent the amounts of each security bought and sold in our portfolio. Then, fill this column with arbitrary amounts (say 0 units of each security). This becomes the starting portfolio from which Excel will begin its process of searching for an optimal solution of the LP. Next, create a cell that contains the cost of the portfolio (the objective function L) by referencing the amounts of each security with their appropriate prices. For the sake of demonstration, I will refer to this cell as $A$1 in our spreadsheet. We must now add to our spreadsheet the constraint equations. In cell $A$2, for instance, we can type the left hand side of the first constraint equation listed above, making sure to reference the appropriate cells for N 1a , N 1b , B3a , B3b , B2a , and B2b . In cell $A$3, we can include our first year's cash flow requirement, namely $20. We can repeat this procedure in cells $B$2 through $C$3 for the cash flow constraints of years 2 and 3. To solve the LP, we invoke the Excel Solver Function and follow the instructions. We want to "Set Cell" $A$1 "to a Minimum," "By Changing" the cell references for the amounts bought and sold of each security. In the box "Subject to the Constraints," we add $A$2>$A$3, and so forth for each annual cash flow. Furthermore, we must add to this box that the cells for the amounts of each security must be greater than or equal to zero. When you click on "Solve," Excel searches for the solution.6 Upon finding one, Excel states that "there is a feasible solution" and stores the appropriate amounts of each security in the cells we have created for them (i.e., Excel replaces the 0's in the amounts column with the solution values). For the Section 1 example, Excel finds that the cheapest portfolio that generates our required cash flows costs 222.7176 (i.e., L = 222.7176). This portfolio is made up of Similarly, we can write the constraint equations for the cash flows in years 2 and 3: -100N 2b - 8B3b - 112B2b + 100N 2a + 8B3a + 112B2a 120 -100N 3b - 108B3b + 100N 3a + 108B3a 108 . We must also incorporate into our LP the fact that the amounts of each security bought or sold must be positive. Altogether, the LP becomes: Minimize: L = -92.81N 1b - 89.97N 2b - 85.14N 3b - 106.07B3b - 111.90B2b + 94.32N 1a + 91.41N 2a + 87.65N 3a + 109.02B3a + 113.70B2a subject to the constraints: -100N 1b - 8B3b - 12B2b + 100N 1a + 8B3a + 12B2a 20 -100N 2b - 8B3b - 112B2b + 100N 2a + 8B3a + 112B2a 120 -100N 3b - 108B3b + 100N 3a + 108B3a 108 N 1a , N 1b , N 2a , N 2b , N 3a , N 3b , B3a , B3b , B2a , B2b 0 . 5 Here N1, N2, N3 represent the number of units to purchase of the 1, 2, and 3 year zero coupon bonds, respectively. The subscripts a and b represent the amount we receive at the ask price and the amount we receive at the bid price. 6 To improve the computational efficiency of Excel, you can select "Options" and check "Assume Linear Model." Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 135 Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 136 Adventures in Debentures Adventures in Debentures the following amounts: N 1b = N 2b = N 3a = N 3b N 1a N 2a B3a = B2a = B2b = B3b = 0.00 = 0.12 = 1.12 = 1.00 . Consider the following universe of securities in a world free of transactions costs: Security 0 d1 0 d2 0 d3 Type 1-year zero 2-year zero 3-year zero 3-year, 8% coupon 2-year, 12% coupon Face Value 100 100 100 100 100 Price 93.57 90.69 86.40 108.05 112.80 Please note that this solution is not entirely obvious. At first glance, we might have assumed that the cheapest portfolio that generates our required cash flows would be equivalent to the portfolio of zeros. We could, for instance, match our cash inflow needs by purchasing .2 units of 0 d1 , 1.2 units of 0 d2 , and 1.08 units of 0 d3 . However, this portfolio would cost us $223.218 which is $0.5004 more expensive than the cheapest portfolio we could purchase. The LP technique has saved us the headache (or wallet-ache) of picking an obvious portfolio which may not be the optimal portfolio. B3 B2 We want to find the cheapest portfolio that delivers no cash flows in years 1, 2 and 3. From Section 1, we learned that we can write this problem as an LP. This LP differs from the one constructed previously in three respects. First, since there are no transactions costs, we can buy and sell the securities at the same price. Therefore, we do not have to differentiate in the objective function and constraint equations between the bid and ask prices of each security. Instead, we let (say) N 1 represent the amount of 0 d1 that we hold in our portfolio. Since this amount can be negative (in the instance when we sell 0 d1 ), we must also remove the constraint that the amounts of each security be strictly positive. Second, unlike in the previous LP, we want to force the portfolio to generate no cash flows in the future. We can accomplish this by replacing the by = in each constraint equation. Third, we want to add a constraint that limits the amount of arbitrage we can make to (say) $1. If we were not to include such a constraint and if arbitrage did exist, the optimal solution for the linear program would be negative infinity. This occurs because the mispricing that the program is taking advantage of to generate arbitrage could be repeated infinitely many times to generate an infinite amount of profits today. Software packages which solve LP's may not be able to handle an optimal solution which is negatively infinite.8 In light of these changes, the LP we wish to solve is the following: 3.) The Minimum Cost Method for Finding Arbitrage -- Some Examples The search for arbitrage can be made within the context of the standard optimization problem by rephrasing the statement "arbitrage is the ability to create positive profits today at no cost in the future." We can think of it as a minimum cost portfolio problem where the portfolio we wish to construct is one that generates no cash flows in the future. In a world without arbitrage opportunities, this portfolio should cost zero dollars today (in fact, it is equivalent to holding no portfolio). However, if arbitrage does exist, then the cost of the portfolio should be negative, meaning there is a positive profit today without future financial obligations.7 This minimum cost portfolio problem, then, can be solved using the same LP technique as discussed in the previous two sections. In this section, we will run through some examples which illustrate using LP's to solve the minimum cost portfolio problem. We will again assume a world of three zeros and two coupon bonds, however, we will concentrate on what happens to arbitrage opportunities when there are transactions costs to buying and selling securities. Example #2 7 Minimize: L = 93.57N 1 + 90.69N 2 + 86.40N 3 + 108.05B3 + 112.80B2 8 If there were no restrictions on the number of securities that one could buy or sell, then the minimum cost would, in fact, be negative infinity. Excel circumvents the negatively infinite solution by incorporating a time limit for Solver to spend searching for an optimal solution. In the case when the minimum value is negative infinity, Solver calculates until its time limit has expired. Then, it reports the largest negative value it has found. The magnitude of this value depends on where Excel starts the search for the solution and on how much time Excel has allotted to solve the program. Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 137 Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 138 Adventures in Debentures Adventures in Debentures subject to the constraints: 100N 1 + 8B3 + 12B2 = 0 100N 2 + 8B3 + 112B2 = 0 100N 3 + 108B3 = 0 L -1 . When we run Excel Solver, we find the optimal solution to be L = 0, which is just the zero portfolio. In other words, we cannot do arbitrage. In this example, this makes sense if we look at how each security is priced. It just so happens that the prices of B3 and B2 match the prices of the synthetic three year and two year coupon bonds which depend only on the prices of the zeros. Example #3 Now, consider the following prices in our universe of securities. Again we assume there are no transactions costs involved in buying or selling these bonds: Example #4 What happens now when transactions costs enter the picture? Suppose the prices of the securities in Examples #2 and #3 are adjusted so that each security has both a bid and ask price.9 Scenarios 1 and 2 display these prices (Price NTC represents the price of each security as shown in Examples #2 and #3): Security Scenario 1: 0 d1 0 d2 0 d3 Price NTC Ask Bid B3 B2 Scenario 2: 0 d1 0 d2 0 d3 93.57 90.69 86.40 108.05 112.80 94.32 91.41 87.65 109.52 113.70 92.81 89.97 85.14 106.57 111.90 Security 0 d1 0 d2 0 d3 Type 1-year zero 2-year zero 3-year zero 3-year, 8% coupon 2-year, 12% coupon Face Value 100 100 100 100 100 Price 93.57 90.69 86.40 105.05 112.80 B3 B2 93.57 90.69 86.40 105.05 112.80 94.32 91.41 87.65 106.52 113.70 92.81 89.97 85.14 103.57 111.90 B3 B2 When we run Excel Solver to solve the appropriate LP (the same LP as in Example #2 but with 105.05 instead of 108.05 in the objective function), we get a value of L = -1. In other words, we can create a portfolio that generates no cash flows in the future but gives us $1 today. Buying a million of these portfolios gives us a million dollars today without any obligation in the future (not bad for a few minutes work)! The intuition behind this result is apparent when we look at the amounts of each bond in our portfolio. We see that our portfolio consists of buying B3 and selling each of the zeros. What does this mean? From Example #2, we know that B3 is mispriced relative to the synthetic B3 created by the zeros (the price of which is 108.05). The portfolio takes advantage of this mispricing by buying B3 and selling its synthetic so that the net cash flow is zero. The program repeats this process until a $1 arbitrage is made. Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 139 When we run the appropriate linear programs assuming transactions costs in each scenario, we again obtain L = 0 in Scenario 1 and L = -1 in Scenario 2. Arbitrage can only be made in Scenario 2. Example #5 In Scenario 2 in the previous example, we were able to create arbitrage profits when there were no transactions costs and when there were transactions costs. Does this happen all of the time? What if the same securities are priced in the following manner? 9 In this example, the bid-ask prices were created so that the average of the bid-ask spread for each security equals the price of the security when there are no transactions costs. Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 140 Adventures in Debentures Adventures in Debentures Consider the following world of four securities that trade free of transactions costs: Security Scenario 3: 0 d1 0 d2 0 d3 B3 B2 Price NTC Ask Bid Security 0 d1 0 d2 0 d3 B3 93.57 90.69 86.40 107.55 112.80 94.32 91.41 87.65 109.02 113.70 92.81 89.97 85.14 106.07 111.90 Type 1-year zero 2-year zero 3-year zero 3-year, 8% coupon Face Value 100 100 100 100 Price 93.57 90.69 86.40 108.05 First, consider what happens in the world without transactions costs. When we run the appropriate linear program we find that the optimal solution is L = -1. This should come as no surprise because we already know that B3 is underpriced by $1 relative to the price of its synthetic. However, when we run the program to find the least cost portfolio assuming these securities trade with transactions costs, we get an optimal solution of L = 0! In other words, transactions costs have absorbed the mispricing of B3. We start with a simple question, can we create a portfolio of 0 d1 , 0 d2 , and 0 d3 that replicates the cash flows of B3 but at a different price? In terms of a linear program, we want to solve the following: Minimize: L = 93.57N 1 + 90.69N 2 + 86.40N 3 subject to the constraints: 100N 1 = 8 100N 2 = 8 100N 3 = 108 L - B3 -1 . Here, L is the cost of replicating B3 today. N 1, N 2, and N 3 are the amounts of 0 d1 , 0 d2 and 0 d3 , respectively. The first three constraints are that the cash flows of the portfolio should match the cash flows of B3. For instance, in year 1, the possible cash flows from the portfolio are only the face value of 0 d1 times N 1. This value should equal 8, the coupon of B3 times its face value of 100. The fourth constraint is similar to the constraint used in the minimum cost portfolio problem in that it limits the amount of arbitrage to $1. Solving this linear program using Excel Solver, we get a minimum of L = 108.05 with N 1 = .08, N 2 = .08 and N 3 = 1.08. Since the value of L is equal to the market price of the security, no arbitrage exists. In fact, in this example, no matter what the market price of B3 is, the linear program will always yield L = 108.05. This makes sense because the price of the synthetic three year coupon bond depends only on the prices of the zero coupon bonds. That is P = .08 93.57 + .08 90.69 + 1.08 86.40 which is just L. Therefore, in a world Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 142 4.) The Replication Method -- A Simple Example Now, it is time to switch gears almost entirely and focus on another method of searching for arbitrage -- the replication method. To motivate studying this method, let us first reiterate the intuition behind how we found arbitrage in the context of the minimum cost portfolio problem. There we defined arbitrage as a portfolio that generates positive profits today and has zero net cash flows in the future. In this section, we want to answer the more general question, when are we able to create arbitrage? Simply put, we can create arbitrage when there is a security in the universe that is mispriced relative to its synthetic. Therefore, a logical way to search for arbitrage is to pick an arbitrary security trading on the market and try to replicate its cash flows with a portfolio of all other marketable securities. Arbitrage does not exist if the cheapest way of replicating the security equals the price of the security on the market. However, if the price of replicating is greater (less) than the price of the security, then one creates arbitrage by buying (selling) the security and selling (buying) the replicated security. This is the essence of the replication problem. Example #6 Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 141 Adventures in Debentures Adventures in Debentures without transactions costs, we only have to compare the market price of B3 to L and buy the cheaper of the two and sell the more expensive in order to do arbitrage. Security 0 d1 0 d2 0 d3 B3 B2 Type 1-year zero 2-year zero 3-year zero 3-year, 8% coupon 2-year, 12% coupon Face Value 100 100 100 100 100 Price 93.57 90.69 86.40 108.05 112.80 5.) The Replication Method -- More Sophisticated Examples The example outlined in the previous section is simplistic beacuse you only need to run the linear program once to find the minimum cost of replicating B3. But, is it true that every time you run a linear program to replicate one bond you will end up with the same replicating portfolio? The answer is no. One reason is that there may be more bonds in the universe than just three zeros and one coupon bond. Suppose that, in addition to the zeros and B3, there is a two year coupon bond, B2, on the market. The minimum cost of replicating B2 is not necessarily the price of the synthetic created by 0 d1 and 0 d2 . It may happen that B3 is mispriced relative to the zeros and that you can use a portfolio which includes the mispriced B3 to replicate B2 at a cheaper price than the price of the synthetic using 0 d1 and 0 d2 . Another reason is that transactions costs play a major role in detemining the minimum cost of replicating B2. In the previous example, we assumed that all bonds could be bought and sold at the same price. In reality, the price at which you buy a security is higher than the price at which you can sell the same security. In the context of the replication problem, there may be a mispricing of either B3 or B2 present, but the ability to make arbitrage profits on this mispricing is limited by the amount of transactions costs. In this section, we will step through a few examples to illustrate what happens to the cost of replicating B2 (and ultimately whether or not one can make arbitrage) as the market prices of B2 and B3 change and as transactions costs are introduced. Example #7 Consider the following universe of securities in a world without transactions costs: Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 143 Here, we want to construct a portfolio that replicates the cash flows of B2. Since we want to achieve the flexibility of having this portfolio include securities 0 d3 and B3, we will use the following linear program: Minimize: L = 93.57N 1 + 90.69N 2 + 86.40N 3 + 108.05B3 subject to the constraints: 100N 1 + 8B3 = 12 100N 2 + 8B3 = 112 100N 3 + 108B3 = 0 L - 112.80 -1 . Of particular importance in this linear program is the third constraint. Since B2 gives no cash flows in year 3 but we include the possibility that our replicating portfolio consists of B3 and 0 d3 , we must force all cash flows we receive in year 3 to be zero. When we solve this linear program, we get a minimum L = 112.80 with N 1 = .12, N 2 = 1.12, and N 3 = B3 = 0. Again, since L is equal to the price of B2, we are unable to create arbitrage. The solution portfolio should not surprise you since we know (from Section 4) that B3 is priced correctly. Therefore, the price of B2 should be (and is) equal to the price of the synthetic created by 0 d1 and 0 d2 . However, this optimal portfolio is not unique. In fact, consider the portfolio which has N 1 = .2, N 2 = 1.2, N 3 = 1.08 and B3 = -1. This portfolio, too, has cash flows that mimic those of B2 and costs 112.80. In fact, this portfolio is just the previous portfolio with the additional feature that B3 is sold and its synthetic purchased. The net effect on the portfolio of this transaction is zero because B3 happens to be priced equal to its synthetic. In fact, we could repeat this process infinitely many times to obtain infinitely many portfolios. Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 144 Adventures in Debentures Adventures in Debentures Example #8 Security Consider what happens when B3 is mispriced. Can we create a portfolio that replicates the cash flows of B2 at a cost less than 112.80? Suppose 0 d1 , 0 d2 , 0 d3 , and B2 are priced as in Example #7, but this time the price of B3 is 105.05. When we run the Solver program, we find that the minimum value of L is 111.80! We can create a $1 arbitrage opportunity by purchasing this portfolio and selling B2 as it is priced on the market. Scenario 2: The intuition behind this result is not that B2 is mispriced, per se. In fact, B2 is correctly priced relative to the zeros. However, since B3 is underpriced, we can replicate B2 by including in our replicating portfolio a purchase of B3 and a sale of its synthetic along with the zeros that mimic the cash flows of B2. Example #9 How do transactions costs affect the ability to do arbitrage when securities are mispriced? As in Examples #4 and #5 of Section 3, we will assume that the bid and ask prices are set such that their averages yield the prices in a world without transactions costs. Consider the following three pricing scenarios: Scenario 3: 0 d1 0 d2 0 d3 0 d1 0 d2 0 d3 Price NTC Ask Bid Scenario 1: 0 d1 0 d2 0 d3 B3 B2 93.57 90.69 86.40 108.05 112.80 93.57 90.69 86.40 105.05 112.80 93.57 90.69 86.40 107.55 112.80 94.32 91.41 87.65 109.52 113.70 94.32 91.41 87.65 106.52 113.70 94.32 91.41 87.65 109.02 113.70 92.81 89.97 85.14 106.57 111.90 92.81 89.97 85.14 103.57 111.90 92.81 89.97 85.14 106.07 111.90 B3 B2 B3 B2 The linear program which optimizes the price of replicating the purchase of B2 (in Scenario 1, for instance) is the following: Minimize: L = 94.32N 1a + 91.41N 2a + 87.65N 3a + 109.52B3a - 92.81N 1b - 89.97N 2b - 85.14N 3b - 106.57B4b subject to the constraints: 100(N 1a - N 1b ) + 8(B3a - B3b ) = 12 100(N 2a - N 2b ) + 8(B3a - B3b ) = 112 100(N 3a - N 3b ) + 108(B3a - B3b ) = 0 L - 111.90 -1 N 1a , N 2a , N 3a , B3a , N 1b , N 2b , N 3b , B3b 0 When we solve the appropriate linear programs for each of the scenarios, we obtain the following results: Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 145 Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 146 Adventures in Debentures Adventures in Debentures Minimize: Scenario 1 2 3 Cost of Portfolio (NTC) L = 112.80 L = 111.80 L = 111.80 Cost of Portfolio (W/TC) L = 113.70 L = 112.70 L = 113.70 L = -92.81N 1b - 89.97N 2b - 85.14N 3b - 106.07B3b - 111.90B2b + 94.32N 1a + 91.41N 2a + 87.65N 3a + 109.02B3a + 113.70B2a subject to constraints: -100N 1b - 8B3b - 12B2b + 100N 1a + 8B3a + 12B2a 20 -100N 2b - 8B3b - 112B2b + 100N 2a + 8B3a + 112B2a 120 -100N 3b - 108B3b + 100N 3a + 108B3a 108 (-100N 3b - 108B3b + 100N 3a + 108B3a - 108)1.06 100 N 1a , N 1b , N 2a , N 2b , N 3a , N 3b , B3a , B3b , B2a , B2b 0 . In other words, arbitrage is obtained in Scenarios 2 and 3 when no transactions costs are present by purchasing the portfolio and selling B2. However, when transactions costs are introduced, arbitrage only exists under Scenario 2. In Scenario 3, the amount of transactions costs absorbs the mispricing of B2. 6.) Extensions The advantages to the LP technique of searching for arbitrage lie in its flexibility. The optimization problems described above can be extended to cover a wide range of similar problems in the field of fixed income securities. This section explores some of these extensions. One obvious extension is that instead of matching cash flows in our constraints, we can apply the technique of LP's to match dollar durations or convexities. Duration and convexity will be discussed later in this course. Example #10 Another extension would allow for cash flow mismatching. Consider Example #1 in the section on basic LP problems. Suppose that the universe of bonds remained the same, however the cash flow requirements also included a $100 cash flow need in year 4. How would we find the minimum cost portfolio given this need and the fact that no bonds generate a cash flow in year 4? This obstacle can be overcome by assuming that one can reinvest the year 3 cash inflows from a portfolio at a conservative reinvestment rate. Suppose, for instance, we honestly believe that we can reinvest year 3 cash flows at one year rate of 6%. The LP we hope to solve becomes: Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 147 Here, the fourth constraint represents the fourth year cash flow which is assumed to be the proceeds from year 3 in excess of the year 3 cash need of $108 reinvested at the 6% reinvestment rate. When we solve this LP, we obtain a minimum cost portfolio of L = 304.9692. To achieve this minimum cost portfolio, our position should be: N 1b = N 2b = N 3a = N 3b N 1a N 2a B3a = B2a = B2b = B3b = 0.00 = 0.0501 = 1.0501 = 1.8735 . Chapter 4: Bond Valuation Using Synthetics, Appendix C Page 148 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Part a. Therefore, Practice Questions with Solutions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." 0.64 = 1 (e0 r0,2 )2 0 r 0,2 = 22.3% . Part b. If you buy A, short B, and short C, you will have no net cash flows in the future. Such a strategy will result in a net profit today of .10. Thus, to generate $2.00 profit today, you need to repeat the above strategy 20 times. That is, buy 20 units of A, short 20 units of B, and short 20 units of C. The above solution is intuitive and easy to find for this simple problem. An alternative approach is to formulate an algebraic statement of the desired position. We want the cash inflow in year 0 (CF0) to be +2.00. We want the cash flow in years 1, 2, and 3 (that is, CF1, CF2, and CF3) to be zero. If we could find such a position, we have found a classic arbitrage. Thus, we want: CF 0 : CF 1 : CF 2 : CF 3 : -2.24NA +1.00NA +1.00NA +1.00NA -1.60NB +1.00NB +0.00NB +1.00NB -0.74NC +0.00NC +1.00NC +0.00NC = = = = 2.00 0.00 0.00 0.00 1. ($$) Assume today is year 0. Consider the following two government securities. Security A pays $1 per year in years 1, 2, and 3. The current value of security A is $2.24. Security B pays $1 per year in years 1 and 3, and it pays nothing in year 2. The current market value of security B is $1.60. Part a. What is the continuously compounded rate of interest (annualized) as of today (or year 0) on a synthetic two year zero-coupon bond? Part b. Assume that Security C pays $1 in year 2. The current price of C is 0.74. Assume that all three securities can be held long or short. Formulate an arbitrage strategy which generates a profit of $2.00 today with no future net liabilities. SOLUTION: Since equations CF1 and CF3 are redundant, we can eliminate CF3 and solve the remaining 3 equations in terms of the 3 unknowns. The solution is: NA = 20 NB = -20 NC = -20 . This alternative formulation provides the same numerical solution as the prior method. 2. ($$$) The prices and cash flows (CF ) from three riskless bonds are as follows: By inspecting the problem, we can see that the purchase of one unit of A and the sale of one unit of B implies 0 Sec. A Sec. B AB 2.24 1.60 0.64 1 1 1 0 2 1 0 1 3 1 1 0 Price CF CF Bond Today Year 1 Year 2 A B C 90 75 155 100 0 100 0 100 100 Assume that short selling is not permitted in the market for these three bonds. Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 150 Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 149 Adventures in Debentures Adventures in Debentures This question is not intended to be hard. The point of the question is to illustrate that different individuals may not perceive the same term structure if arbitrage is possible. Part a. Is there a single set of discount prices which can rationalize the three prices quoted above? Keep in mind that the lack of a single set of discount prices implies arbitrage is possible if the short sale restriction is not present. Part b. Henry Shortman wishes to construct a portfolio which offers cash flows of 200 in year 1 and 100 in year 2. What portfolio should he choose, and what is its cost? Part c. John Longfellow wishes to construct a portfolio which offers cash flows of 100 in year 1 and 200 in year 2. What portfolio should he choose, and what is its cost? Part d. Starting with the portfolio position described in part (b), what is the marginal cost to Henry Shortman, and the associated annualized rate of interest (continuously compounded), of achieving an incremental cash flow of 100 in year 1? In year 2? Keep in mind that you can adjust some of the current bond holdings to achieve the desired incremental cash flow. Part e. Starting with the portfolio position described in part (c), what is the marginal cost to John Longfellow, and the associated annualized rate of interest (continuously compounded), of achieving an incremental cash flow of 100 in year 1? In year 2? Keep in mind that you can adjust some of the current bond holdings to achieve the desired incremental cash flow. Part f. Sketch the curves describing the annualized rates (continuously compounded) faced by Shortman and Longfellow. How do you account for the differences? SOLUTION: Part a. If there were interest rates which exactly equate prices and present values, then there must be a set of discount factors which solves: 90 = 100d1 75 = 100d2 155 = 100d1 + 100d2 . Quite clearly, there is no such pair of values for d1 and d2 . The right hand side of the third equation equals the sum of the right hand side of the first two. However, the sum of the left hand sides of the first two equations does not equal the left hand side of the third equation. Part b. The cheapest portfolio for Mr. Shortman is to hold 1 bond A and 1 bond C. The price of this is 245. (The alternative is to buy 2 units of A and 1 unit of B, but the price of this is 255.) Part c. Mr. Longfellow should hold 1 unit of B and 1 unit of C at cost of 230. Part d. To increase his cash flow in year 1 by 100, Mr. Shortman should buy another Bond A at 90. The rate associated with this is 10.53%. To increase his cash flow in year 2 by 100, Mr. Shortman should sell his Bond A (for 90) and buy another bond C (at 155). The net cost of this is 155 - 90 = 65, and the associated spot rate is given by: 65 = 100 e-2r2 , so r2 is 21.54%. Part e. To increase his cash flow in year 1, Mr. Longfellow should sell his bond B (for 75) and buy a bond C for 155; a net cost of 80. The spot rate associated with this is given by: 80 = 100 e-r1 , so r1 is 22.31%. Finally, to increase his cash flow in year 2 by 100, Mr. Longfellow should buy an additional bond B for 75. The spot rate associated with this is: 75 = 100 e-2r2 , Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 151 Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 152 Adventures in Debentures Adventures in Debentures so r2 is 14.38%. Bond Part f. The spot rate curves faced by Mr. Shortman and Mr. Longfellow are given by: a b c d Price 650 850 250 2,300 K1 100 100 0 600 K2 150 100 0 650 K3 200 100 0 700 K4 250 100 0 750 K5 300 1,100 500 800 . ... . .... . 25 .. . ....... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... .... ... .... ...... ..... ... .... ... ..... ... ..... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. .. ... ... .. ... ... ... .. ... ... ... .. ... ... ... . . ... ... ... ... ... ... ... ... . ................. ................ .. .. Part a. Even though there are 5 time periods involving future cash flows and only 4 securities, someone suggested that a synthetic version of Bond d could be created by buying 1 unit of Bond a, buying 5 units of Bond b, and shorting 10 units of Bond c. If it is possible to create such a synthetic, demonstrate whether or not this particular combination works. If it is not possible to create such a synthetic, explain why not. Part b. In the context of this problem, is it possible to demonstrate whether or not Bond d is mispriced relative to the other bond prices? If not, explain why not. If it is possible, demonstrate the a trading strategy to exploit the mispricing. SOLUTION: Rates (continuously 20 compounded) 15 10 0.5 1.0 1.5 Maturity (Years) 2.0 Part a. 1 Kat + 5 1 100 + 5 1 150 + 5 1 200 + 5 Kbt - 10 Kct = 1Kdt 100 - 10 100 - 10 100 - 10 100 - 10 0 = 600 0 = 650 0 = 700 0 = 750 ? for t = 1, 2, 3, 4, 5 which equals Kd1 which equals Kd2 which equals Kd3 which equals Kd4 which equals Kd5 The reason that these curves look so different is that Bond C is under priced relative to Bonds A and B. When Bond C in combination with an existing portfolio of bonds can be used to achieve the desired cash flows, the implied interest rate is high because Bond C has a low price. However, when either Bond A or B must be used to achieve the desired cash flow, the implied rate is low because the prices of these two bonds are high relative to Bond C. Since these two individuals have different portfolios based on parts b and c, they purchase Bond C in different ways to achieve the desired cash flows. 1 250 + 5 1 300 + 5 1,100 - 10 500 = 800 Thus, the suggested combination of a, b, and c is a synthetic version of d. 3. ($) The following table describes the cash flow characteristics of 4 bonds. All four bonds mature on year 5; today is year 0. (Note Kt is the cash flow the security pays to its owner in period t.) Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 153 Part b. The cost of the synthetic is (1 650) + (5 850) - (10 250) = 2,400, which is more costly than bond d. To create an arbitrage we should short the synthetic and go long bond d. For example, to generate an arbitrage profit of $100, we need to buy 1 unit of d, buy 10 units of c, short 1 unit of a, and short 5 units of b. 4. ($$) Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 154 Adventures in Debentures Adventures in Debentures Today is year 0. You operate a company which has outstanding a 2 year coupon-bearing bond. The bond pays $10 on year 1 and $110 on year 2. To keep the problem simple, we will assume you only issued one unit of this bond. Your firm has just been purchased by another firm. As part of the restructuring, it is desirable to set aside default-free assets in a separate account to guarantee repayments on this bond. To guarantee the repayments on this bond, the assets in this separate account should have net cash flows on years 1 and 2 equal to the payments on this bond. The following table describes the current asking prices and issue prices for some securities as well as the cash flows that each security will generate. The asking price is the price you would have to pay if you decided to purchase the security. The issue price is the amount of money that you would raise if you issued the security in the bond market. K1 is the cash flow paid on year 1. K2 is the cash flow paid on year 2. All of the numbers in the following table are on a per unit basis. to consider: Strategy 1 2 3 N1 10 0 -100 N2 110 100 0 NAnnuity 0 10 110 Cost (10 .9) + (110 .8) = 97 (100 .8) + (10 1.6) = 96 (-100 .85) + (110 1.6) = 91 All 3 strategies can be computed by just inspecting the problem. All 3 strategies delivered the desired cash flows in the future. Clearly, strategy #3 is the best, for it is the cheapest. 5. ($$) This question studies the economic significance of the payment of accrued interest. You will need to understand the notions of "flat" versus "full" prices. Today is September 27, 1995. Today you purchase a government bond with face value of $1,000.00. The annual coupon yield is 20%. The coupons are paid on a semi-annual basis. The bond matures on December 27, 1996. The remaining coupons will be paid on December 27, 1995; June 27, 1996; and December 27, 1996. The last date where a coupon was paid was on June 27, 1995. You purchased this bond for immediate cash settlement. On September 27, 1995, the discount function is as follows: For $1 to be received on October 27, 1995, the current price is 0.99. For $1 to be received on December 27, 1995, the current price is 0.97. For $1 to be received on June 27, 1996, the current price is 0.94. For $1 to be received on September 27, 1996, the current price is 0.92. For $1 to be received on December 27, 1996, the current price is 0.90. Part a. Assume that the seller does not receive the accrued interest. That is, you as the buyer receive the accrued interest, and the buyer of the bond does not have to rebate the accrued interest to the seller. How much is the bond worth to the buyer? Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 156 Bond a b c Issue Price $0.85 $0.75 $1.55 Asking Price $0.90 $0.80 $1.60 K1 1 0 1 K2 0 1 1 In this separate account which is dedicated to your liability consisting of the constant coupon bond, you may only have a nonzero position in any two of the above three securities. What is the best investment strategy (i.e., how many units of each bond)? Why? SOLUTION: Since you may only hold 2 of the 3 securities, there are only 3 investment strategies Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 155 Adventures in Debentures Adventures in Debentures Part b. Assume that the seller does receive the accrued interest. Further, the accrued interest is calculated in the usual way based on an actual/actual calendar with the seller receiving a pro rata share of the next coupon based on the time that has past since the last coupon payment What is the total payment the buyer is willing to make to own this bond? (The total payment should include the accrued interest the buyer must pay the seller.) Part c. Assume that the seller does receive accrued interest. However, the accrued interest is calculated to reflect the compounding of the coupon yield based on an actual/actual calendar. What is the total payment the buyer is willing to make to own this bond? (The total payment should include the accrued interest the buyer must pay the seller.) Part d. Now assume today is September 27, 1996. Also assume the discount function on September 27, 1996 is the same as the discount function described above for September 27, 1995. Assume that the seller does receive the accrued interest. Further, the accrued interest is calculated in the usual way based on an actual/actual calendar with the seller receiving a pro rata share of the next coupon based on the time that has past since the last coupon payment. What is the total payment the buyer is willing to make to own this bond? (The total payment should include the accrued interest the buyer must pay the seller.) SOLUTION: Price when seller does not receive accrued interest = (.97 100) + (.94 100) + (.90 1,100) = $1,181.00 Part b. Price when seller does receive accrued interest =- 92 92 + 91 100 + (.97 100) + (.94 100) + (.90 1,100) = -50.27 + 1,181.00 = $1,130.73 Total cash from buyer = 1,130.73 + 50.27 = 1,181.00 CONCLUSION: when comparing parts a and b, it should be clear that accrued interest has no real economic impact. The seller of the security gets the same total revenue from the buyer whether the buyer pays a single amount or whether we split the buyer's payments into two pieces -- the flat price and the accrued interest. Part c. Price when seller 92/ (91+92) .2 does receive com- - 1 1,000 =- 1+ pounded accrued 2 interest + (.97 100) + (.94 100) + (.90 1,100) = -49.082 + 1,181.00 Part a. = $1,131.92 Total cash from buyer = 1,131.92 + 49.082 = 1,181.00 6/27/95 9/27/95 12/27/95 $100 6/27/96 $100 12/27/96 $1,100 92 days 91 days CONCLUSION: The fact that accrued interest is a linear approximation (and ignores compounding) has no real economic significance. The seller receives the same total revenue from the buyer when accrued interest reflects compounding and when it does not. Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 158 Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 157 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Part d. -92 91 + 92 100 + (.97 1,100) = 1,016.73 92 91 + 92 100 = 1,067.00 The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Even if you decide not to submit your answer to this problem set, you should review my solution only after you attempt the questions to know if you could do them or not. Many times my solution is obvious AFTER you see it, but you need to know if the solution was obvious to you BEFORE you are told the solution. If you submit your answers to this problem set, please keep the following in mind: I sometimes fall behind where I expect to be in the lectures and an assignment is due before I get to some of the relevant material. If you believe the lectures have not yet covered the material necessary to answer a particular question and the required reading provides no guide, then you should indicate this on your solution and skip the question. Names of students along with their course and section numbers should be clear. The number of students in a study group should be less than or equal to five. Homework should be stapled. The final numerical answer should be "flagged" in some manner. Boxing, highlighting, and/or underlining the number are appropriate. Pages of spreadsheet printouts should be kept to a minimum. Only the essential information should be incorporated. You should not spend an excessive amount of time trying to solve any particular question. If you cannot complete a question, just describe what you tried to do. Questions Total cash from buyer = 1,016.73 + NOTE: the point of this part of the question is to emphasize that accrued interest is on the coupon only, not the face value. Chapter 4: Bond Valuation Using Synthetics, Practice Questions with Solutions Page 159 Chapter 4: Bond Valuation Using Synthetics, Questions Page 160 Adventures in Debentures Adventures in Debentures 1. ($) This question should be straightforward. We will rely on your answers for an example in a later session in the course. Please take the time to do these calculations. Using an electronic spreadsheet like Excel should allow you to complete this question quickly. Consider a constant coupon bond, which pays an annual coupon of $1009.09. These coupon are paid on time periods 1, 2, . . ., 20. The face value of this bond is $9363.03, and it is paid on time period 20. Part a. Find the full price of this bond at time periods 0, 0.5, and 1. (You should assume that if you buy the bond on time period 1 that the coupon paid on this date goes to the seller.) To value this bond on these 3 dates, use the following yield curve: Maturity 1/365 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 Yield 8.0000% 8.0825% 8.1632% 8.2422% 8.3194% 8.3950% 8.4688% 8.5408% 8.6111% 8.6797% 8.7465% 8.8116% 8.8750% 8.9366% 8.9965% 9.0547% 9.1111% 9.1658% 9.2188% 9.2700% 9.3194% Maturity Yield 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 9.3672% 9.4132% 9.4575% 9.5000% 9.5408% 9.5799% 9.6172% 9.6528% 9.6866% 9.7188% 9.7491% 9.7778% 9.8047% 9.8299% 9.8533% 9.8750% 9.8950% 9.9132% 9.9297% 9.9444% The yields in the prior table are continuously compounded and annualized. Part b. Now assume the yield curve described in part a shifts down by 100 basis points for all maturities. That is, in this part assume r1 = 7.1632%, r2 = 7.3194%, and so on. Find the full price of this bond at time periods 0, 0.5, and 1. (You should assume that if you buy the bond on time period 1 that the coupon paid on this date goes to the seller.) Chapter 4: Bond Valuation Using Synthetics, Questions Page 161 Chapter 4: Bond Valuation Using Synthetics, Questions Page 162 Adventures in Debentures Adventures in Debentures Part c. Now assume the yield curve described in part a shifts up by 100 basis points for all maturities. That is, in this part assume r1 = 9.1632%, r2 = 9.3194%, and so on. Find the full price of this bond at time periods 0, 0.5, and 1. (You should assume that if you buy the bond on time period 1 that the coupon paid on this date goes to the seller.) Part d. Summarize your answers from the last 3 parts of this question by creating a 3 dimensional bar chart. The z axis should be the full price of the bond, the x and y axes should be the overnight rate and the passage of time (in either order). Passage of time is the difference between the time period for the valuation and today's time period. Since today's time period is 0, the passage of time is equal to the time period. The easiest way to do this 3 dimensional bar chart is to fill in the following table in an Excel spreadsheet and then ask Excel to chart the results: Price CF CF Bond Today Year 1 Year 2 A B C 935 800 960 1,000 0 100 1,000 1,100 Do the prices of these bonds permit arbitrage, i.e., is it possible to obtain a free lunch by trading in these bonds? If so, how would you take advantage of the opportunity? Also, if a free lunch is available, where would you eat it and what would you order? Note: Throughout these exercises, you will find questions involving arbitrage like this one. While it is always profitable to find arbitrage and to know how to exploit it, the real point of these questions involving arbitrage is not to train you how to trade in such situations. The arbitrage-based questions are to demonstrate how fundamental our valuation approach is. Market prices must converge back to those prices which preclude arbitrage. Passage of Time (in Years) 0.0 Short-Term Rate: 7.00% 8.00% 9.00% 0.5 1.0 3. ($$$$) In this problem you are to continue your fantasy that you are an Arbitrage Seeking Bond Trader (ASBT). Your fantasy is now a bit more complicated. You now have four bonds to analyze. Each is a default free government bond. The first column of the following table gives the current price of each of the four. Again assume that you can take long and short positions in these bonds at these prices. The remaining columns of the table show the cash flows (CF ) produced by the bonds at the end of years 1, 2, and 3. All bonds mature at or before the end of year 3. Bond 2. ($$$) In this problem you are to fantasize that you are a bond trader looking for arbitrage opportunities. The first column of the following table gives the current prices of three government (default free) bonds. Assume that you can take long and short positions in these bonds at the given prices. (If you short a bond you receive the current price but you must pay out all future cash flows.) The remaining columns of the table show the cash flows (CF ) produced by the bonds at the end of years 1 and 2. All bonds mature at or before the end of year 2. Chapter 4: Bond Valuation Using Synthetics, Questions Page 163 Price Today 100.20 93.00 92.85 121.20 CF CF CF Year 1 Year 2 Year 3 10 100 5 20 10 105 20 110 A B C D 120 Do the prices of these bonds permit arbitrage? If so, how would you take advantage of the opportunity? This is a challenging problem. You may find it difficult. Don't Chapter 4: Bond Valuation Using Synthetics, Questions Page 164 Adventures in Debentures Adventures in Debentures spend too much time working on it. You certainly shouldn't miss lunch trying to solve it. The goal is to get a free lunch. HINT: Question 2 is a "warm-up" for Question 3. If you cannot do Question 2, you should not try to do Question 3. Years till Maturity 1 2 3 4 Price 0.9091 0.8000 0.7143 0.6250 4. ($$$) You manage a pension fund for a major corporation. All the assets of the pension fund are in marketable financial securities with a current total value of $110 (million). You hired an actuary to project the future benefits that the fund is obligated to pay. While no one is retiring in the next year, after that time there will be an increasing number of retirements reflecting the demographics of the work force. The projected liabilities are: How many of each zero should you purchase? What is the total cost of this dedicated portfolio? Part b. This part is the same as the previous part except you must set up the dedicated portfolio without the use of the zeros. In this part you must rely on the following four bonds. (Bond C is a graduated payment mortgage with no default risk, and Bond D is a fully amortizing constant payment bond.) Years till Liability Payment (millions) 1 2 3 4 0 20 45 80 Bond A B C D Price Today 98.0874 99.0805 104.4055 106.6940 CF CF CF CF Year 1 Year 2 Year 3 Year 4 11 12 10 35 11 12 12 35 111 12 15 35 0 112 120 35 You have decided to rearrange the assets of the fund so as to minimize the chances that the value of the assets are not sufficient to meet the liabilities. After considering the alternatives, you decide to develop a dedicated portfolio to perfectly immunize the fund. A dedicated portfolio is one where the future cash flows of the liabilities match the future cash flows of the assets. (The match is in terms of the dollar amount of the flows as well as the timing of the flows.) Part a. In this part, you are to setup a dedicated portfolio strategy involving zero coupon bonds. To the extent that the value of the assets exceeds the liabilities, you may place this surplus in the stock market. The current prices of zeros with face value of $1 follow: Chapter 4: Bond Valuation Using Synthetics, Questions Page 165 How many units of each bond should you purchase? What is the total cost of this dedicated portfolio? Part b is straightforward, but the calculations are quite tedious. You should do the calculations using an electronic spreadsheet like Microsoft Excel. 5. ($$$$) You are a bond trader, and there has been a power failure in New York City. You and other traders have taken to a street corner in order to keep trading. You can buy or sell the following bonds with the following cash flows (or CF ) at the following prices. All bonds mature at or before the end of year 3. Chapter 4: Bond Valuation Using Synthetics, Questions Page 166 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Bond A B C Price Today 100.00 95.00 112.00 CF CF CF Year 1 Year 2 Year 3 5 4 7 5 4 7 105 104 107 Chapter Quickly you make a back of the envelope calculation and take a position to lock in an arbitrage profit of $2.00 today and no future net cash flows. What did you do? 5 Interpreting Bond Yields Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Announcements and Assignments . . . . . . . . . . . . . . 175 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 176 A: The Problem with Redemption Yields . . . . . . . . . 204 B: Par Yields . . . . . . . . . . . . . . . . . . . . . . . . 214 Practice Questions with Solutions . . . . . . . . . . . . . . 219 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Chapter 4: Bond Valuation Using Synthetics, Questions Page 167 Chapter 5: Interpreting Bond Yields Page 168 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Preface to Interpreting Bond Yields 1.) Suggested Preparation Before Reviewing this Chapter. "Money is better than poverty, if only for financial reasons." -- Woody Allen, Without Feathers Some of the numerical examples in this chapter are taken from a prior lecture titled "Bond Valuation Using Synthetics." You should review the examples in that prior chapter before attending this session. 2.) Materials Needed for the Lecture. Some of the numerical examples in this lecture rely on the information from the chapter titled "Data for a Recurring Illustration." You may find it convenient to bring the lecture notes from this chapter to class as a convenient reference. 3.) Summary of Chapter. Parts of this chapter could be a review for people trained in finance, for we revisit a well known issue concerning when a yield to maturity will equal the realized holding period return. We will also address a topic that should be new to most people. We will provide a formal comparison of the maximum differences among yields on annuities, zero coupon bonds, and constant coupon bonds -- when all three types of instruments are correctly priced. This formal comparison demonstrates the extent of yield differences among equally desirable investments. Today's chapter provides some motivation for an upcoming chapter titled "Bond Values and the Passage of Time." Chapter 5: Interpreting Bond Yields Page 169 Chapter 5: Interpreting Bond Yields, Preface Page 170 Adventures in Debentures Adventures in Debentures You should review the lecture notes in advance of the session. If you already understand the material, please feel free not to attend the lecture. 6.) Supplemental Reading. After attending this session, you may find the following reading useful: Thomas Klaffky and Robert W. Kopprasch. 1990. "Analysis of Treasury Zero Coupon Bonds." Chapter 6 of The Handbook of U.S. Treasury and Government Agency Securities. Probus Publishing Company. Pages 115 143. 4.) Road Map for Chapter. The topics in this chapter will be organized as follows: 1.) Explain the calculation of yield to maturity. 2.) Develop yield curves based on annuities, zero coupon bonds, and constant coupon bonds. 3.) Interpret yield as a holding period return: comparing bonds with same coupon yields (perhaps zero) but with different maturities. 4.) Interpret yield as a holding period return: comparing bonds with different coupon yields but with same maturities. In the following article, several practical examples are provided to demonstrate some of the problems with yields. Thomas Klaffky. 1982. "The New World of Coupon Stripping." August. Bond Portfolio Analysis Group of Salomon Brothers. 7.) Other Assignments. I expect to have assigned seating in this class. You should be sitting in a seat that you want to use for the rest of the semester. 5.) Required Reading. After attending this sessions, you should read: Appendix A, "The Problem with Redemption Yields." Appendix B, "Par Yields." The following buzz words will be used in the lecture notes, the readings, and/or the problem sets: Annuity, annuity yield curve, constant coupon yield curve, coupon yield, current yield, holding period return, internal rate of return (or IRR), par yield curve, redemption yields, riding the yield curve, spot rate (or yield) curve, term structure of interest rates, and yield to maturity. Chapter 5: Interpreting Bond Yields, Preface Page 172 8.) New Vocabulary Used in this Chapter. You will need to review Appendix B titled "Par Yields" in order to complete some of the problem set questions. Also, some of the discussion in this chapter is closely related to the appendix titled "Alternative Interpretations of Forward Rates," which is part of the chapter on "Forward Contracts." Chapter 5: Interpreting Bond Yields, Preface Page 171 Adventures in Debentures Adventures in Debentures 9.) Clarifying the Yield Terminology. In fixed income securities, there are three different types of yields.1 A yield can refer to a coupon yield, yield to maturity, or current yield. A coupon yield equals the total coupons paid in one year divided by the face value of the bond. A current yield equals the total coupons paid in one year divided by the market value of the bond. In this course, we will rarely use the concept of current yield. A yield to maturity is the single rate which when used to discount all future cash flows to the present equates the current market price of the bond to the sum of its discounted future cash flows. (In the lecture notes, we will provide an explicit equation (as well as a numerical example) which defines yield to maturity.) bT is the (continuously compounded) yield to maturity for a standard constant coupon bond paying k dollars per period until date T with face value of $1. Now find bT such that the left and right hand sides of the following equation are equal: T BT = ( t=1 ke-tbT ) + e-T bT , where BT is the price of a constant coupon bond paying k dollars per period until date T with face value of $1. If kT is the (annually compounded) par yield for a T period bond, then kT = (1 - dT )/AT . To convert to a continuously compounded par yield, k T , k T = ln(1 + (1 - dT )/AT ) . 10.) Summary of Important Equations. The (continuously compounded) yield to maturity, y, for any fixed income security finds the value of y which equates the right and left hand sides of the following equation: Bond Price = K1 (e-y ) + K2 (e-2y ) + + KT (e-T y ) , where Kt is the cash flow in period t. aT is the (continuously compounded) yield to maturity for a standard annuity paying one dollar per period until date T . Now find aT such that the left and right hand sides of the following equation are equal: T 11.) Acknowledgments. Some of the exercises in the problem set questions were taken from a course that I co-taught with Charles Jacklin, Paul Pfleiderer, and William Sharpe at the Stanford Business School. AT = t=1 e-taT , where AT is the price of a standard annuity paying one dollar per period until date T . In other words, AT = T dt . t=1 1 Of course, there are even more differences among yields if we account for the various conventions that can be used to quote the yield -- for example, the compounding frequency, the day count convention, etc. See the chapter titled "The Grammar of Fixed Income Securities" for a discussion of these conventions. Chapter 5: Interpreting Bond Yields, Preface Page 173 Chapter 5: Interpreting Bond Yields, Preface Page 174 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Announcements and Assignments ecture notes L for Interpreting Bond Yields A Road Map 1.) Explain the calculation of yield to maturity. 2.) Develop yield curves based on annuities, zero coupon bonds, and constant coupon bonds. 3.) Interpret yield as a holding period return: comparing bonds with same coupon yields (perhaps zero) but with different maturities. 4.) Interpret yield as a holding period return: comparing bonds with different coupon yields but with same maturities. Chapter 5, Announcements and Assignments Page 175 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 176 Adventures in Debentures Adventures in Debentures Definition of Yield to Maturity The yield to maturity 1 is that constant discount rate which equates the "present value" of future cash flows with the current market price. That is, the (continuously compounded) yield to maturity solves2 the following equation: Bond Price = K1(e-y ) + K2(e-2y ) + + KT (e-T y ) Example Calculation of a Bond Yield Consider a five year bond with an annual coupon of 12 percent per year. The face value of the bond is $100. The current price for this bond is $111.36. To calculate the yield to maturity for this bond, we need to solve the following equation for y: 111.36 =12(e-y ) + 12(e-2y ) + 12(e-3y )+ 12(e-4y ) + 112(e-5y ) where Kt = coupon in period t and y = (continuously compounded) yield to maturity. You will discover that if you set y equal to 0.0869 (or 8.69%) in the above equation the equality holds.3 1 2 Sometimes yield to maturity is called the internal rate of return or the redemption yield. Or the (annually compounded) yield to maturity solves the following equation: Bond Price = where y = (annually compounded) yield to maturity. KT K2 K1 + + + (1 + y) (1 + y)2 (1 + y)T 3 Many calculators and spreadsheets will compute the annually compounded yield as a built-in function. In this case, you may find it easier to compute the annually compounded yield first and then transform the annually compounded yield into the continuously compounded yield. For this example, the annually compounded yield is 9.07%. Thus, the continuously compounded yield equals ln(1 + .0907), or 8.69%. Page 178 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 177 Chapter 5: Interpreting Bond Yields, Lecture Notes Adventures in Debentures Adventures in Debentures Possible Reasons for Calculating Yields 1.) Indicates the price of a security. 2.) Could suggest a mispricing of a security relative to other available securities if the mispricing is large enough. 3.) Predicts the return on a security under some assumptions. Tracking the Milestones 1.) Explain the calculation of yield to maturity. 2.) Develop yield curves based on annuities, zero coupon bonds, and constant coupon bonds. 3.) Interpret yield as a holding period return: comparing bonds with same coupon yields (perhaps zero) but with different maturities. 4.) Interpret yield as a holding period return: comparing bonds with different coupon yields but with same maturities. Chapter 5: Interpreting Bond Yields, Lecture Notes Page 179 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 180 Adventures in Debentures Adventures in Debentures Basic Types of Yield Curves 1.) Zero coupon yield curve. (Sometimes called the spot rate curve or the term structure of interest rates.) 2.) Annuity yield curve. 3.) Constant coupon yield curves. (Note there is a different yield curve for each coupon rate.) 4.) Par yield curve. 5.) Forward rate curves. Given any one of the above five, it is possible to construct the remaining four. Calculating Annuity Yields Assume the continuously compounded yields on zero coupon bonds are r1 = 8.1632%, r2 = 8.3194%, and r3 = 8.4688%. Then the price of a three year annuity paying one dollar each year for the next three years is: 1e-1.081632 + 1e-2.083194 + 1e-3.084688 =0.92161 + 0.84672 + 0.77564 =2.5440 . To calculate the yield to maturity on this three year annuity, we need to find the value of a3 that equates the left and right hand sides of the following equation: 2.5440 = 1e-1a3 + 1e-2a3 + 1e-3a3 . The solution is a3 equals 8.3607%.4 4 The general algebraic formulation for the T period annuity follows. Let AT equal the price of a standard annuity paying one dollar per period until date T . Recall dt = e-trt . Now find aT such that the left and right hand sides of the following equation are equal: T T AT = t=1 dt = t=1 e-taT . Chapter 5: Interpreting Bond Yields, Lecture Notes Page 181 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 182 Adventures in Debentures Constructing an Annuity Yield Curve Characteristics of the Annuities t Price rt dt Yr 1 Cash 1 1 1 1 . . . 1 1 1 . . . . . . 1 1 1 . . . 1 1 1 0 1 0 0 0 0 0 0 0 0 0 . . . 1 Yr 2 Cash Yr 3 Cash Yr 4 Cash Yr 40 Cash Yield Chapter 5: Interpreting Bond Yields, Lecture Notes 1 2 3 4 . . . 40 9.1111% 0.02614 9.8102 . . . . . . . . . 8.6111% 0.70861 3.2526 8.4688% 0.77564 2.5440 8.3194% 0.84672 1.7683 8.1632% 0.92161 0.9216 8.1632% 8.2644% 8.3607% 8.4522% . . . 9.4995% Page 183 NOTE: The spot rate and the annuity yield are continuously compounded. Dollar Value 1 0.8 0.6 0.4 0.2 Continuously Compounded Yield 0.1 0.09 0.085 0.095 0.08 Adventures in Debentures 0 0 Current Current 10 10 Chapter 5: Interpreting Bond Yields, Lecture Notes Discount Term Structure 20 Maturity 30 40 20 Maturity 30 Function 40 Page 184 Adventures in Debentures Adventures in Debentures Current Continuously Compounded Yield 0.1 Annuity Yield Curve Calculating Yields on Constant Coupon Bonds Assume the continuously compounded yields on zero coupon bonds are r1 = 8.1632%, r2 = 8.3194%, and r3 = 8.4688%. Then the price of a three year constant coupon bond paying a ten percent annual coupon each year for the next three years is: 0.10e-1.081632 + 0.10e-2.083194 + 1.10e-3.084688 =.10(0.92161 + 0.84672 + 0.77564) + 0.77564 =(0.10 2.5440) + 0.77564 =1.0300 . To calculate the yield to maturity on this three year constant coupon bond, we need to find the value of b3 that equates the left and right hand sides of the following equation: 1.0300 = 0.10e-1b3 + 0.10e-2b3 + 1.10e-3b3 . The solution is b3 equals 8.4499%.5 annuity yield curve 5 0.095 0.09 0.085 0.08 0 10 20 Maturity 30 40 Upper and Lower Bounds Continuously Compounded Yield 0.1 zero yield curve 0.095 on Yields 0.09 0.085 The general algebraic formulation for the T period constant coupon bond paying k dollars per period follows. Let BT equal the price of a constant coupon bond paying k dollars per period until date T with face value of $1. Recall AT = T dt . Now find bT such that the left t=1 and right hand sides of the following equation are equal: T 0.08 0 10 20 Maturity 30 40 BT = (kAT ) + dT = ( t=1 ke-tbT ) + e-T bT . Chapter 5: Interpreting Bond Yields, Lecture Notes Page 185 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 186 Adventures in Debentures Constructing a Constant Coupon Yield Curve Characteristics of Bonds with Coupon Yield of 10% t Price rt dt Yr 1 Cash 1.10 0.10 0.10 0.10 . . . 0.10 0.10 0.10 . . . . . . 0.10 0.10 1.10 . . . 0.10 0.10 1.10 0.00 1.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 . . . 1.10 Yr 2 Cash Yr 3 Cash Yr 4 Cash Yr 40 Cash Yield Chapter 5: Interpreting Bond Yields, Lecture Notes 1 2 3 4 . . . 40 9.1111% 0.02614 1.0072 . . . . . . . . . 8.6111% 0.70861 1.0339 8.4688% 0.77564 1.0300 8.3194% 0.84672 1.0236 8.1632% 0.92161 1.0138 8.1632% 8.3120% 8.4499% 8.5769% . . . 9.4649% Page 187 NOTE: The spot rate and the yield on the coupon bond are continuously compounded. Continuously Compounded Yield 0.1 0.09 0.085 0.095 0.08 0 10 annuity coup yield 20 Maturity 30 40 = 10 % yield curve Continuously Compounded Yield 0.1 0.09 0.085 0.095 0.08 0 10 Adventures in Debentures zero yield curve Chapter 5: Interpreting Bond Yields, Lecture Notes Page 188 20 Maturity Yield Curve for 10 % Constant Upper and Lower Bounds on Yields 30 40 Coup Bond Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Explain the calculation of yield to maturity. 2.) Develop yield curves based on annuities, zero coupon bonds, and constant coupon bonds. 3.) Interpret yield as a holding period return: comparing bonds with same coupon yields (perhaps zero) but with different maturities. 4.) Interpret yield as a holding period return: comparing bonds with different coupon yields but with same maturities. Yields Versus Holding Period Returns Just comparing yields among bonds with different maturities is not complete. At best, yields represent the holding period return on capital if the bond is held till maturity. When comparing the yields on short- versus long-term bonds, the investment strategy for the short-term bond is not complete -- that is, how will the capital returned on a the short-term bond be invested until the long-term bond matures? One way to compare the returns across investments is to equalize the holding periods. To do this, we need the relation between yields to maturity and holding period returns. The holding period return is just the growth rate on the initial capital invested until the end of some holding period. This growth rate should reflect any coupons received as well as the market value of the bond at the end of the holding period. Assuming no coupons are paid within the holding period, the holding period return on a bond is equal to its yield if the bond's yield is the same at the beginning and the end of the holding period.6 6 If a coupon is received within the holding period, then the holding period return is equal to the yield if the bond's yield is the same at the beginning and the end of the holding period and any coupons received can be reinvested at that yield until the end of the holding period. Page 190 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 189 Chapter 5: Interpreting Bond Yields, Lecture Notes Adventures in Debentures Adventures in Debentures Calculating a Holding Period Return When the Yield is Stable Consider again the five year bond with an annual coupon of 12 percent per year. The face value of the bond is $100. The current price for this bond is $111.36. Recall that the continuously compounded yield on this bond is 8.69%. Assume you buy this bond and sell it one year later. Assume one year from now this bond still yields 8.69% (with continuous compounding); under this assumption, the value of this bond in one year is: 12(e-.0869) + 12(e-2.0869) + 12(e-3.0869) + 112(e-4.0869) = 109.46 . Under these circumstances, the holding period return (continuously compounded) is:7 ln((109.46 + 12)/111.36) = 8.69% . As expected, the holding period return is the same as the yield if the yield for the bond is the same at the beginning and at the end of the holding period. 7 With annually compounded yields the results are identical. That is, the annually compounded yield is 9.07%, so the value of the bond at the end of the holding period is: 12 12 112 12 + + + = 109.46 . (1 + .0907)1 (1 + .0907)2 (1 + .0907)3 (1 + .0907)4 Under these circumstances, the holding period return (annually compounded) is: 109.46 + 12.00 - 111.36 = 9.07% 111.36 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 191 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 192 Adventures in Debentures Adventures in Debentures Stable Yields on Bonds Versus Stable Term Structure Consider investing in zero coupon bonds. If the holding period return equal the yield, then the yield on a T year zero coupon bond at the beginning of the holding period must be the same as the yield on the T - h year zero coupon bond at the end of the holding period. (h is the length of the holding period.) How must the term structure of interest rates change to accomodate this equality between holding period returns and yields? If the term structure is upward sloping at the beginning of the holding period, then the term structure needs to upward sloping but at a higher level at the end of the holding period. If the term structure is downward sloping at the beginning of the holding period, then the term structure needs to downward sloping but at a lower level at the end of the holding period. If the term structure is flat at the beginning of the holding period, then the term structure needs to be flat and at the same level at the end of the holding period. Consider the following illustration . . . Chapter 5: Interpreting Bond Yields, Lecture Notes Page 193 A Hypothetical Term Structure Where Holding Period Returns Equal Yields Current Maturity (years) 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 Term Struc 8.1632% 8.3194% 8.4688% 8.6111% 8.7465% 8.8750% 8.9965% 9.1111% 9.2188% 9.3194% 9.4132% 9.5000% 9.5799% Future Term Struc 8.3194% 8.4688% 8.6111% 8.7465% 8.8750% 8.9965% 9.1111% 9.2188% 9.3194% 9.4132% 9.5000% 9.5799% 9.6528% Implied Change + + + + + + + + + + + + + Holding Period Return 8.1632% 8.3194% 8.4688% 8.6111% 8.7465% 8.8750% 8.9965% 9.1111% 9.2188% 9.3194% 9.4132% 9.5000% 9.5799% Chapter 5: Interpreting Bond Yields, Lecture Notes Page 194 Adventures in Debentures Adventures in Debentures Current Maturity (years) 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 Term Struc 9.6528% 9.7188% 9.7778% 9.8299% 9.8750% 9.9132% 9.9444% 9.9688% 9.9861% 9.9965% 10.0000% 9.9965% 9.9861% 9.9688% 9.9444% 9.9132% Future Term Struc 9.7188% 9.7778% 9.8299% 9.8750% 9.9132% 9.9444% 9.9688% 9.9861% 9.9965% 10.0000% 9.9965% 9.9861% 9.9688% 9.9444% 9.9132% 9.8750% Implied Change + + + + + + + + + + - Holding Period Return 9.6528% 9.7188% 9.7778% 9.8299% 9.8750% 9.9132% 9.9444% 9.9688% 9.9861% 9.9965% 10.0000% 9.9965% 9.9861% 9.9688% 9.9444% 9.9132% In the chapter titled "Bond Values and the Passage of Time," we will discuss this idea in greater detail. If the term structure is stable and upward sloping, then the holding period return is greater than the yield to maturity. If the term structure is stable and downward sloping, then the holding period return is less than the yield to maturity. If the term structure is stable and flat, then the holding period return is equal to the yield to maturity. Holding Period Returns When Term Structure Is Stable Consider investing in zero coupon bonds and holding this investment for one year. If the term structure of interest rates is the same at the beginning and the end of the holding period, what is the relation between the holding period return and the yield to maturity? Chapter 5: Interpreting Bond Yields, Lecture Notes Page 195 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 196 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Explain the calculation of yield to maturity. 2.) Develop yield curves based on annuities, zero coupon bonds, and constant coupon bonds. 3.) Interpret yield as a holding period return: comparing bonds with same coupon yields (perhaps zero) but with different maturities. 4.) Interpret yield as a holding period return: comparing bonds with different coupon yields but with same maturities. Comparing Yields on Bonds with Same Maturity Now we want to compare yields to maturity on coupon bonds where the coupons differ but the maturities are the same. Since the maturities of the coupon bonds match, we can analyze these bonds with comparable holding periods -- for example, these bonds can be held till maturity. However, the comparison is still problematic because the complete investment strategy is not clear. How must we reinvest the intermediate coupons in order to have a holding period return equal the yield to maturity? The yield to maturity on a coupon bearing bond is a holding period return if (1) the bond is held till maturity; (2) all coupons received prior to maturity are reinvested until the maturity; and (3) this reinvestment rate is the yield to maturity. An illustration of the above claim follows8 . . . 8 Consider a T period bond, which has coupons equal to K1 , K2 , . . ., and KT . The current price of the bond is B, and its yield to maturity (continuously compounded) is y. By definition of a yield, it must be: B = K1 e-1y + K2 e-2y + + KT e-T y . Chapter 5: Interpreting Bond Yields, Lecture Notes Page 197 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 198 Adventures in Debentures Adventures in Debentures Comparing Yields on Bonds with Same Maturity: A Numerical Illustration Consider three 5 year bonds; each bond has a face value of $100. All bonds mature on the same date. All bonds pay annual coupons at the same point in time. The coupons, current prices, and yields (continuously compounded) for the three bonds are: BOND COUPON PRICE YIELD(%) A B C 12 7 2 111.36 91.86 72.37 8.6863% 8.7065% 8.7333% Calculating a Holding Period Return 1 Here we examine Bond A and assume that all intermediate coupons are reinvested at its yield (8.6863%) until the maturity of the bond. The holding period return for such a scenario is: Yr 1 2 3 4 5 Reinvestment Rate 8.6863% 8.6863% 8.6863% 8.6863% na Cash Flow 12 12 12 12 112 Future Value 12 (e.0868634) = 16.99 12 (e.0868633) = 15.57 12 (e.0868632) = 14.28 12 (e.0868631) = 13.09 112 1= 112.00 After simple rearrangement: BeT y = K1 e(T -1)y + K2 e(T -2)y + + KT . Total Future Value: 171.92 The holding period return (continuously compounded) is: ln(Total FV/Initial Price) ln(171.92/111.36) = = 8.6863%. 5 5 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 200 Clearly, the right hand side of the above expression establishes that when the intermediate coupons are reinvested at rate y, then your accumulated wealth from an investment of B with a return of y is given on the left hand side. Chapter 5: Interpreting Bond Yields, Lecture Notes Page 199 Adventures in Debentures Adventures in Debentures Calculating a Holding Period Return 2 Here we examine Bond C and assume that all intermediate coupons are reinvested at its yield (8.7333%) until the maturity of the bond. The holding period return for such a scenario is: Comparing Yields on Bonds with Same Maturity: A Summary of Problems When comparing yields on bonds with the same maturity but different coupon yields, the linkage between the yield and the holding period return is not clear. There are 2 problems: 1.) The yield is the holding period return if the intermediate coupons can be reinvested at that yield. But, of course, there are other scenarios about the future term structure that may be (more) realistic. 2.) The yield is the holding period return if the intermediate coupons can be reinvested at that yield. However, why would bonds with different yields earn different reinvestment rates? This presents a logical flaw when comparing bonds solely on the basis of their yields. Yr 1 2 3 4 5 Reinvestment Rate 8.7333% 8.7333% 8.7333% 8.7333% na Cash Flow 2 2 2 2 102 Future Value 2 (e.0873334) = 2 (e.0873333) = 2 (e.0873332) = 2 (e.0873331) = 102 1= 2.84 2.60 2.38 2.18 102.00 Total Future Value: 112.00 The holding period return (continuously compounded) is: ln(Total FV/Initial Price) ln(112.00/72.37) = = 8.7333%. 5 5 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 201 Chapter 5: Interpreting Bond Yields, Lecture Notes Page 202 Adventures in Debentures Adventures in Debentures Worksheet Appendix A: The Problem with Redemption Yields The attached article is titled "The Problem with Redemption Yields," and it was written by Stephen Schaefer. It appeared in the July-August, 1977 issue of the Financial Analysts Journal (pages 59 67). This article is one of my favorites in the course. It clearly demonstrates that yields on various securities will differ even when all securities are correctly priced. Further, it shows the direction of the differences among yields as well as the magnitude. Chapter 5: Interpreting Bond Yields, Lecture Notes Page 203 Chapter 5: Interpreting Bond Yields, Appendix A Page 204 Adventures in Debentures Adventures in Debentures Chapter 5: Interpreting Bond Yields, Appendix A Page 205 Chapter 5: Interpreting Bond Yields, Appendix A Page 206 Adventures in Debentures Adventures in Debentures Chapter 5: Interpreting Bond Yields, Appendix A Page 207 Chapter 5: Interpreting Bond Yields, Appendix A Page 208 Adventures in Debentures Adventures in Debentures Chapter 5: Interpreting Bond Yields, Appendix A Page 209 Chapter 5: Interpreting Bond Yields, Appendix A Page 210 Adventures in Debentures Adventures in Debentures Chapter 5: Interpreting Bond Yields, Appendix A Page 211 Chapter 5: Interpreting Bond Yields, Appendix A Page 212 Adventures in Debentures Adventures in Debentures Appendix B: Par Yields A par yield is a coupon yield that is necessary for a constant coupon bond to sell for its par (or face) value; equivalently, it is the yield to maturity on a constant coupon bond that sells for par. When you are asked to compute a par yield, you should think about the problem as computing a coupon yield, not a yield to maturity -- otherwise, the calculation is very difficult. 1.) Computing a Par Yield as a Coupon Yield. The general algebraic formulation for the T period par bond paying kT dollars per period follows 1 = kT d1 + kT d2 + + (1 + kT )dT , where dt is the price of a discount bond paying $1 on date t. Since the price of a T year annuity, AT , can be written as d1 + d2 + + dT , the prior expression can be rewritten as: 1 = (kT AT ) + dT , or kT = (1 - dT )/AT . To convert to continuously compounding: k T = ln(1 + (1 - dT )/AT ) . Consider a numerical example. Assume the continuously compounded yields on zero coupon bonds are r1 = 8.1632%, r2 = 8.3194%, and r3 = 8.4688%. First, we want to find the annual coupon (k3 ) that is appropriate for a three year constant coupon Chapter 5: Interpreting Bond Yields, Appendix A Page 213 Chapter 5: Interpreting Bond Yields, Appendix B Page 214 Adventures in Debentures Adventures in Debentures bond to sell for par ($ 1) : 1 =k3 e-1.081632 + k3 e-2.083194 + (1 + k3 )e-3.084688 =k3 (0.92161 + 0.84672 + 0.77564) + 0.77564 =(k3 2.5440) + 0.77564 , or, 1 - .77564 k3 = 2.5440 = 8.8192% . Par Yield (cont comp) 8.1632% 8.3129% 8.4518% 8.5798% Par Yield (ann comp) 8.5056% 8.6682% 8.8192% 8.9586% Annuity Price 0.9216 1.7683 2.5440 3.2526 The prior numerical example can be extended for any maturity based the yields for zero coupon bonds. The second column in the next table provides yields on zero coupon bonds. (The discount function in the third column, the annuity price in the fourth column, and the annuity yield in the fifth column are explained in the lecture notes.) Based on these numbers and the methodology in the previous section, the par yield for a range of maturities are reported in the last two columns of the next table using either annually compounded or continuously compounded rates. The next figure graphs the continuously compounded par yield in the last column of the table on the vertical axis and the maturity of the par bond (which is given in the first column of the table) on the horizontal axis. As the second figure illustrates, the par yield is bounded between the zero coupon yield curve and the annuity yield curve. Since a par bond is a constant coupon bond, this is not surprising.1 1 0.92161 0.84672 0.77564 0.70861 8.1632% 8.3194% 8.4688% 8.6111% While a par bond is a constant coupon bond, a par yield curve is not a constant coupon yield curve. A constant coupon yield curve graphs the yields for a particular coupon rate against maturity, but a par yield curve represents the yield on a par bond across maturities but the coupon across different maturities need not the be same. Chapter 5: Interpreting Bond Yields, Appendix B Page 215 Chapter 5: Interpreting Bond Yields, Appendix B 40 1 2 3 4 t . . . 9.1111% rt . . . 0.02614 dt . . . 9.8102 2.) Constructing a Par Yield Curve. . . . . . . k 3 = ln(1 + .088192) = 8.4518% . NOTE: The spot rate and the annuity yield are continuously compounded. Page 216 Constructing a Par Yield Curve 8.1632% 8.2644% 8.3607% 8.4522% Annuity Yield 9.4995% Second, for any bond selling at par, its coupon yield (annually compounded) must equal its annually compounded yield to maturity. (See below for a discussion of this.) Thus, the annually compounded yield to maturity on the above bond is also 8.8192%. Third, to compute the continuously compounded yield to maturity, we merely convert from the annually compounded yield in the usual way: 9.9270% . . . 9.4646% . . . Adventures in Debentures Adventures in Debentures Current Continuously Compounded Yield 0.1 Par Yield Curve 3.) Computing a Par Yield as a Yield to Maturity. It is well known that a bond that sells for par will have a yield to maturity equal to its coupon yield. This section demonstrates this result. The proof is straightforward, although somewhat tedious. I have included the following proof for completeness, but it will not be part of any examination. Consider a constant coupon bond with a face value of $1, a coupon yield of k, and a maturity of T years. If this bond sells for par (or its face value of $1), the following equation is true by definition: 1= 1+k k k + + , + 1 + y (1 + y)2 (1 + y)T 0.095 0.09 0.085 0.08 0 10 20 Maturity 30 40 where y is the (annualized) yield to maturity based on annual compounding. Given this definition, one can show that the coupon yield, k, must equal the yield to maturity, y: 1 1 1 1 + + + + 1=k 1 + y (1 + y)2 (1 + y)T (1 + y)T k = 1- 1 (1 + y)T + + 1 1 1 + + + 1 + y (1 + y)2 (1 + y)T 1 (1+y)T -1 Upper and Lower Bounds Continuously Compounded Yield 0.1 zero yield curve par yield curve 0.095 annuity on Yields Let ST = 1 1+y + 1 (1+y)2 , then: 1 1 + + 1+y (1 + y)T -1 1 1 1 + + ST = + 1+y (1 + y)T -1 (1 + y)T 1 (1 + y)ST - ST = 1 - = yST (1 + y)T (1 + y)ST = 1 + or ST = (1 + y)T - 1 . y(1 + y)T 0.09 yield curve 0.085 Thus, we can use yST and ST to rewrite the above equation for k: k = [yST ] 0 10 20 Maturity 30 40 0.08 1 ST =y . Chapter 5: Interpreting Bond Yields, Appendix B Page 217 Chapter 5: Interpreting Bond Yields, Appendix B Page 218 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Practice Questions with Solutions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." VA = VB = 24.00 = 21.72 . (e0.1 ) 70.80 = 64.06 . (e0.1 ) 1. ($) Assume that you have $100 to invest today, and the current rate for borrowing and lending is 10% (annualized with continuously compounded). You call your broker, and he tells you that two newly issued bonds are being sold today. Bond A costs $20 and will return $24 in one year. Bond B costs $60 and will return $70.80 in one year. Your broker says that there is high demand for each of these new issues. The broker must ration these securities among his many customers. He will allow you to buy only one unit of Bond A or one unit of Bond B, but not both. Part a. Calculate the continuously compounded yield (annualized) for each bond. Bond B is more undervalued than Bond A on a per bond basis. If you could purchase 3 of Bond A or 1 of Bond B, you would three units of A. However, your broker will ration only one of A or one of B. While the rate of return on A dominates that of B, you can put more dollars into B and earn a smaller return on a larger base. Even though the percentage return on B is smaller, its dollar return is larger. 2. ($$) Demonstrate by way of a simple numerical example that the realized holding period return (continuously compounded) on an investment is equal to the continuously compounded yield only under certain assumptions. (These assumptions include that the reinvestment rate on the coupons is equal to the yield and that the bond is held till maturity.) If the bond is sold prior to maturity, what must be assumed about the sale price in order for the realized, continuously compounded return to equal the continuously compounded yield? SOLUTION: Part b. Which bond do you pick to purchase and why? SOLUTION: Your solution could involve an example of your choosing. My example follows. Consider a 10 year bond with an annual coupon yield of 20% and face value of $100. Assume the current term structure is given by 24.00 yA = 18.23% . (eyA ) rt = .06 + .01t for t = 1, 2, . . . , 10. The current market price for such a bond is: 10 The yield for each bond is: 20 = 70.80 60 = y yB = 16.55% . (e B ) B = 129.63 = Now calculate the present value of purchasing one or the other bond with your $100. Chapter 5: Interpreting Bond Yields, Practice Questions with Solutions Page 219 t=1 20 (ert )t + 100 . (er10 )10 Chapter 5: Interpreting Bond Yields, Practice Questions with Solutions Page 220 Adventures in Debentures Adventures in Debentures The yield to maturity is the solution to the following: 10 129.63 = t=1 20 (ey )t + 100 (ey )10 y = 13.33% . If the bond is held till maturity and the coupons are reinvested at the yield, the future value from this investment is: 20(e0.1333 )9 + 20(e0.1333 )8 + + 20(e0.1333 ) + 120 = 491.40 . 3. ($$) The reinvestment of coupon income can represent a substantial portion of the total income from holding a bond. Consider a 30 year bond with an annually compounded coupon yield (annualized) of nine percent. When the annualized reinvestment rate (continuously compounded) for all coupons is 6%, calculate the percentage of total income that is due to reinvestment. When the annualized reinvestment rate (continuously compounded) for all coupons is 12%, calculate the percentage of total income that is due to reinvestment. Note: total income is coupon income plus reinvestment income (i.e., the interest on interest). SOLUTION: The realized, continuously compounded return on this bond is ln or on annualized basis: 1 10 491.40 = 133.3% 129.63 491.40 = 13.33% . 129.63 There are several ways to solve this problem. One approach which is perfectly acceptable is to work through a numerical calculation which may be done with the aid of an electronic spreadsheet. What follows is more of an algebraic solution to this problem. However, no matter how you solve this problem, the moral of the story is the same. Reinvestment income can be a very important part of the total income of a bond-- especially when the maturity is long or reinvestment rates are high. Thus, yields to maturity could be quite misleading about the realized return. Furthermore, reinvestment risk can be substantial from investing in coupon bonds. term structure, one can calculate the future price of the bond under this scenario: 9 ln If the bond's yield stays the same after one year has elapsed, then the price of the bond in one year will be: 9 100 20 + B = (ey )t (ey )9 t=1 when y = 0.1333, then B (the prime indicates a future price) will be: 9 B = 20 (e0 r 1,t+1 t ) + 100 (e0 r 1,10 9 ) . 20 (e.1333 )t + 100 (e.1333 )9 t=1 = 128.11 = B . Based on the above equation, one can impute B = 119.03. Under this scenario, the one year holding period return would be: B +K -B 119.03 + 20 - 129.63 = = 7.25% . B 129.63 7.25% is equivalent to 7% with continuous compounding; that is, ln B +K 119.03 + 20 = ln = 7.00% . B 129.63 t=1 If the bond is sold after one year, the one year holding period return is: 128.11 + 20 - 129.63 B +K -B = = 14.26% . B 129.63 14.26% is equal to 13.33% with continuous compounding; that is,1 continuously compounded return = ln 1 B +K 128.11 + 20 = ln = 13.33% . B 129.63 This is a useful place to make a digression on forward rates that we will discuss in the upcoming lectures . . . If the term structure next year is consistent with the implied forward rates based on the current Just as we will show in an upcoming chapter, the one year holding period return is 0 r0,1 , not the original yield of 13.33%. This scenario verifies our interpretation of the forward rate as a break-even rate. Chapter 5: Interpreting Bond Yields, Practice Questions with Solutions Page 221 Chapter 5: Interpreting Bond Yields, Practice Questions with Solutions Page 222 Adventures in Debentures Adventures in Debentures Let k be the coupon yield (annually compounded) and F be the dollar face (or par) value of the bond; thus, K = kF is the coupon in dollar terms. For simplicity, we will assume throughout this solution that the term structure is flat and the interest rate equals r for all maturities. Total coupons from a T period bond with annual coupon payments is Total income (with reinvestment of coupons) of a bond is given by: kF (er )T -1 + kF (er )T -2 + + kF (er )1 + kF . T t=1 K = T kF . three payments over the next three years. You put the actual security in your safety deposit box, and you are looking at a photocopy of the certificate. The problem is that your K-tel copying machine left a smudge on your copy, and you cannot make out the final scheduled payment. However, you know that the current market value of the security is $3,258.27, the security pays $700 in one year, and $1,100 in two years, and that the security has a continuously compounded yield (annualized) equal to 8%. Your bank is closed, so you do not have access to your safety deposit box. What is the amount of the payment that you are supposed to receive at the end of three years? SOLUTION: The percent of total income due to coupon income is: kF T T t=1 kF (er )t-1 = T T r t-1 t=1 (e ) . Solve the following equation for K: 3,258.27 = K 1,100 700 + + (e0.08 ) (e0.08 )2 (e0.08 )3 K 2,129.00 . = The sum in the denominator can be simplified,2 so the last expression becomes: T (er - 1) . r T (e ) - 1 The above fraction represents the portion of total income due to coupon income. Since total income is due to both coupon income and reinvestment income, one minus the above expression gives the percentage of total income due to reinvestment income. That is, T (er - 1) . 1- r T (e ) - 1 For this problem, T = 30 and r = .06, then substituting into the last equation implies that the percentage of total income due to reinvestment is 63.26%. If T = 30 and r = .12, then the percentage is 89.26%. As you can see from the above equation, the percentage of total income attributed to reinvestment income is not affected by the coupon yield or the face value of the bond. 5. ($$) In the table below a discount function for five years is presented. d1 d2 d3 d4 d5 .90 .78 .63 .51 .40 (d1 is the price of a dollar to be delivered one year from today, d2 is the price for a dollar to be delivered two years from today, etc.) Assume that GMP INC. can invest in a bond which will produce guaranteed payments of 30,000 dollars at the end of each of the next five years. You are not told the amount of money that must be invested today to buy this bond, but you are told that the bond's continuously compounded yield is 13.98%. Given the above discount function, is it possible to say whether this is a worthwhile bond to buy? If so, is it worthwhile? Now assume there is another bond available to GMP. This produces 45,000 dollars in one year and nothing thereafter. Again assume that you are not told the investment required today to purchase this bond but you are told that the bond earns an annualized yield (continuously compounded) of 12.66%. Is it possible to say whether this is a worthwhile investment to undertake? If so, is it worthwhile? Again use the discount function given above in answering this question. Chapter 5: Interpreting Bond Yields, Practice Questions with Solutions Page 224 4. ($$) You recently bought a security issued by the State of Nevada that makes exactly 2 This simplification can be seen by defining: S = (er )T -1 + (er )T -2 + + (er ) + 1 . Then (e )S - S = (e ) - 1, and solving this equation for S gives the desired result to be substituted into the denominator. r r T Chapter 5: Interpreting Bond Yields, Practice Questions with Solutions Page 223 Adventures in Debentures Adventures in Debentures SOLUTION: Although we are not given the investment required to buy this bond, we can easily calculate what it must be since we know the yield. Let V be the investment. Then V satisfies V = 30,000 30,000 30,000 30,000 30,000 + + + . + (e0.1398 ) (e0.1398 )2 (e0.1398 )3 (e0.1398 )4 (e0.1398 )5 i.e., 12.66% versus 10.54%. Only in a very simple situation like this one can such a comparison be easily and meaningfully made. In all cases the appropriate and always correct criterion is based on discounting the future cash flows based on the properly constructed discount function. In other words V is the value of 30,000 dollars received each year for five years when all cash flows are discounted at 13.98%. The initial investment must therefore be 100,554.23 dollars.3 To determine if this is a worthwhile bond, we need to determine the market value of 30,000 dollars received with certainty in each of the next five years. We do this by using the discount function. The present value of the future stream of cash flows produced by the bond is (.9)30,000 + (.78)30,000 + (.63)30,000 + (.51)30,000 + (.40)30,000 = 27,000 + 23,400 + 18,900 + 15,300 + 12,000 = 96,600 . If GMP purchases this bond, it is paying 100,554.23 dollars for something which is only worth 96,600 dollars. This is truly a bad deal. The net value is negative (-3,954.23 = 96,600 - 100,554.23). For those who still might believe that the yield is a useful number, we pose the following question: against what rate will you compare the 13.98% yield? The annualized, continuously compounded rate on a one year zero coupon bond is 10.54%. The yield is greater than this. The annualized, continously compounded rate on a three year zero coupon bond is 15.40%. The yield is less than this. The second bond produces only one cash flow, 45,000 dollars after one year and has a 12.66% yield. This means that the initial investment must be 39,648.87. The current value of 45,000 is (using the fact that d1 = .9) equal to 40,500. The value is thus greater than the investment required so the net value is positive. This is a good deal and the bond should be purchased. (Note that in this case where there is only one cash flow it is possible to compare the yield to a rate on zero coupon bond -- 3 We can use the yield to calculate the initial investment only because someone knows the initial investment. This person used the initial investment and the future cash flows to calculate the yield and then reported this number to us rather than the amount that needs to be invested. The yield is a derived number. In general if there are M cash flows in a cash flow stream (including the cash received or paid out today) and we are given the yield and M - 1 of the cash flows, we can always calculate the amount of the remaining cash flow. Chapter 5: Interpreting Bond Yields, Practice Questions with Solutions Page 225 Chapter 5: Interpreting Bond Yields, Practice Questions with Solutions Page 226 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Questions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Even if you decide not to submit your answer to this problem set, you should review my solution only after you attempt the questions to know if you could do them or not. Many times my solution is obvious AFTER you see it, but you need to know if the solution was obvious to you BEFORE you are told the solution. If you submit your answers to this problem set, please keep the following in mind: I sometimes fall behind where I expect to be in the lectures and an assignment is due before I get to some of the relevant material. If you believe the lectures have not yet covered the material necessary to answer a particular question and the required reading provides no guide, then you should indicate this on your solution and skip the question. Names of students along with their course and section numbers should be clear. The number of students in a study group should be less than or equal to five. Homework should be stapled. The final numerical answer should be "flagged" in some manner. Boxing, highlighting, and/or underlining the number are appropriate. Pages of spreadsheet printouts should be kept to a minimum. Only the essential information should be incorporated. You should not spend an excessive amount of time trying to solve any particular question. If you cannot complete a question, just describe what you tried to do. 1. ($) The term structure based on government strips is flat at 10% (annualized with continuously compounded) for all maturities. Assume throughout this problem that you can borrow or lend at these rates. Three non-government (but riskless) bonds are available for purchase; all three bonds sell for $100. Bond A is a two year zero. Bond A pays $550 in year 2. Bond B and C are one year zeros. Bond B pays $225 in year 1, and Bond C pays $450 in year 1. (Note Bonds A, B, and C were not used in determining that the term structure based on government strips is flat.) Part a. Calculate the annualized yield (in continuously compounded terms) for each bond. Show that the yield is not a reliable guide for investment decisions. That is, show that the bond with the highest yield is not the most undervalued bond. Part b. A possible investment involves the purchase of both Bonds A and C. Another possible investment involves the purchase of both Bonds B and C. For each of these alternatives, calculate the annualized yield (continuously compounded) on the portfolio which holds both A and C. Also calculate the annualized yield (continuously compounded) on the portfolio which holds both B and C. In comparing these two portfolios, show that yield is not a reliable guide for investment decisions. Also show that valuation based on synthetics "add" while yields do not. That is, show that the net present value of each component can be added up to get the net present value for the portfolio. Also show that the yield on a portfolio is not equal to an obvious weighted average of the yield of its components. 2. ($$$) In the appendix to this lecture, there is an article titled "The Problem with Redemption Yields" by Stephen Schaefer. Use a spreadsheet program like Excel and create a figure like Figure 3 and another figure like Figure 4 from the article. It is not necessary that you exactly replicate the numbers in the Figures in the article. You only need to get figures with the same type of shape. Your solution should include the plots you created as well as a short discussion of how you did it. The purpose of this exercise is to demonstrate that yields to maturity are not reliable guides for investment decisions. If you construct the figures in the appropriate way you will find that we have a variety of possible yields on correctly priced bonds. Chapter 5: Interpreting Bond Yields, Questions Page 227 Chapter 5: Interpreting Bond Yields, Questions Page 228 Adventures in Debentures Adventures in Debentures 3. ($$$) The current par yield curve is given below. Maturity (in years) 1 2 3 4 5 Par Yield 10% 15% 20% 23% 25% A "plain vanilla bond" refers to any zero coupon bond, any annuity, or any constant coupon bond. Yields on zero coupon bonds are not negative for any maturity. You should assume you can buy or sell a zero coupon bond with any maturity from 1 to 30 years. You can also buy or sell an annuity with any maturity from 1 to 30 years. In addition, you may buy or sell any bond that is mentioned within a part to this question. Determine whether the following statements are true or false. Do not provide an explanation. Part a. When rt > at > bt (k), then an arbitrage must exist. Part b. When rt > bt (k) > at , then an arbitrage is not possible. Part c. Assume arbitrage does not exist. If the yields on any 2 plain vanilla bonds with same maturity are equal, then the yield on all plain vanilla bonds with that same maturity must be equal. Part d. Assume arbitrage does not exist. The yield curve for annuities must flatten out for long term annuities. Part e. Assume arbitrage does not exist. It is possible for the yields on plain vanilla bonds for all possible maturities and all possible coupon yields to be equal. Part f. Assume arbitrage does not exist. The term structure of interest rates (or the spot rate curve) is upward sloping. The yield on a zero coupon bond with a single repayment of $1,978.92 on year t may be less than rt . Part g. Assume arbitrage does not exist. If the term structure of interest rates (or the spot rate curve) is upward sloping, then the yield curve for any plain vanilla bond must be flat or upward sloping. Part h. Assume arbitrage does not exist. The par yield curve must fall on or between the yield curve for zero coupon bonds and the annuity yield curve. Assume that the par yields are quoted on an annualized basis with annual compounding. Using the information in the above par yields,1 what is the value (as of period 0) of a bond that has the following cash flows: K1 = $10, K2 = $10, and K3 = $110, where Kt is the cash paid in period t? 4. ($$$$) Today is year 0. In all parts of this question, you should assume there is no cost to buy or sell a bond. Also assume all bonds are without default risk. Further, ignore any tax considerations. If a bond makes cash payments to its holder, the payments occur once per year. All yields are annualized with annual compounding. rt is the yield on a zero coupon bond that matures on year t. (This zero coupon bond pays $1 on year t.) at is the yield on an annuity which matures on year t; the annuity pays $1 per year. (The first annuity payment is on year 1.) bt (k) is the yield on a constant coupon bond which matures on year t with annual coupon payments; the face value of the bond is $100. (The coupon yield on this bond is k, and the first coupon payment is on year 1.) 1 A par bond is a constant coupon bond which has a current market value equal to the face (or par) value of the bond. The par yield curve indicates the coupon yield that must be provided in order for such a constant coupon bond to sell for par. A constant coupon bond provides annual coupon payments of equal dollar amount. At maturity the constant coupon bond pays the usual annual coupon payment plus the face value of the bond. The first coupon will be paid in 1 year from today. Chapter 5: Interpreting Bond Yields, Questions Page 229 Chapter 5: Interpreting Bond Yields, Questions Page 230 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Part i. Assume arbitrage does not exist. Consider the yield on a bond which pays $100 on year 1 and $50 on year 2. This yield must fall on or between the yield curve for zero coupon bonds and the annuity yield curve. Part j. Assume arbitrage does not exist. Assume the term structure of interest rates (or the spot rate curve) is not flat. The annuity yield curve may cross the spot rate curve more than one time. Chapter 6 Bond Values and the Passage of Time Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Announcements and Assignments . . . . . . . . . . . . . . 238 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 239 A: Thetas and Time Profiles for Coupon-Bearing Bonds . 264 Practice Questions with Solutions . . . . . . . . . . . . . . 274 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Chapter 5: Interpreting Bond Yields, Questions Page 231 Chapter 6: Bond Values and the Passage of Time Page 232 Adventures in Debentures Adventures in Debentures and back on the vagrant coastal breezes, he was snagged by Larry Walters Goes for a Flight1 You've heard of Rickenbacker and Lindbergh and Doolittle. You've heard of Yeager. But have you heard of Larry Walters? Probably not. Yet Walters -- like another relatively unsung hero, the legendary D.B. Cooper -- was made of that special stuff that separates aviation legends from the common run of folk. power lines in Long Beach, and led away in handcuffs. He is reported to have said, by way of explanation of his exploit, "man can't just sit around." Walters subsequently fell on hard times, became bankrupt, and died by his own hand in 1993. But his memory survives as a model of those qualities of independence, vision, and disregard for common caution without which aviation would never have come into being. In 1982 Walters, a truck driver by trade, bought a bunch of weather balloons at a surplus store. He filled them with helium and tied them to a lawn chair. He provided himself with a two-way radio, a parachute, some jugs of water, and an air rifle, and then cut his conveyance loose from the bumper of his car, which was anchoring it to the ground. Take a moment to imagine the thrill and terror of that ascent, transforming a man surrounded by the normal appurtenances of life -- garden, house, sport -- into a speck floating in an infinite space. Had he rigged up some sort of seat belt? Did the chair tip and wobble? Did he call out to the ant-like figures below? We don't know. It is clear, however, that he violated FARs by passing through Los Angeles TCA without a transponder or a clearance. Two passing jetliners reported to controllers that they had seen a man with a gun seated in a deck chair at 11,000 feet. A helicopter went up to take a look. Walters had planned to descend by shooting out the balloons with his pellet gun, one at a time. He had deflated 10 of them this way, when he accidentally dropped the gun. Evidently 10 was enough. After being carried out to sea 1 This story appeared in Flying Magazine. It was retold on "Car Talk" on National Public Radio. Chapter 6: Bond Values and the Passage of Time Page 233 Chapter 6: Bond Values and the Passage of Time Page 234 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures 1.) Introduce a measure of the return from the passage of time (theta) based on the discount function. Preface to Bond Values and the Passage of Time 1.) Materials Needed for the Lecture. Some of the numerical examples in this lecture rely on the information from the chapter titled "Data for a Recurring Illustration." You may find it convenient to bring the lecture notes from this chapter to class as a convenient reference. 2.) Give an approximate measure of theta. 3.) Provide additional interpretations of theta. 4.) Calculate a theta for a portfolio of fixed income securities. 5.) Apply theta to calculate the profit and loss of a bond portfolio under a particular scenario about the term structure. 4.) Required Reading. After attending this session, you should read: 2.) Summary of Chapter. This chapter studies how the passage of time (holding constant the term structure of interest rates) affects the values of fixed income securities. Here we will introduce theta (), which is a measure of the anticipated capital gain (or loss) on a security if interest rates remain at their current level. Theta quantifies that portion of the bond return due to passage of time as opposed to capital gains (or losses) due to changes in interest rates. While theta is a straightforward concept, it will turn out to be very important later in the course when we attempt to understand convexity. In a prior chapter we discussed some of the problems in interpreting bond yields. We concluded that a bond yield is generally not a good indicator of the potential holding period return for a bond even when the term structure of interest rates remains constant. Our discussion of theta could be motivated by the problems in interpreting bond yields, for theta is an appropriate measure of the potential gain or loss on a fixed income security when the term structure remains stable. Appendix A, "Thetas and Time Profiles for Coupon-Bearing Bonds." 5.) New Vocabulary Used in this Chapter. The following buzz words will be used in the lecture notes, the readings, and/or the problem sets: Cost of carry (or carry), curve trading, riding the yield curve, slide effect, theta (), theta profile, and time profile. 6.) Clarifying Some Terminology. Throughout the course, we will talk about the implications associated with a stable term structure of interest rates. The term structure of interest rates has many synonyms in this course including the spot rate curve, yields on zero coupon bonds, and sometimes just interest rates. Keep in mind that a stable term structure does not necessarily imply that the yield to maturity on a given instrument stays constant over time. For example, if the term 3.) Road Map for Chapter. The topics in this chapter will be organized as follows: Chapter 6: Bond Values and the Passage of Time, Preface Page 235 Chapter 6: Bond Values and the Passage of Time, Preface Page 236 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons structure is upward sloping, then the yield on a particular zero coupon bond will decrease over time as that zero coupon bond approaches its maturity -- even if the term structure of interest rates remains constant over time. Clearly, a stable term structure need not imply that the yield to maturiy on a particular coupon bearing bond remains constant as that particular bond approaches its maturity. Announcements and Assignments 7.) Summary of Important Equations. dt - dt+s s = minus the slope of discount function s0 t = lim Q p = i=1 Ni i Chapter 6: Bond Values and the Passage of Time, Preface Page 237 Chapter 6, Announcements and Assignments Page 238 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures ecture notes L for Bond Values and the Passage of Time A Road Map 1.) Introduce a measure of the return from the passage of time (theta) based on the discount function. 2.) Give an approximate measure of theta. 3.) Provide additional interpretations of theta. 4.) Calculate a theta for a portfolio of fixed income securities. 5.) Apply theta to calculate the profit and loss of a bond portfolio under a particular scenario about the term structure. Potential Questions to Consider 1.) Consider the term structure described in the chapter titled "Data for a Recurring Illustration." If the term structure is stable, which zero coupon bond should you buy? 2.) If the term structure is stable, should you buy bonds with high yields and sell bonds with low yield spreads to exploit the yield spread? 3.) Assume the term structure is downward sloping. If the term structure remains stable, could you lose money by buying long term bonds? Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 239 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 240 Adventures in Debentures Adventures in Debentures Graphical Explanation of Theta Bond Values and the Passage of Time 1 Current Discount Function Assume you purchase today a zero coupon bond that matures in 12.5 years. The current price is 0.303431. Ten years from now you sell this bond before it matures. Assume the price of a 2.5 year bond is 0.810686 in 10 years. (Note this is also the current price of a 2.5 year bond which implies interest rates on 2.5 year bonds are the same in 10 years as they are today.) Then your capital gain per year is: 0.810686 - 0.303431 = .0507255 . 12.5 - 2.5 Clearly, if the term structure is constant, then the shape of the current discount function measures the effect of the passage of time. As the holding period becomes smaller, the time appreciation per year is minus the slope of the discount function. i (theta) is the notation we will use to represent the effect of time (holding the term structure constant) on the value of security i when the holding period is infinitesimal. Consider the following graphs . . . 0.8 Dollar Value 0.6 0.4 0.2 0 0 5 10 15 20 25 30 Maturity (in years) Current Discount Function 1 0.8 Dollar Value 0.6 0.4 0.2 0 0 5 10 15 20 25 30 Maturity (in years) Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 241 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 242 Adventures in Debentures Adventures in Debentures Current Discount Function 1 Tracking the Milestones 1.) Introduce a measure of the return from the passage of time (theta) based on the discount function. 2.) Give an approximate measure of theta. 0.8 Dollar Value 0.6 0.4 3.) Provide additional interpretations of theta. 0.2 0 4.) Calculate a theta for a portfolio of fixed income securities. 0 5 10 15 20 25 30 Maturity (in years) Current Discount Function 1 5.) Apply theta to calculate the profit and loss of a bond portfolio under a particular scenario about the term structure. 0.8 Dollar Value 0.6 0.4 0.2 0 0 5 10 15 20 25 30 Maturity (in years) Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 243 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 244 Adventures in Debentures Adventures in Debentures Approximating Theta: A Numerical Illustration Consider a zero coupon bond that pays one dollar in 12.5 years. Using the same discount function as before, it turns out that d12.5 = $0.303431. By using other points on this same discount information, we can approximate the capital gain for this 12.5 year zero coupon bond. For example, Improving the Theta Approximation: A Numerical Illustration Continuously Maturity (in years) 12.00 12.25 12.40 12.45 12.49 12.50 Compounded Yield 9.5000% 9.5206% 9.5328% 9.5368% 9.5400% 9.5408% 9.5416% 9.5448% 9.5488% 9.5606% 9.5799% Annually Compounded Yield 9.9659% 9.9886% 10.0019% 10.0064% 10.0099% 10.0108% 10.0116% 10.0151% 10.0195% 10.0325% 10.0537% .319819022 .311525059 .306645033 .305034251 .303751315 .303431369 .303111738 .301836359 .300249195 .295534491 .287830952 Discount 12.5 d12 - d13 0.319819 - 0.287831 = = 0.0319881 . 13 - 12 13 - 12 However, we could improve this approximation with more detailed information about this discount function . . . 12.51 12.55 12.60 12.75 13.00 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 245 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 246 Adventures in Debentures Adventures in Debentures Based on this more detailed information about the discount function, d12.75 - d12.25 = 0.0319811 . 12.75 - 12.25 Putting the Discount Function under a Microscope Current Discount Function 0.32 0.315 Dollar Value 0.31 0.305 0.3 0.295 12.5 - 12.5 d - d12.40 - 12.60 = 0.0319792 . 12.60 - 12.40 - d12.45 d - 12.55 = 0.0319789 . 12.55 - 12.45 - d12.49 d - 12.51 = 0.0319788 . 12.51 - 12.49 12.5 12.5 0.29 12 12.2 12.4 12.6 12.8 13 Maturity (in years) Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 247 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 248 Adventures in Debentures Adventures in Debentures Why Approximate Theta? Later we will develop ideas comparable to theta for measuring the approximate capital gain (or loss) due to changes in interest rates. One such measure is dollar duration (or delta). For dollar duration, unlike theta, we will have a simple analytical formula for measuring the potential gain due to changes in interest rates. In contrast, we have no simple formula for theta, and we will be forced to approximate it by examining the numerical slope of the discount function. We will not develop a simple formula for theta, for we would need a general function to describe all possible term structures. To generate this general discount function (which can be done) requires many additional assumptions -- many of which are unrealistic. The lack of such a model requires a numerical approximation to theta. The good news is that our measure of theta is not model dependent. It is more general than our equation for calculating dollar duration. Tracking the Milestones 1.) Introduce a measure of the return from the passage of time (theta) based on the discount function. 2.) Give an approximate measure of theta. 3.) Provide additional interpretations of theta. 4.) Calculate a theta for a portfolio of fixed income securities. 5.) Apply theta to calculate the profit and loss of a bond portfolio under a particular scenario about the term structure. Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 249 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 250 Adventures in Debentures Adventures in Debentures Thetas for More Maturities (1) Mat. 1/365 0.5 1.0 1.5 2.0 2.5 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 (2) Yield 8.0000% 8.0825% 8.1632% 8.2422% 8.3194% 8.3950% 8.4688% 8.6111% 8.7465% 8.8750% 8.9965% 9.1111% 9.2188% 9.3194% 9.4132% 9.5000% 9.5799% (3) Price .999781 .960393 .921611 .883704 .846717 .810686 .775643 .708614 .645761 .587135 .532721 .482445 .436186 .393787 .355066 .319819 .287831 (4) Theta .080000 .078407 .076705 .074908 .073029 .071083 .069081 .064956 .060742 .056512 .052329 .048244 .044300 .040527 .036949 .033581 .030432 (5) (2) (3) .079982 .077624 .075233 .072837 .070442 .068057 .065688 .061019 .056482 .052108 .047926 .043956 .040211 .036699 .033423 .030383 .027574 Page 251 (1) Mat. 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 (2) Yield 9.6528% 9.7188% 9.7778% 9.8299% 9.8750% 9.9132% 9.9444% 9.9688% 9.9861% 9.9965% 10.0000% 9.9965% 9.9861% 9.9688% 9.9444% 9.9132% 9.8750% (3) Price .258880 .232745 .209204 .188045 .169060 .152056 .136849 .123263 .111142 .100339 .090718 .082156 .074542 .067775 .061763 .056426 .051690 (4) Theta .027506 .024802 .022315 .020038 .017963 .016077 .014369 .012827 .011438 .010191 .009072 .008070 .007174 .006375 .005662 .005025 .004458 (5) (2) (3) .024989 .022620 .020455 .018484 .016695 .015074 .013609 .012288 .011099 .010030 .009072 .008213 .007444 .006756 .006142 .005594 .005104 NOTE: maturity (or mat.) is quoted in years, and the yield is continuously compounded. Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 252 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Adventures in Debentures Adventures in Debentures Comparing Theta with Yields If the yield curve is flat at r, then it is straightforward to show1 for any zero coupon bond that matures on date: t: t = r dt . However, when the yield curve is not flat, theta must reflect not only the "time effect," but also the "slide effect." The slide effect is just the idea that the appropriate discount may be lower or higher than the current discount rate as a given zero coupon bond gets closer to maturity, assuming the term structure stays constant. The slide effect provides additional return to the passage of time (holding constant the term structure of interest rates) if the current term structure is upward sloping; that is, t > rt dt . The slide effect provides a negative offset to the return from the passage of time (holding constant the term structure of interest rates) if the current term structure is downward sloping; that is, t < rt dt . When the yield curve is upward sloping, investors talk about "riding the yield curve." This trading strategy reflects the enhanced return an investor receives from the slide effect -- assuming the term structure of interest rates remains stable over time. In general, investments which focus on yield spreads among different fixed income securities are called "curve trading." Theta is a useful way to quantify the performance of a curve trades. 1 When the yield curve is flat, dt = e-tr , where r is the same value for any maturity. The first derivative of this function with respect to maturity is -re-tr = -rdt . Since theta reflects the passage of time, not an increase in maturity, we need to reverse the sign of the derivative, which is the equation in the text above. Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 253 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 254 Adventures in Debentures Adventures in Debentures Riding the Yield Curve 0.12 Continuously Compounded Yield A Theta Profile For the typical shape of the discount function, the time effect for short-term zero coupon bonds is very important. In fact, for short-term bonds the time effect can be large enough so that capital losses are unlikely even if interest rates make a dramatic increase. Based on the discount function given earlier . . . 0.11 0.1 0.09 0.08 Current Theta Profile 0.08 0.07 0 5 If you buy a five year zero coupon bond, its price today is: 0 d5 Theta Value 10 15 20 Maturity Hin yearsL 25 30 0.06 0.04 =e -50 r0,5 . 0.02 If you sell this zero coupon bond in one year and the term structure has not changed, its price next year is: 1 d5 = e-41r1,5 = e-40r0,4 . 0 0 5 10 15 20 25 30 Maturity (in years) Clearly, your holding period return benefits from the stable and upward sloping term structure of interest rates. Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 255 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 256 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Introduce a measure of the return from the passage of time (theta) based on the discount function. 2.) Give an approximate measure of theta. 3.) Provide additional interpretations of theta. 4.) Calculate a theta for a portfolio of fixed income securities. 5.) Apply theta to calculate the profit and loss of a bond portfolio under a particular scenario about the term structure. Calculating the Time Effect for Portfolios The time effect for portfolios can be calculated as the weighted sum of the time effect for each component of the portfolio, where the weights reflect the number of units of that component that are held in the portfolio. That is, for a portfolio p containing n different securities (where the portfolio holds Ni units of security i), the theta of the portfolio is: p = N 1 1 + N2 2 + . . . + Nn n . If security i is a liability, not an asset, in the portfolio than Ni is less than zero. Calculating the time appreciation for a coupon-bearing bond is very similar to that for a portfolio of zero coupon bonds. After this session, review the appendix titled "Thetas and Time Profiles for Coupon-Bearing Bonds" for more details. Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 257 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 258 Adventures in Debentures Adventures in Debentures Calculating the Time Effect for Portfolios: A Numerical Example Consider a portfolio (or balance sheet) with assets that include 10 units of a 2 year zero, 5 units of a 9 year zero, and 3 units of a 30 year zero. Furthermore, there is a single liability consisting of 7 units of a 20 year zero. The market value of the net equity is: (10 .846717) + (5 .436186) + (3 .051690) - (7 .136849) = $9.85 The theta of the net equity is: + = (10 .073029) + (5 .044300) (3 .004458) - (7 .014369) $0.86 Tracking the Milestones 1.) Introduce a measure of the return from the passage of time (theta) based on the discount function. 2.) Give an approximate measure of theta. 3.) Provide additional interpretations of theta. 4.) Calculate a theta for a portfolio of fixed income securities. 5.) Apply theta to calculate the profit and loss of a bond portfolio under a particular scenario about the term structure. Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 259 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 260 Adventures in Debentures Adventures in Debentures Evaluating a Long-Short Postion 1 Consider an investment position motivated by the belief that interest rates will drop in the future. To exploit this belief, a trader takes a long position in 12 year zero coupon bonds and funds this position by taking a short position in 1 year zero coupon bonds. What is the "cost of carry" of this position if the term structure of interest rates fails to change? Current prices and thetas for the relevant securities: d12 = 0.319819 , 12 = 0.033581 , d1 = 0.921611 , 1 = 0.076705 . Say you invest $100,000 in the 12 year zero coupon bonds. That is, you buy N12 = 312,676.86 units of these bonds. To fund this investment, you short $100,000 in the 1 year zero coupon bonds. That is, you short N1 = -108,505.65 units of these bonds. Thus, the theta of this position is: Evaluating a Long-Short Postion 2 Consider another investment position motivated by the high yield on the 24 year zero coupon bond. A trader takes a long position in 24 year zero coupon bonds and funds this position by taking a short position in 20 year zero coupon bonds. Current prices and thetas for the relevant securities: d24 = 0.090718 , 24 = 0.009072 , d20 = 0.136847 , 20 = 0.014369 . Say you invest $100,000 in the 24 year zero coupon bonds. That is, you buy N24 = 1,102,317.07 units of these bonds. To fund this investment, you short $100,000 in the 20 year zero coupon bonds. That is, you short N20 = -730,743.09 units of these bonds. Thus, the theta of this position is: p = (N24 24) + (N20 20) = (1,102,317.07 .009072) - (730,743.09 .014369) = -$499.83 . Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 261 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 262 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Worksheet Appendix A: Thetas and Time Profiles for Coupon-Bearing Bonds During the class session, we focused on measuring the time appreciation for zero coupon bonds under the assumption that time passes but the term structure of interest rates remain constant. In this appendix, we extend this analysis to coupon-bearing bonds. 1.) Thetas and Coupon-Bearing Bonds: A Solution Using Synthetics. A portfolio of zeros with the same cash flows as a coupon-bearing bond is an example of a replicating portfolio or a synthetic security. (We sometimes refer to this as the LegoTM approach to valuation.) One can apply this idea to calculate the theta of a coupon-bearing bond. Since a coupon-bearing bond can be thought of as a portfolio of zero coupon bonds and since we know the theta of a portfolio is the weighted sum of the thetas of the components, we know: T coupon bond = t=1 Kt t , where Kt is the cash (coupon and/or face value) received in period t and t is the theta for the zero coupon bond that pays $1 on period t. The above formula may not be appropriate at the moment a coupon is paid to the bond holder, for theta may not be well defined at that point in time. Theta measures the slope of the time profile. As we will see in below, the time profile is quite jagged at the moment the coupon is paid, so its slope is not well defined. 2.) Calculating Thetas for Coupon-Bearing Bonds: A Numerical Illustration. As an illustration, consider a 20 year coupon-bearing bond with face value of $9363.03 Chapter 6: Bond Values and the Passage of Time, Lecture Notes Page 263 Chapter 6: Bond Values and the Passage of Time, Appendix A Page 264 Adventures in Debentures Adventures in Debentures and with annual coupon payments of $1009.09. The next coupon will be paid in one year from today. This bond has no risk of default. To calculate the theta for this bond, we will need the "theta profile" for the current discount function. The following table provides the relevant information: Continuously Maturity (in years) 1/365 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 Compounded Yield 8.0000% 8.1632% 8.3194% 8.4688% 8.6111% 8.7465% 8.8750% 8.9965% 9.1111% 9.2188% 9.3194% 9.4132% 9.5000% 9.5799% 9.6528% 9.7188% 9.7778% 9.8299% 9.8750% 9.9132% 9.9444% 0.999781 0.921611 0.846717 0.775643 0.708614 0.645761 0.587135 0.532721 0.482445 0.436186 0.393787 0.355066 0.319819 0.287831 0.258880 0.232745 0.209204 0.188045 0.169060 0.152056 0.136849 0.080000 0.076705 0.073029 0.069081 0.064956 0.060742 0.056512 0.052329 0.048244 0.044300 0.040527 0.036949 0.033581 0.030432 0.027506 0.024802 0.022315 0.020038 0.017963 0.016077 0.014369 Discount Theta bearing bond (20 ) as: 20 = 1009.091 + 1009.092 + . . . + 1009.0919 + 10,372.1220 = (1009.09 0.076705) + (1009.09 0.073029) + . . . + (1009.09 0.016077) + (10,372.12 0.014369) = 972.54 where t is the theta for a zero coupon bond that pays $1 when it matures in t periods (that is, it is minus the slope of the discount function at maturity t). The approach we followed for this 20 year coupon-bearing bond can also be applied to other financial contracts (e.g., forwards, bond options, bonds with embedded options, floating rate notes, and swaps) as long as we know how to create synthetic alternatives. (Of course, we need the thetas for the components of the synthetic.) In these cases, we could calculate the theta by finding the theta of the synthetic alternative. (This will become more obvious in future sessions.) 3.) Time Profiles for a Coupon-Bearing Bond. A time profile plots the value of a fixed income security over time -- holding constant the current term structure. The slope of the time profile is equal to theta. In the case of a zero coupon bond, a time profile can be constructed directly from the current discount function. If the term structure of interest rates remains stable, then the future discount function will be the same as the current discount function. Under this assumption, the future price of a zero coupon bond can be read from the current discount function merely by determining the remaining time till maturity of this bond as time passes. To illustrate a time profile, consider a 12.5 year zero coupon bond with a face value of one dollar. The current price of this bond is 0.303431, and the current discount function is described by the prior table. Based on this prior table, we know the price of this bond will be 0.319819 in 6 months from now when the bond has 12 years to mature. The prior table also indicates that the bond will have a price of 0.355066 in 18 months from now when the bond has 11 years to mature. Clearly, we can compute Chapter 6: Bond Values and the Passage of Time, Appendix A Page 266 Based on the above information we can calculate the theta of the 20 year coupon Chapter 6: Bond Values and the Passage of Time, Appendix A Page 265 Adventures in Debentures Adventures in Debentures the bond's price at any future date using the current discount function as long as the term structure of interest rates remains stable. Figure 1 provides the time profile for this particular bond based on the discount function in the prior table. Essentially, the time profile for a zero coupon bond with a face value of one dollar is just created from the discount function by reversing the horizontal axis of the discount function. Figure 1: The time profiles that follow distinguish between the full versus the flat price. Note the full price is the present value of the security. The flat price is the full price less the accrued interest.2 Figure 2 provides the time profile of the full price of the 20 year coupon-bearing bond when the term structure has a short-term rate of 8%. Between two coupon payments, the bond price appreciates assuming the term structure does not change. This appreciation reflects the fact that the present value of less distant payments is greater than the present value of more distant payments assuming interest rates are the same. As expected, the full price of the coupon-bearing drops on the moment the coupon is paid by the borrower; the size of the drop is just equal to the coupon payment. The jaggedness of the time profile means the slope of the time profile is not well defined at those points, so theta cannot be measured at the moment any coupon is paid. Figure 2: Time Profile: 12.5 Yr Zero 1 0.9 Dollar Value 0.8 0.7 0.6 0.5 Time Prof: Full Price of 20 Yr Bond Annual Coup=1009.09 & Face=9363.03. (term structure scenario: r=8%) 13500 13000 0 2 4 6 8 10 12 dollar value 0.4 0.3 Passage of Time (in years) Now consider a time profile for a coupon-bearing bond. As in the prior section, this bond matures in 20 years and has annual coupon payments. The coupon is $1009.09, and the face value is $9363.03. The following pages provide a time profile when the overnight rate is 8%, 6%, and 13%. The 8% case corresponds to the term structure given in the prior table. The 6% case is the same as the prior table except the rates are 200 basis points lower for all maturities. Similarly, the 13% case corresponds to a situation where the term structure is 500 basis points higher for all maturities than the term structure in the earlier table.1 1 12500 12000 11500 11000 10500 10000 9500 0 5 10 15 20 time passage (in years) In other words, we are examining situations where the term structure makes uniform shifts. We will spend some time on this uniform shift assumption when we discuss dollar duration in future 2 sessions. However, time profiles can be calculated whether or not the term structure makes uniform shifts. For more details on accrued interest, flat prices, and full prices, see the appendix titled "Interest Rate Quotes and Conventions." Chapter 6: Bond Values and the Passage of Time, Appendix A Page 267 Chapter 6: Bond Values and the Passage of Time, Appendix A Page 268 Adventures in Debentures Adventures in Debentures The full price of this coupon-bearing bond at the moment before it matures is $10,372.12 = $1,009.09 + $9,363.03. The time profile in Figure 2 reflects this value when the time passage is 20 years. Figure 3 is the same as Figure 2 except the flat price, not the full price, of the coupon-bearing bond is plotted. Clearly, the flat price is less jagged since the accrued interest is removed. The flat price of this bond at the moment before it matures is just $9,363.03. The time profile in Figure 3 reflects this values when the time passage is 20 years. Figure 3: which is selling at a premium. Figure 4: Time Prof: Full Price of 20 Yr Bond Annual Coup=1009.09 & Face=9363.03. (term structure scenario: r=6%) 13500 13000 dollar value Time Prof: Flat Price of 20 Yr Bond Annual Coup=1009.09 & Face=9363.03. (term structure scenario: r=8%) 13000 12500 dollar value 12000 11500 11000 10500 10000 9500 9000 0 5 10 15 20 12500 12000 11500 11000 10500 10000 9500 0 5 10 15 20 time passage (in years) The reader should note that in all graphs in this appendix the scale of the vertical axis is the same, for the difference between the maximum and minimum values is the same across the graphs. This makes comparing the graphs straightforward. Figure 5 is similar to Figure 4. However, Figure 5 provides a picture of the flat price, not the full price. Since the coupon-bearing bond has a high coupon yield relative to current interest rates, the flat price is at a premium relative to the face value. However, as the bond approaches maturity the premium disappears assuming the term structure does not change over time. time passage (in years) Figure 4 provides a time profile for the same coupon-bearing bond as in the prior figures. However, Figure 4 is based on a term structure where rates are 200 basis points lower than in the earlier two figures. Given the lower interest rates, the couponbearing bond clearly sells out at a premium relative to its face value of $9,363.03. However, as the coupon-bearing bond approaches maturity, the premium must be eliminated because at maturity the bond must sell for its face value plus coupon (for the full price) or the face value (for the flat price). Figure 4 shows the full price trending down (ignoring the jaggedness between coupon payments) for this bond Chapter 6: Bond Values and the Passage of Time, Appendix A Page 269 Chapter 6: Bond Values and the Passage of Time, Appendix A Page 270 Adventures in Debentures Adventures in Debentures Figure 5: Figure 6: Time Prof: Flat Price of 20 Yr Bond Annual Coup=1009.09 & Face=9363.03. (term structure scenario: r=6%) 13000 12500 dollar value Time Prof: Full Price of 20 Yr Bond Annual Coup=1009.09 & Face=9363.03. (term structure scenario: r=13%) 10500 10000 dollar value 9500 9000 8500 8000 7500 7000 12000 11500 11000 10500 10000 9500 9000 0 5 10 15 20 6500 0 5 10 15 20 time passage (in years) Figure 6 is similar to Figure 2 and Figure 4. In all three graphs, the full price of the same coupon-bearing bond is provided over time. However, in Figure 6 the term structure of interest rates is much higher than the term structures for either Figure 2 or Figure 4. The term structure in Figure 6 is high relative to the coupon yield of the coupon-bearing bond. As a result, the price of the coupon-bearing bond is at a discount relative to the face value plus the final coupon payment. Clearly, the full price of this coupon bearing bond must be equal to the face value plus the coupon payment just prior to maturity. This explains why the full price drifts up (in a jagged way) as time passes (or as the bond approaches maturity). time passage (in years) Figure 7 provides the flat price of the same coupon-bearing bond whereas Figure 6 presented the full price. Figure 7 uses the same term structure of interest rates as was used in Figure 6. Figure 7 is less jagged than Figure 6, but in both figures the time appreciation is positive for this bond selling at a discount. Chapter 6: Bond Values and the Passage of Time, Appendix A Page 271 Chapter 6: Bond Values and the Passage of Time, Appendix A Page 272 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Figure 7: Time Prof: Flat Price of 20 Yr Bond Annual Coup=1009.09 & Face=9363.03. (term structure scenario: r=13%) 9500 9000 dollar value 8500 8000 7500 7000 6500 6000 5500 0 5 10 15 20 Practice Questions with Solutions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." 1. ($$) This question checks your ability to interpret time profiles. Unless the questions state to the contrary, all graphs in this question assume that future term structures based on continuously compounded rates (annualized) make uniform shifts. Part a. Compare Figures 1 and 2. Which Figure has the higher value of ? No explanation time passage (in years) Chapter 6: Bond Values and the Passage of Time, Appendix A Page 273 Chapter 6: Bond Values and the Passage of Time, Practice Questions with Solutions Page 274 Adventures in Debentures Adventures in Debentures for your answer should be provided. Part b. Calculate the approximate in Figure 1 when time passage is 0.2 years. Part c. Assume Figures 1 and 2 are based on zero coupon bonds with the same face value but different maturities. Indicate whether Figure 1 or 2 is more likely for the long term zero coupon bond. No explanation for your answer should be provided. Part d. Both Figures 3 and 4 provide a time profile (using the full price) for coupon-bearing bonds. However, the implicit coupon payments in Figure 3 are not the same as in Figure 4. Which Figure has the higher coupon? No explanation for your answer should be provided. Figure 1. 8000 dollar value 7950 7900 7850 7800 7750 0 0.1 0.2 0.3 0.4 time passage (in years) Figure 2. 4100 4075 dollar value Calculate the approximate coupon payment for the bond in Figure 3. Part e. 4050 4025 4000 3975 3950 0 0.1 0.2 0.3 0.4 Part f. If the values for the two bonds used in Figures 3 and 4 were based on flat prices, not full prices, would the flat price alternative to Figure 3 drift up or drift down? Would the flat price alternative to Figure 4 drift up or drift down? No explanation for your answer should be provided. time passage (in years) Chapter 6: Bond Values and the Passage of Time, Practice Questions with Solutions Page 275 Chapter 6: Bond Values and the Passage of Time, Practice Questions with Solutions Page 276 Adventures in Debentures Adventures in Debentures Part g. Calculate the approximate in Figure 4 when the passage of time is 4.5 years. SOLUTION: Figure 3. 10500 10000 9500 9000 8500 0 1 2 3 4 5 time passage (in years) Figure 4. 11200 dollar value 11000 10800 10600 10400 10200 0 1 2 3 4 5 time passage (in years) Part a. Figure 1 has a higher than Figure 2. (Capital gain in Figure 1 over .4 years is 8,050 - 7,750 = 300 while capital gain in Figure 2 over .4 years is 4,100 - 3,950 = 150.) Part b. is the slope of the time profile = dollar value 7,950-7,820 .3-.1 = 140 .2 = 700 per year. Part c. Usually, we expect to be greater for short-term bonds than long-term bonds. Therefore, Figure 1 is the short-term bond, while Figure 2 is the long-term bond. Part d. Figure 4 implies a higher coupon payment than Figure 3. (Just compare the size of vertical drops in Figures 3 and 4, and note vertical drop is larger in Figure 4.) Part e. Vertical drop in Figure 3 is 500, so coupon is 500. Part f. Flat price version of Figure 3 would drift up since bond is selling at a discount relative to its face value. Conversely, flat price vesion of Figure 4 would drift down. Part g. is the slope of the time profile around 4.5 years, so: 880 11,000 - 10,120 = = 880 . = 5-4 1 2. ($$$) The following table provides the current term structure based on zero coupon bonds. The second column of the table indicates the annualized rates with continuous comChapter 6: Bond Values and the Passage of Time, Practice Questions with Solutions Page 277 Chapter 6: Bond Values and the Passage of Time, Practice Questions with Solutions Page 278 Adventures in Debentures Adventures in Debentures pounding. Maturity (in years) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 20.0 rs 9.50% 10.00% 10.25% 10.50% 10.75% 11.00% 11.10% 11.20% 14.00% Part b. Currently, the theta of the 20 year zero coupon bond is .005. Given this theta value, approximate the current price of a zero coupon bond which matures in 19 years and 3 months. (You should assume that the face value of this zero coupon bond is one dollar.) Part c. Draw a plausible discount function where your approximation in part b above is too high. SOLUTION: Part a. At t = 0, +20 e-1.10 - 80 e-2.1050 + 120 e-3.1100 = 39.52 . At t = .5, +20 e-.5.095 - 80 e-1.5.1025 + 120 e-2.5.1075 = 42.19 . At t = 1.5, Consider an individual's balance sheet consisting of a single asset and a single liability The cash flows of the asset are K1 = $20, K2 = $20, and K3 = $120. The liability has one cash outflow of $100 to be paid out in year 2. Part a. Consider a time profile for the market value of the net equity of this balance sheet. Give the three values on the time profile curve corresponding to 3 points on the horizontal axis. Use today (t = 0), 6 months from now (t = .5), and 18 months from now1 (t = 1.5) as the three points on the horizontal axis. In other words, provide three points on the following graph: -80 e-.5.095 + 120 e-1.5.1025 = 26.61 . . . . ... .... . . . . . . . . . . . . . . Value of Net Equity ? ? .. ... . . 42.19 39.52 Value of Net Equity 26.61 .. . .... . . ... 0 ? . . .... ... . .5 time (t) 1.5 0 .5 1.5 time (t) Part b. 1 You should assume that the coupon received on year 1 was immediately consumed, and it was not reinvested. r20 = 14% d20 = 1 e-20.14 = .0608101 Page 280 Chapter 6: Bond Values and the Passage of Time, Practice Questions with Solutions Page 279 Chapter 6: Bond Values and the Passage of Time, Practice Questions with Solutions Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons 20 = .005 d19.25 d20 + (3/4 20 ) = .0608101 + (3/4 .005) = .0645601 = Questions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Even if you decide not to submit your answer to this problem set, you should review my solution only after you attempt the questions to know if you could do them or not. Many times my solution is obvious AFTER you see it, but you need to know if the solution was obvious to you BEFORE you are told the solution. If you submit your answers to this problem set, please keep the following in mind: I sometimes fall behind where I expect to be in the lectures and an assignment is due before I get to some of the relevant material. If you believe the lectures have not yet covered the material necessary to answer a particular question and the required reading provides no guide, then you should indicate this on your solution and skip the question. Names of students along with their course and section numbers should be clear. The number of students in a study group should be less than or equal to five. Homework should be stapled. The final numerical answer should be "flagged" in some manner. Boxing, highlighting, and/or underlining the number are appropriate. Pages of spreadsheet printouts should be kept to a minimum. Only the essential information should be incorporated. You should not spend an excessive amount of time trying to solve any particular question. If you cannot complete a question, just describe what you tried to do. Part c. . . . ... . .... . . . . dt . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. . ... ... . . ... ... ... .... . . .... ..... ...... . . ............. ............ ..... ..... .. .. . .... . . .. . . . .... ... .... . .... . ..... ...... . ......... .............. . ................... ........... . . . . ...... ..... . .. . .. 20 Chapter 6: Bond Values and the Passage of Time, Practice Questions with Solutions Page 281 Chapter 6: Bond Values and the Passage of Time, Questions Page 282 Adventures in Debentures Adventures in Debentures 1. ($$) Today is year 0. The table below provides information about the discount function as well as the value of theta: Part d. Consider a balance sheet which contains 4 securities. On the asset side, there are 4,500 units of a 2 year zero and 2,000 units of a 3 years zero. On the liability side, there are 1,200 units of a 6 month zero and 700 units of a 1 year zero. All zero coupon bonds pay $1.00 on the date that they mature. What is the theta of the net equity for this balance sheet? Part e. Based on your value for theta for the net equity, what is the approximate wealth after you hold this portfolio for 2 months (from today) assuming the discount function does not change. Maturity 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 2.5 3.0 Discount 0.960393 0.952569 0.944779 0.937022 0.929299 0.921611 0.883704 0.846717 0.810686 0.775643 Theta 0.078407 0.078075 ??? 0.077398 0.077053 0.076705 0.074908 0.073029 0.071083 0.069081 2. ($$$$) Assume you forecast that today's yields on zero coupon bonds will remain constant over the near term (say, one week). You wish to structure a portfolio of fixed income securities to exploit this forecast. In structuring your portfolio, you are not concerned about any risk due to changes in interest rates. Your goal is to maximize your future wealth in the near term (say, one week). In structuring your portfolio, you may use any of the instruments in the following list. (We will refer to this list as your trading universe.) To keep the problem simple, you may hold fractional units of any security in the list. Part a. Give a good approximate value for theta for the bond that matures in 0.7 years. Part b. Consider a two year coupon bond with an annual coupon yield of 30%. The face value is $1000.00. The coupon payments are semi-annual. The next coupon occurs 6 months from today. If the bond is sold, assume the seller does not receive accrued interest. What is the theta for this coupon bond? Part c. Based on your value for theta, what is the approximate capital gain or loss from holding this security for 1 month from today. (Assume the discount function is the same next month as it is now.) Chapter 6: Bond Values and the Passage of Time, Questions Page 283 ID A B C D E F Price 10.29 655.53 1,071.20 9,058.66 86.24 93.05 Yield 7.87% 8.52% 8.21% 9.57% 7.68% 7.47% Theta 0.83 56.85 91.54 864.62 6.72 6.89 The column headings in the above table are: "ID" is the identification code for each security in the trading universe. "Price" is the current price of the security. Since we are ignoring trading costs, this price is the price at which the security can be bought. If you decide to short the security, then the price represents the amount of money you receive by shorting one unit of the security. "Yield" is quoted as annualized number using annual compounding. "Theta" is our usual measure of time appreciation quoted Chapter 6: Bond Values and the Passage of Time, Questions Page 284 Adventures in Debentures Adventures in Debentures on a per year basis. Part a. Assume your current wealth is $100,000.00. You may take long or short positions using the securities in your trading universe. However, you are restricted from excessive leverage -- the market value of all your liabilities may not exceed $50,000.00. Now report the portfolio that you would hold to obtain the goals described in the first paragraph of this question. Your answer should contain a table that looks like: ID A B C D E F Number of Units market value of that investment is in excess of $40,000.00. Now repeat part a with this new investment restriction that requires diversification. Be sure to report your answer in the tabular format suggest in part a above. Part d. NOTE: If we have not discussed the chapter titled "Forward Contracts" before this problem set is due, you may skip this part of this question. However, you will still be responsible for the material for purposes of the exam. Now assume your current wealth is $100,000.00 as in part a. Now assume your trading universe has expanded. In addition to the trading universe in part a, you may also go long or short a forward contract. The forward contract matures in 24 months; upon maturity a 5 year zero coupon bond with a face value of $1000 is delivered. The forward price is 649.13. This forward contract has a market value of zero with a theta of 9.04. While you may go long or short this forward contract, you are restricted not to have more than 100 forward contracts in your portfolio. The leverage constraint in part a is still applicable; however, the leverage restriction does not account for any forward contracts. As discussed in part c, there are also diversifcation restrictions with respect to the rest of the trading universe. Now repeat part c with this expanded list of securities in the trading universe. Be sure to indicate clearly whether a position is long or short. (You may indicate a short position using a negative value for the number of units.) Part b. Now assume your current wealth is zero. Let security "G" in the following table be the forward contract. Your answer should Now repeat part a with this new assumption about current wealth. Be sure to report your answer in the tabular format suggest in part a above. Part c. Now assume your current wealth is $100,000.00 as in part a. The leverage restriction of part a continues to apply to this question. However, unlike part a, your investment portfolio must satisfy a diversification constraint -- you may not have a position (long or short) in any security where the Chapter 6: Bond Values and the Passage of Time, Questions Page 285 Chapter 6: Bond Values and the Passage of Time, Questions Page 286 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons contain a table that looks like: ID A B C D E F G Be sure to indicate clearly whether a position is long or short. (You may indicate a short position using a negative value for the number of units.) Part e. Now compute the thetas of the portfolios that you constructed in parts a through d above. Compare the relative magnitudes of these four thetas. Justify the relative magnitudes of these four values. Number of Units Chapter 7 Forward Contracts Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Announcements and Assignments . . . . . . . . . . . . . . 295 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 296 A: Forward Contracts: More Synthetics . . . . . . . . . . 317 B: Alternative Interpretations of Forward Rates . . . . . 323 Practice Questions with Solutions . . . . . . . . . . . . . . 349 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Chapter 6: Bond Values and the Passage of Time, Questions Page 287 Chapter 7: Forward Contracts Page 288 ...
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