Fixed Income Mathematics
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Table of contents
- Cover
- Copyright Pageiv
- Contentsv
- Chapter 1. Introduction„Who this Book is for and What it Hopes to Accomplish1
- Historical Background„The Big Change in Investment, Loan, and Money Management1
- What this Book Hopes to Accomplish3
- What Sort of Problems Might this Book Help You to Solve?4
- Who this Book is Meant to Address4
- The Mathematical Knowledge Required for this Book5
- The Role of Examples and Problems in this Book6
- Chapter 2. Interest, Its Calculation, and Return on Investment7
- A General Introduction to Interest7
- How to Compute Interest8
- Notation9
- Percentage Rate and Time Period10
- Return on Investment11
- Analysis of Investments or Returns without Explicit Money Values, and Intangible Investments and Ret14
- Chapter Summary16
- Chapter 3. Compound Interest19
- What is Compound Interest?20
- Using Compound Interest Tables21
- Looking at the Compound Interest Tables23
- Compounding within a Period23
- The Equations for Compound Interest–Compounding within a Period24
- Continuous Compounding: How it Works and When it Applies25
- The Derivation of the Equations for Continuous Compounding26
- What is a Mathematical Model?28
- Some Famous Mathematical Models29
- Reasons for Using Continuous Functions in Financial Models30
- A Business Example of Use of Continuous Functions30
- Further Reflections on Approach 331
- Computing i, Given S, Snt, T, and N31
- Accuracy Requirements36
- Legal Requirements for Accuracy36
- An Example from Compound Interest36
- Using Tables and Interpolating between Values37
- The Rule of 7237
- A Zero Interest Rate?37
- Negative Interest Rates?38
- Real and Nominal Rates38
- Chapter Summary39
- Suggestions for Further Study41
- Chapter 4. Present Values51
- What is Present Value?52
- The Equation for Present Value52
- The General Equation for Present Value53
- The Present Value Tables54
- Using Present Values to Make Project Decisions54
- Example of Project Analysis55
- Using Different Interest Rates in the Analysis57
- The Equations for Flow of Funds Analysis57
- The Various Number Systems and What They Mean58
- Solving Polynomial Equations61
- Practical Considerations in Using Calculators and Computers to Solve Polynomial Equations62
- Using the Bisection Method to Find Real Solutions63
- What if the Exponents are not Integers?64
- Chapter Summary64
- Suggestions for Further Study67
- Chapter 5. Annuities Certain77
- What is an Annuity Certain?77
- Examples of Annuities Certain78
- Why Annuities Certain are Important78
- The Equation for the Present Value of an Annuity Certain79
- A Look at the Tables for an Annuity Certain80
- Solving for the Interest Rate, Given the Annuity Certain and Its Cost80
- The Perpetuity80
- The Annuity Due81
- Further Comments82
- Analysis and Calculation of Some Combination Annuities Certain83
- Chapter Summary83
- Chapter 6. Bond Price Calculation103
- What is a Bond?104
- How Bonds are Described105
- How to Read a Bond Market Report105
- What is a Call Feature?106
- What is a Put Option?108
- Discount Securities108
- The General Equation for Computing a Bond Price, Given the Yield108
- A Note on Yield110
- A Note on Accrued Days in the Settlement Period (A in Equation 6.1) and Dated Date111
- Analysis of the Equation111
- Standards of Accuracy114
- Pricing Zero Coupon Bonds115
- Pricing to a Call Feature115
- Examples of Bond Price Calculations116
- Amortization of Premium and Accrual of Discount117
- Accrual of Discount117
- Amortization of Premium119
- A Portfolio Management Interlude120
- Pricing a Bond to a Call121
- A Look at a Basis Book123
- Basic Rules for Prices and Yields123
- The Shape of the Price-Yield Curve124
- Dirty Price and Clean Price125
- Chapter Summary126
- Suggestion for Further Study128
- Chapter 7. The Future Value (or Amount) of an Annuity131
- Uses of the Future Value of an Annuity131
- The Equation for the Future Value of an Annuity132
- The Tables for Amount of Annuity133
- Some Investment Policy Implications133
- Chapter Summary133
- Chapter 8. Accrued Interest143
- What is Accrued Interest?143
- Bonds that Accrue Interest144
- The Equation for Accrued Interest144
- Chapter Summary146
- Chapter 9. Discount Yield149
- Discount Yield150
- Calculation of the Discount and Price for Discount Securities150
- What is a Treasury Bill?150
- The Day-Count Conventions for T-Bills151
- Price Calculations for Discount Municipal Securities152
- Bond Equivalent Yield (BEY) What It Means and How to Compute It152
- Derivation of the Bond Equivalent Yield Equations153
- Why We Care about Bond Equivalent Yield (BEY)154
- A Historical Note155
- Taxation of Income from Treasury Bills155
- Chapter Summary155
- Chapter 10. Calculations for Other Securities159
- Certificates of Deposit159
- Repurchase Agreements160
- Uses of Repos and Reverse Repos161
- Pricing Repos161
- Chapter Summary161
- Class Project161
- Chapter 11. Quotations and Bond Market Reports163
- Long-Term Instruments163
- Discount Instruments165
- Chapter Summary166
- Chapter 12. Types of Yields167
- Nominal (or Coupon) Yield168
- Discount Yield168
- Current Yield168
- True (or Bond) Yield169
- A Method to Compute Approximate Bond Yield, Given Price170
- Chapter Summary171
- Chapter 13. Sources of Return, Total Return, and Interest on Interest173
- Examples174
- Sources of Return174
- How to Analyze this Problem174
- Some Observations on These Examples177
- Problems179
- Chapter 14. Volatility and Its Measures181
- What is Volatility?181
- Why do We Care about Volatility?182
- What Volatility Measures Can do for You182
- Measures of Volatility183
- Properties of Volatility183
- Bond Investment Management Interlude185
- Measuring Volatility by Measuring Price Change per Unit Change in Yield185
- Chapter Summary185
- Chapter 15. Duration187
- Historical Background188
- How Long will a Flow of Funds be Outstanding?188
- Modified Duration192
- Payback Interlude192
- A Plea for Payback193
- Calculation of Macaulay Duration and Modified Duration193
- Using Modified Duration to Predict Price Change195
- Dollar Duration196
- A Pictorial View of Duration196
- A Misconception about Duration198
- Portfolio Duration198
- Computing Portfolio Duration199
- Using Duration as a Portfolio Management Tool200
- Duration for Bonds with Embedded Options200
- Negative Duration201
- Problems with Duration as a Measure201
- Chapter Summary201
- Chapter 16. Convexity205
- Convexity206
- Convexity Calculation207
- Addition for Convexity207
- Modified Prices for Convexity208
- Dollar Convexity208
- Meaning of Convexity208
- A Misconception about Convexity209
- Portfolio Convexity209
- Chapter Summary209
- Chapter 17. The Mathematical Development of Duration, Convexity, and the Equation to Predict New Bon211
- Derivation of Duration212
- Negative Duration214
- Derivation of Convexity214
- Negative Convexity216
- Taylor’s Series Expansion216
- Reasons for the Equations and Additional Factors Introduced in the Previous Two Chapters217
- Chapter 18. Probability and Some Applications to Finance219
- Elementary Concepts in Probability: A Review219
- Examples of Probabilities220
- Independent Events220
- The Gambler’s Fallacy220
- Probability as a Mathematical Model221
- Use of the Word PopulationŽ221
- Sources of Probabilities221
- Probability Distribution Functions226
- The Binomial Distribution226
- Continuous Probability Distributions227
- The Normal Distribution229
- Statistics and Statistical Analysis230
- Measures of Central Tendency231
- Measure of Dispersions232
- Applications to Insurance233
- Chapter Summary234
- Suggestions for Further Reading and Study237
- Chapter 19. The Term Structure of Interest Rates, the Expectations Hypothesis, and Implied Forward R239
- The Term Structure of Interest Rates240
- Shapes of Yield Curves240
- The Expectations Hypothesis241
- Implied Spot Rates and Bootstrapping a Spot Yield Curve243
- Computing the Spot Rates244
- Calculation of Spot Rates244
- Using the Treasury Spot Rate in the Treasury Market245
- Using Spot Rates with Other Bonds246
- Implied Future Forward Rates247
- Other Term Structure Hypotheses248
- Risk Premium Hypothesis248
- Liquidity Preference Hypothesis248
- Market Segmentation Hypothesis248
- Discussion of the Various Hypotheses249
- Chapter Summary249
- Suggestions for Further Study251
- Chapter 20. Variable and Uncertain Cash Flows253
- Valuing a Varying Series of Cash Flows Using the Same Interest Rate, Varying Interest Rates, and Pro254
- Sources of the Probabilities You Use256
- Sources of the Interest Rates You Use256
- Applying Probability Concepts to Value a Variable or Uncertain Flow of Funds257
- Different Size Payments with Different Probabilities of Being Paid at the Same Time258
- Applying These Concepts to Life Insurance259
- Discussion of the Life Insurance Application261
- Using Different Interest Rates262
- How to Compute an Annual Premium for the Insurance262
- Calculating Life Insurance Company Reserves263
- Explanation of Year 2 Income and Expenses264
- Applying These Totals to Actual Insurance Company Operations264
- Reserves for Other Insurance Companies265
- Computing the Value of a Pension265
- Applications to Project Analysis268
- Applications to Bonds: Weighted Average Duration and Effective Duration268
- The Advantages and Disadvantages of Effective Duration270
- Chapter Summary271
- Suggestions for Further Study272
- Chapter 21. Mortgage-Backed Securities273
- What is a Mortgage?273
- How a Level-Payment Self-Amortizing Mortgage Works274
- The Equation for Level-Payment, Self-Amortizing Mortgages275
- Variable Rate Mortgages276
- Points277
- Mortgage Pools278
- Pass-Through Securities278
- Pay-Through Securities (Collateralized Mortgage Obligations (CMOS))278
- Cash Flows for Mortgages280
- Prepayment Models281
- Mortgage-Backed Investment Management: Application of Duration and Probability Concepts282
- Chapter 22. Futures Contracts287
- Cash, Forward, and Futures Trades287
- The Cross Hedge290
- The Need for Hedging Management291
- The Futures Contract291
- Settlement of a Futures Contract292
- Financial Futures292
- Hedging with Financial Futures293
- Cost of Carry294
- Conversion Factors295
- Conversion Factor Equation CBOT U.S. 2-Year Treasury Note298
- Conversion Factor Equation CBOT U.S. 5-Year Treasury Note300
- Conversion Factor Equation CBOT U.S. 10-Year Treasury Note300
- Conversion Factor Equation CBOT U.S. 30-Year Treasury Bond301
- Understanding the Equations for Computing Conversion Factors302
- Understanding Deliverable Grades of Treasury Securities304
- Web Sites304
- Chapter Summary305
- Chapter 23. Options307
- What is an Option?308
- Purposes of Options309
- Factors that Determine Option Prices311
- Black-Scholes Options Pricing Model312
- The Assumptions for Black-Scholes312
- Understanding These Assumptions313
- An Immediate Problem with Black-Scholes for Bonds314
- Hedging and Hedging Ratios (The Greeks)315
- The Put-Call Parity Relationship315
- Hedging Ratios (The Greeks)316
- Another Mathematical Approach to Continuous Functions, as Part of the Development of the Black-Schol317
- Other Approaches Fractal Analysis318
- Chapter Summary319
- Suggestions for Further Study321
- Index323
An introduction to common fixed income instruments and mathematics, this book offers explanations, exercises, and examples without demanding sophisticated mathematics. Not only does the author use his business and teaching experience to highlight the fundamentals of investment and management decision-making, but he also offers questions and exercises that suggest the applicability of fixed income mathematics. Written for the reader with a general mathematics background, this self-teaching book is suffused with examples that also make it a handy reference guide. It should serve as a gateway to financial mathematics and to increased competence in business analysis.
* An easy-to-understand introduction to the mathematics of common fixed income instruments
* Offers students explanations, exercises, and examples without demanding sophisticated mathematics
* Uses international comparisons to illustrate how interest is compounded
* An easy-to-understand introduction to the mathematics of common fixed income instruments
* Offers students explanations, exercises, and examples without demanding sophisticated mathematics
* Uses international comparisons to illustrate how interest is compounded