Fixed Income Mathematics

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收藏 2010-03-18

1 Introduction—Who this Book is for and What it Hopes to
Accomplish
Historical Background—The Big Change in Investment, Loan,
and Money Management 1
What this Book Hopes to Accomplish 3
What Sort of Problems Might this Book Help You to Solve? 4
Who this Book is Meant to Address 4
The Mathematical Knowledge Required for this Book 5
The Role of Examples and Problems in this Book 6
2 Interest, Its Calculation, and Return on Investment
A General Introduction to Interest 7
How to Compute Interest 8
Notation 9
Percentage Rate and Time Period 10
Return on Investment 11
Analysis of Investments or Returns without Explicit Money Values,
and Intangible Investments and Returns 14
Chapter Summary 16
3 Compound Interest
What is Compound Interest? 20
Using Compound Interest Tables 21
Looking at the Compound Interest Tables 23
Compounding within a Period 23
The Equations for Compound Interest–Compounding within
a Period 24Continuous Compounding: How it Works and When it
Applies 25
The Derivation of the Equations for Continuous Compounding 26
What is a Mathematical Model? 28
Some Famous Mathematical Models 29
Reasons for Using Continuous Functions in Financial Models 30
A Business Example of Use of Continuous Functions 30
Further Reﬂections on Approach 3 31
Computing i, Given S, SNT, T, and N 31
Accuracy Requirements 36
Legal Requirements for Accuracy 36
An Example from Compound Interest 36
Using Tables and Interpolating between Values 37
The Rule of 72 37
A Zero Interest Rate? 37
Negative Interest Rates? 38
Real and Nominal Rates 38
Chapter Summary 39
Suggestions for Further Study 41
4 Present Values
What is Present Value? 52
The Equation for Present Value 52
The General Equation for Present Value 53
The Present Value Tables 54
Using Present Values to Make Project Decisions 54
Example of Project Analysis 55
Using Different Interest Rates in the Analysis 57
The Equations for Flow of Funds Analysis 57
The Various Number Systems and What They Mean 58
Solving Polynomial Equations 61
Practical Considerations in Using Calculators and Computers to Solve
Polynomial Equations 62
Using the Bisection Method to Find Real Solutions 63
What if the Exponents are not Integers? 64
Chapter Summary 64
Suggestions for Further Study 67
5 Annuities Certain
What is an Annuity Certain? 77
Examples of Annuities Certain 78
Why Annuities Certain are Important 78
The Equation for the Present Value of an Annuity Certain 79
A Look at the Tables for an Annuity Certain 80
Solving for the Interest Rate, Given the Annuity Certain and
Its Cost 80
The Perpetuity 80
The Annuity Due 81
Further Comments 82
Analysis and Calculation of Some Combination Annuities Certain 83
Chapter Summary 83
6 Bond Price Calculation
What is a Bond? 104
How Bonds are Described 105
How to Read a Bond Market Report 105
What is a Call Feature? 106
What is a Put Option? 108
Discount Securities 108
The General Equation for Computing a Bond Price,
Given the Yield 108
A Note on Yield 110
A Note on Accrued Days in the Settlement Period
(A in Equation 6.1) and Dated Date 111
Analysis of the Equation 111
Standards of Accuracy 114
Pricing Zero Coupon Bonds 115
Pricing to a Call Feature 115
Examples of Bond Price Calculations 116
Amortization of Premium and Accrual of Discount 117
Accrual of Discount 117
Amortization of Premium 119
A Portfolio Management Interlude 120
Pricing a Bond to a Call 121
A Look at a Basis Book 123
Basic Rules for Prices and Yields 123
The Shape of the Price-Yield Curve 124
Dirty Price and Clean Price 125
Chapter Summary 126
Suggestion for Further Study 128
7 The Future Value (or Amount) of an Annuity
Uses of the Future Value of an Annuity 131
The Equation for the Future Value of an Annuity 132
The Tables for Amount of Annuity 133
Some Investment Policy Implications 133
Chapter Summary 133
8 Accrued Interest
What is Accrued Interest? 143
Bonds that Accrue Interest 144
The Equation for Accrued Interest 144
Chapter Summary 146
9 Discount Yield
Discount Yield 150
Calculation of the Discount and Price for Discount Securities 150
What is a Treasury Bill? 150
The Day-Count Conventions for T-Bills 151
Price Calculations for Discount Municipal Securities 152
Bond Equivalent Yield (BEY): What It Means and How to
Compute It 152
Derivation of the Bond Equivalent Yield Equations 153
Why We Care about Bond Equivalent Yield (BEY) 154
A Historical Note 155
Taxation of Income from Treasury Bills 155
Chapter Summary 155
10 Calculations for Other Securities
Certiﬁcates of Deposit 159
Repurchase Agreements 160
Uses of Repos and Reverse Repos 161
Pricing Repos 161
Chapter Summary 161
Class Project 161
11 Quotations and Bond Market Reports
Long-Term Instruments 163
Discount Instruments 165
Chapter Summary 166

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ibanker

2010-3-18 22:46:46

12 Types of Yields
Nominal (or Coupon) Yield 168
Discount Yield 168
Current Yield 168
True (or Bond) Yield 169
A Method to Compute Approximate Bond Yield, Given Price 170
Chapter Summary 171
13 Sources of Return, Total Return, and Interest on Interest
Examples 174
Sources of Return 174
How to Analyze this Problem 174
Some Observations on These Examples 177
Problems 179
14 Volatility and Its Measures
What is Volatility? 181
Why do We Care about Volatility? 182
What Volatility Measures Can do for You 182
Measures of Volatility 183
Properties of Volatility 183
Bond Investment Management Interlude 185
Measuring Volatility by Measuring Price Change per Unit Change
in Yield 185
Chapter Summary 185
15 Duration
Historical Background 188
How Long will a Flow of Funds be Outstanding? 188
Modiﬁed Duration 192
Payback Interlude 192
A Plea for Payback 193
Calculation of Macaulay Duration and Modiﬁed Duration 193
Using Modiﬁed Duration to Predict Price Change 195
Dollar Duration 196
A Pictorial View of Duration 196
A Misconception about Duration 198
Portfolio Duration 198
Computing Portfolio Duration 199
Using Duration as a Portfolio Management Tool 200
Duration for Bonds with Embedded Options 200
Negative Duration 201
Problems with Duration as a Measure 201
Chapter Summary 201
16 Convexity
Convexity 206Convexity Calculation 207
Addition for Convexity 207
Modiﬁed Prices for Convexity 208
Dollar Convexity 208
Meaning of Convexity 208
A Misconception about Convexity 209
Portfolio Convexity 209
Chapter Summary 209
17 The Mathematical Development of Duration, Convexity,
and the Equation to Predict New Bond Prices,
Given Yield Changes
Derivation of Duration 212
Negative Duration 213
Derivation of Convexity 214
Negative Convexity 215
Taylor’s Series Expansion 215
Reasons for the Equations and Additional Factors Introduced in
the Previous Two Chapters 216
18 Probability and Some Applications to Finance
Elementary Concepts in Probability: A Review 219
Examples of Probabilities 220
Independent Events 220
The Gambler’s Fallacy 220
Probability as a Mathematical Model 221
Use of the Word “Population” 221
Sources of Probabilities 221
Probability Distribution Functions 226
The Binomial Distribution 226
Continuous Probability Distributions 227
The Normal Distribution 229
Statistics and Statistical Analysis 230
Measures of Central Tendency 231
Measure of Dispersions 232
Applications to Insurance 233
Chapter Summary 234
Suggestions for Further Reading and Study 237
19 The Term Structure of Interest Rates, the Expectations
Hypothesis, and Implied Forward Rates
The Term Structure of Interest Rates 240
Shapes of Yield Curves 240The Expectations Hypothesis 241
Implied Spot Rates and Bootstrapping a Spot Yield Curve 243
Computing the Spot Rates 244
Calculation of Spot Rates 244
Using the Treasury Spot Rate in the Treasury Market 245
Using Spot Rates with Other Bonds 246
Implied Future Forward Rates 247
Other Term Structure Hypotheses 248
Risk Premium Hypothesis 248
Liquidity Preference Hypothesis 248
Market Segmentation Hypothesis 248
Discussion of the Various Hypotheses 249
Chapter Summary 249
Suggestions for Further Study 251
20 Variable and Uncertain Cash Flows
Valuing a Varying Series of Cash Flows Using the Same Interest Rate,
Varying Interest Rates, and Probabilities 254
Sources of the Probabilities You Use 256
Sources of the Interest Rates You Use 256
Applying Probability Concepts to Value a Variable or Uncertain
Flow of Funds 257
Different Size Payments with Different Probabilities of Being Paid
at the Same Time 258
Applying These Concepts to Life Insurance 259
Discussion of the Life Insurance Application 261
Using Different Interest Rates 262
How to Compute an Annual Premium for the Insurance 262
Calculating Life Insurance Company Reserves 263
Explanation of Year 2 Income and Expenses 264
Applying These Totals to Actual Insurance Company Operations 264
Reserves for Other Insurance Companies 265
Computing the Value of a Pension 265
Applications to Project Analysis 268
Applications to Bonds: Weighted Average Duration and Effective
Duration 268
The Advantages and Disadvantages of Effective Duration 270
Chapter Summary 271
Suggestions for Further Study 272
21 Mortgage-Backed Securities
What is a Mortgage? 273
How a Level-Payment Self-Amortizing Mortgage Works 274The Equation for Level-Payment, Self-Amortizing Mortgages 275
Variable Rate Mortgages 276
Points 277
Mortgage Pools 278
Pass-Through Securities 278
Pay-Through Securities (Collateralized Mortgage Obligations
(CMOS)) 278
Cash Flows for Mortgages 280
Prepayment Models 281
Mortgage-Backed Investment Management: Application of
Duration and Probability Concepts 282
22 Futures Contracts
Cash, Forward, and Futures Trades 287
The Cross Hedge 290
The Need for Hedging Management 291
The Futures Contract 291
Settlement of a Futures Contract 292
Financial Futures 292
Hedging with Financial Futures 293
Cost of Carry 294
Conversion Factors 295
Conversion Factor Equation: CBOT U.S. 2-Year Treasury Note 298
Conversion Factor Equation: CBOT U.S. 5-Year Treasury Note 300
Conversion Factor Equation: CBOT U.S. 10-Year Treasury Note 300
Conversion Factor Equation: CBOT U.S. 30-Year Treasury Bond 301
Understanding the Equations for Computing Conversion Factors 302
Understanding Deliverable Grades of Treasury Securities 304
Web Sites 304
Chapter Summary 305
23 Options
What is an Option? 308
Purposes of Options 309
Factors that Determine Option Prices 311
Black-Scholes Options Pricing Model 312
The Assumptions for Black-Scholes 312
Understanding These Assumptions 313
An Immediate Problem with Black-Scholes for Bonds 314
Hedging and Hedging Ratios (The Greeks) 315
The Put-Call Parity Relationship 315
Hedging Ratios (The Greeks) 316

Another Mathematical Approach to Continuous Functions,
as Part of the Development of the Black-Scholes Model 317
Other Approaches: Fractal Analysis 318
Chapter Summary 319
Suggestions for Further Study 321
Index 323