kellison theory of interest

8/11/2019 Kellison Theory of Interest

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Contents

1

The measurement of interest

1

1 1

Introduction 1

1.2

1.3

1 4

Simple interest

1 5

Compound interest 7

1 6

Present value 10

1.7

1.8

1.9

1 10

Varying interest 28

1 11

Summary of results 29

The accumulation and amount functions 1

The effective rate

of

interest 4

The effective rate of discount 12

Nominal rates of interest and discount

17

Forces of interest and discount 22

2 Solution of problems in interest 35

2.1 Introduction 35

2.2 Obtaining numerical results 35

2.3 Determining time periods 38

2.4 The basic problem

40

2.5 Equations

of

value 42

2.6 Unknown time

2.7

2.8 Practical examples 51

Unknown rate

of

interest 48

3

Basicannuities

58

3.1 Introduction 58

3.2 Annuity-immediate 59

3.3 Annuity-due 62

3.4

3 5

Perpetuities 69

3 6

Annuity values on any date

65

Nonstandard terms and interest rates 71

x



xii

ontents

3.7 Unknown time 72

3.8 Unknown rate of interest 75

3.9 Varying interest 80

3 10 Annuities not involving compound interest 82

4

More general annuities

9

4.1 Introduction 95

4.2 Annuities payable at a different frequency than interest is

convertible 95

4.3 Further analysis of annuities payable less frequently than interest is

convertible 97

4.4 Further analysis of annuities payable more frequently than interest

is convertible 102

4.5 Continuous annuities 107

4.6 Basic varying annuities 109

4.7

4.8

4.9 Summary of results 121

More general varying annuities 117

Continuous varying annuities 119

5

Yieldrates 128

5 1

5.2

5 3

5.4

5 5

5.6

5.7

5.8

5.9

Introduction 128

Discounted cash flow analysis 129

Uniqueness of the yield rate 133

Reinvestment rates 136

Interest measurement of a fund 140

Time-weighted rates of interest 145

Portfolio methods and investment year methods 149

Capital budgeting 152

More general borrowingAending models 156

6

Amortization schedules and sinking funds

166

6.1

6.2

6.3

6.4

6 5

6.6

6 7

6.8

Introduction 166

Finding the outstanding loan balance 167

Amortization schedules 169

Sinking funds 175

Differing payment periods and interest conversion periods

18

1

Varying series

of

payments 185

Amortization with continuous payments 189

Step-rate amounts of principal 192



ontents

xiii

7

Bonds and other securities

7.1 Introduction 204

7.2 Types of securities 204

7.3 Price of a bond 208

7.4 Premium and discount 213

7.5 Valuation between coupon payment dates 220

7.6 Determination of yield rates 225

7.7 Callable bonds 230

7.8 Serial bonds 232

7.9 Some generalizations 234

7.10 Other securities 236

7.11 Valuation of securities 238

2 4

8

Practical applications


8.2 Truth in lending 248

8.3 Real estate mortgages 255

8.4 Approximate methods 260

8.5 Depreciation methods 270

8.6 Capitalized cost 277

8.7 Short sales 279

8.8 Modern financial instruments 282

48

9

More advanced financial analysis

294


9.2

9.3

9.4 Recognition of inflation 298

9.5

9.6 Yield curves 309

9.7 Interest rate assumptions 313

9.9 Immunization 320

9.10 Matching assets and liabilities 326

An economic rationale for interest 295

Determinants of the level of interest rates 296

Reflecting risk and uncertainty 302

O9.8 Duration 315

1

Stochastic approaches to interest

336

10 1

Introduction 336

10.2 Independent rates of interest 337

10.3 Dependent rates

of

interest 345



xiv

ontents

10.4

An asset pricing model 349

10 5

An option pricing model 355

10 6

A random walk model 361

10 7 Scenario testing 365

10 8 A review of the literature 37

Appendix I

Appendix I1

Table

of

compound interest functions 376

Table numbering the days

of

the year

393

Appendix I11

Basic mathematical review 394

Appendix IV

Statistical background

396

Appendix V

Iteration methods 399

Appendix

VI Further analysis

of

varying annuities 4 1

Appendix VII Illustrative mortgage loan amortization schedule 4 2

Appendix VIII

Full immunization 4 6

Appendix IX

Derivation of the variance of an annuity 4 8

Appendix X Derivation of the Black-Scholes formula

41

Bibliography 412

Answers to the exercises

415

Glossaryofnotation

431

Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . o . . . D . .39

kellison theory of interest

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