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
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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
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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
8.1 Introduction 248
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
9.1 Introduction 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
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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
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