Financial Mathematics For Actuarial Science-The Theory of Interest

395 Pages·2020·6.444 MB·\395

by  Richard James Wilders (Author)

FINANCIAL MATHEMATICS
FOR ACTUARIAL SCIENCE
THE THEORY OF INTEREST
Richard James Wilders
Financial
Mathematics for
Actuarial Science
The Theory of Interest

Financial
Mathematics for
Actuarial Science
The Theory of Interest
Richard James Wilders
North Central College
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Contents
Preface xi
Author xiii
1 Overview and Mathematical Prerequisites 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Calculators and Computers . . . . . . . . . . . . . . . . . . . 1
1.2.1 Sequences and Series . . . . . . . . . . . . . . . . . . . 6
1.3 Approximation Techniques . . . . . . . . . . . . . . . . . . . 9
1.3.1 Newton’s Method (Also Called the Newton-Raphson
Method) . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Approximations Using Taylor Series . . . . . . . . . . 11
1.4 Exponents and Logarithms . . . . . . . . . . . . . . . . . . . 14
Exercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Measuring Interest 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Accumulation and Amount Functions . . . . . . . . . . 19
2.3 The Effective Rate of Interest . . . . . . . . . . . . . . . . . 22
2.4 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Other Accumulation Functions . . . . . . . . . . . . . . . . . 32
2.7 Present and Future Value: Equations of Value . . . . . . . . 36
2.8 Nominal and Effective Rates of Interest . . . . . . . . . . . . 40
2.9 Discount Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.9.1 Nominal Rate of Discount . . . . . . . . . . . . . . . . 51
2.10 Forces of Interest and Discount . . . . . . . . . . . . . . . . . 54
2.10.1 Varying Rates of Interest . . . . . . . . . . . . . . . . 57
Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Solving Problems in Interest 67
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Measuring Time Periods: Simple Interest
and Fractional Time Periods . . . . . . . . . . . . . . . . . . 67
3.3 Fractional Time Periods . . . . . . . . . . . . . . . . . . . . . 69
3.4 Equations of Value at any Time . . . . . . . . . . . . . . . . 70
vii
viii Contents
3.5 Unknown Time . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6 Doubling Time . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6.1 The Rule of 72 . . . . . . . . . . . . . . . . . . . . . . 82
3.7 Finding the Rate of Interest . . . . . . . . . . . . . . . . . . 82
3.8 Deposits and Withdrawals: Cash Flow Problems . . . . . . . 86
Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 92
4 Annuities 97
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Fixed Term Annuities-Immediate with Constant Payments . 98
4.3 Fixed Term Annuities-Due . . . . . . . . . . . . . . . . . . . 108
4.4 The Value of an Annuity at any Date . . . . . . . . . . . . . 114
4.4.1 The Value of an Annuity prior to its Inception . . . . 114
4.4.2 The Value of an Annuity after the Final Payment
Is Made . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.3 The Value of an Annuity at any Time between the
First and Last Payments. . . . . . . . . . . . . . . . . 117
4.5 Perpetuities . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.6 Non Integer Time Periods . . . . . . . . . . . . . . . . . . . . 122
4.7 Unknown Time . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.8 Unknown Rate of Interest . . . . . . . . . . . . . . . . . . . . 124
4.9 Varying Rates of Interest . . . . . . . . . . . . . . . . . . . . 125
4.9.1 Present Value of Annuities with Varying Interest
Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.9.2 Accumulated Value of Annuities under Varying
Interest Rates . . . . . . . . . . . . . . . . . . . . . . . 126
4.10 Annuities Payable at Different Frequencies than Interest Is
Convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.11 Alternative Method: Annuities Payable Less
Frequently than Interest Is Convertible . . . . . . . . . . . . 131
4.12 Annuities Paid More Frequently than Interest Is Converted . 135
4.13 Perpetuities Paid More Frequently than Interest Is Convertible 138
4.14 Annuities with Varying Payments . . . . . . . . . . . . . . . 139
4.15 Varying Perpetuities . . . . . . . . . . . . . . . . . . . . . . . 142
4.16 Varying Annuities Paid Less Frequently than Interest Is
Convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.17 Continuous Annuities . . . . . . . . . . . . . . . . . . . . . . 146
Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 148
5 Amortization Schedules and Sinking Funds 153
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2 Amortization Method . . . . . . . . . . . . . . . . . . . . . . 153
5.3 Amortization of a Loan . . . . . . . . . . . . . . . . . . . . . 156
5.4 Methods for Computing the Loan Balance . . . . . . . . . . 157
Contents ix
5.5 Allocation of Loan Payments between Principal and Interest 161
5.6 Formulas for the Balance, Amount to Interest, and Amount to
Principal at any Time . . . . . . . . . . . . . . . . . . . . . . 165
5.7 Examples Using the TI BA II Plus to Create
Amortization Tables . . . . . . . . . . . . . . . . . . . . . . . 170
5.8 Creating Annualized Amortization Tables
Using Excel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.9 How to Construct the Sinking Fund Schedule
in Excel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.10 Amortization with Non-Standard Payments . . . . . . . . . . 186
5.11 Truth in Lending . . . . . . . . . . . . . . . . . . . . . . . . 189
5.12 Real Estate Loans – Home Loans . . . . . . . . . . . . . . . 191
5.13 Balloon and Drop Payments . . . . . . . . . . . . . . . . . . 195
5.14 Problems Involving Loans . . . . . . . . . . . . . . . . . . . . 196
5.15 Important Concepts from this Chapter . . . . . . . . . . . . 198
Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 198
6 Yield Rates 207
6.1 Discounted Cash Flow Analysis . . . . . . . . . . . . . . . . 207
6.1.1 Using the TI BA II Plus Cash Flow Worksheet
to Compute Internal Rate of Return . . . . . . . . . . 211
6.2 Uniqueness of the Yield Rate . . . . . . . . . . . . . . . . . . 213
6.3 Reinvestment . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.4 Interest Measurement of a Fund . . . . . . . . . . . . . . . . 220
6.4.1 Dollar-Weighted Estimate for the IRR . . . . . . . . . 223
6.4.2 Time-Weighted Estimate for the IRR . . . . . . . . . 226
6.5 Selling Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
6.6 Investment Year and Portfolio Methods . . . . . . . . . . . . 233
Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 236
7 Bonds 241
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.2 Basic Bond Terminology . . . . . . . . . . . . . . . . . . . . 244
7.3 Pricing a Bond . . . . . . . . . . . . . . . . . . . . . . . . . . 247
7.4 Premium and Discount . . . . . . . . . . . . . . . . . . . . . 252
7.5 Determination of Yield Rates . . . . . . . . . . . . . . . . . . 260
7.6 The Term Structure of Interest . . . . . . . . . . . . . . . . . 262
7.7 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 271
7.8 Converting from Forward Rates to Spot Rates . . . . . . . . 273
7.9 Price of a Bond between Coupon Payments . . . . . . . . . . 279
7.10 Callable Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 280
7.11 Bonds with Varying Payments . . . . . . . . . . . . . . . . . 283
Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . 285