Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy

Preface xi

Acknowledgments xv

List of Abbreviations xvii

1. Background Mathematics 1

1.1 Arithmetic, Notation, and Formulas 1

1.2 Minus (Negative) Signs 5

1.3 Lists and Subscripted Variables 6

1.4 Changes 9

1.5 Exponents 11

1.6 Summations 12

1.7 Graphs and Charts 13

1.8 Approximations 18

1.9 Rates—Average and Instantaneous 19

1.10 Inequalities and Ranges of Numbers 22

Problems 23

2. Compound Interest 26

2.1 Some Mathematics 30

2.2 My Website Spreadsheet 31

2.3 Online Calculators 32

2.4 Scaling 32

2.5 Proration—Working Inside a Compounding Interval 33

2.6 Initial Charges and Effective Interest Rate 34

2.7 In the Limit—Continuous Compounding 35

Problems 37

3. Loan Amortization and Savings 38

3.1 Loans 38

3.2 Calculating the Payment Amount 44

3.3 Paying off a Loan Very Slowly 45

3.4 My Website Spreadsheet 46

3.5 A Note About Total Interest for a Year 47

3.6 Online Calculators 48

3.7 Loans with First Payment Due Immediately 48

3.8 Irregular Payments 49

3.9 Regular Savings 50

Problems 52

4. Mortgages 54

4.1 Online Calculators 55

4.2 Fixed Rate Mortgages 55

4.3 Adjustable Rate Mortgages (ARMs) 57

4.4 Balloon Loans 59

4.5 Up-Front Costs 59

Problems 60

5. Prepayment Penalties 61

5.1 Rule of 78 63

5.2 Other Prepayment Penalties 66

Problems 68

6. Credit Cards 69

6.1 Credit Card Statements 70

6.2 Transfers 73

6.3 Payment Allocation 75

6.4 Daily Balance 76

6.5 Some Calculation Examples 76

6.6 Grace Period 81

6.7 Changing Interest Rates 81

6.8 A Bankruptcy Spiral 81

6.9 Minimum Payment 83

6.10 Other Interest Calculation Approaches 83

6.11 Debit Cards—Something Completely Different 83

Problems 84

7. Present Value 86

7.1 Online Calculators 88

7.2 Doing It with My Spreadsheet 88

7.3 The Effect of Interest Rates on Present Value Calculations 90

7.4 Why This All Matters 91

7.5 A Very Involved Example: Writing Your Own Spreadsheet 91

7.6 Future Value 93

7.7 Present Value of Prepayment Penalties 94

Problems 95

8. Comparing Loans 96

8.1 Up-Front Costs 99

8.2 Adjustable Rate Mortgages (ARMs) 100

8.3 A Few Last Words 102

Problems 103

9. Taxation and Inflation 105

9.1 Understanding Personal Federal Income Tax Rates 105

9.2 Online Tax Calculators 111

9.3 Taxation of Earned Interest 111

9.4 Deductible Interest 112

9.5 Deferred Taxation Savings 112

9.6 Online Deferred Taxation Plan Calculators 115

9.7 Inflation 115

Problems 118

10. Life Insurance 119

10.1 What Is an Insurance Policy? 119

10.2 Probability 120

10.3 Introduction to the Life Tables 126

10.4 Expected Values 128

10.5 Term Insurance 130

10.6 Time Payments 133

10.7 Decreasing Term Insurance 134

10.8 Insurance for the Rest of Your Life 134

10.9 Whole Life Insurance 136

10.10 Breaking Down the Year 139

Problems 142

11. Annuities 144

11.1 A Benchmark Savings Plan 145

11.2 Immediate Annuity with Period Certain 145

11.3 Deferred Annuities 148

11.4 Life Annuities 150

11.5 Payments for Couples 153

11.6 Online Calculators 156

11.7 Variable Annuities 156

Problems 157

12. Reverse Mortgages and Viatical Settlements 158

12.1 Reverse Mortgages 158

12.2 Viatical Settlements 162

Problems 163

13. Investing: Risk versus Reward 164

13.1 Stocks 165

13.2 Portfolios 167

13.3 Calculators 171

13.4 Dollar Cost Averaging 173

13.5 Short Sales 174

13.6 Stock Dividends 174

13.7 Bonds 175

13.8 Options 175

13.9 Online Calculators and Listings 179

13.10 Ponzi Schemes and Other Scams 180

Problems 185

Reference 186

14 Gambling 187

14.1 Probability and Odds 187

14.2 Probability and Expected Return 188

14.3 Pari-Mutuel Betting 193

Problems 197

15 Spreadsheet Calculators 198

15.1 Introduction to the Spreadsheets 198

15.2 Some Programming Notes 205

16 Solutions 207

16.1 Chapter 1 207

16.2 Chapter 2 210

16.3 Chapter 3 213

16.4 Chapter 4 215

16.5 Chapter 5 216

16.6 Chapter 6 218

16.7 Chapter 7 220

16.8 Chapter 8 221

16.9 Chapter 9 223

16.10 Chapter 10 226

16.11 Chapter 11 229

16.12 Chapter 12 233

16.13 Chapter 13 234

16.14 Chapter 14 237

Index 240