Mathematical Finance

M. J. ALHABEEB, PhD, is Professor of Economics and Finance at the University of Massachusetts Amherst. A recipient of the Academy of Educational Leadership's Outstanding Teaching Award for Innovative and Creative Teaching, Dr. Alhabeeb has been teaching finance and various courses in economics for more than thirty years.

An introduction to the mathematical skills needed to understand finance and make better financial decisions
 
Mathematical Finance enables readers to develop the mathematical skills needed to better understand and solve financial problems that arise in business, from small entrepreneurial operations to large corporations, and to also make better personal financial decisions. Despite the availability of automated tools to perform financial calculations, the author demonstrates that a basic grasp of the underlying mathematical formulas and tables is essential to truly understand finance.
 
The book begins with an introduction to the most fundamental mathematical concepts, including numbers, exponents, and logarithms; mathematical progressions; and statistical measures. Next, the author explores the mathematics of the time value of money through a discussion of simple interest, bank discount, compound interest, and annuities. Subsequent chapters explore the mathematical aspects of various financial scenarios, including:
* Mortgage debt, leasing, and credit and loans
* Capital budgeting, depreciation, and depletion
* Break-even analysis and leverage
* Investing, with coverage of stocks, bonds, mutual funds, options, cost of capital, and ratio analysis
* Return and risk, along with a discussion of the Capital Asset Pricing Model (CAPM)
* Life annuities as well as life, property, and casualty insurance
 
Throughout the book, numerous examples and exercises present realistic financial scenarios that aid readers in applying their newfound mathematical skills to devise solutions. The author does not promote the use of financial calculators and computers, but rather guides readers through problem solving using formulas and tables with little emphasis on derivations and proofs.
 
Extensively class-tested to ensure an easy-to-follow presentation, Mathematical Finance is an excellent book for courses in business, economics, and mathematics of finance at the upper-undergraduate and graduate levels. The book is also appropriate for consumers and entrepreneurs who need to build their mathematical skills in order to better understand financial problems and make better financial choices.

Preface xv
 
UNIT I MATHEMATICAL INTRODUCTION 1
 
1 Numbers, Exponents, and Logarithms 3
 
1.1. Numbers, 3
 

1.2. Fractions, 3
 
1.3. Decimals, 5
 
1.4. Repetends, 6
 
1.5. Percentages, 7
 
1.6. Base Amount, Percentage Rate, and Percentage Amount, 8
 
1.7. Ratios, 9
 
1.8. Proportions, 10
 
1.9. Aliquots, 10
 
1.10. Exponents, 11
 
1.11. Laws of Exponents, 11
 
1.12. Exponential Function, 12
 
1.13. Natural Exponential Function, 13
 
1.14. Laws of Natural Exponents, 14
 
1.15. Scientific Notation, 15
 
1.16. Logarithms, 15
 
1.17. Laws of Logarithms, 16
 
1.18. Characteristic, Mantissa, and Antilogarithm, 16
 
1.19. Logarithmic Function, 18
 
2 Mathematical Progressions 20
 
2.1. Arithmetic Progression, 20
 
2.2. Geometric Progression, 23
 
2.3. Recursive Progression, 26
 
2.4. Infinite Geometric Progression, 28
 
2.5. Growth and Decay Curves, 29
 
2.6. Growth and Decay Functions with a Natural Logarithmic Base, 34
 
3 Statistical Measures 35
 
3.1. Basic Combinatorial Rules and Concepts, 35
 
3.2. Permutation, 37
 
3.3. Combination, 40
 
3.4. Probability, 41
 
3.5. Mathematical Expectation and Expected Value, 44
 
3.6. Variance, 46
 
3.7. Standard Deviation, 48
 
3.8. Covariance, 49
 
3.9. Correlation, 50
 
3.10. Normal Distribution, 52
 
Unit I Summary 54
 
List of Formulas 55
 
Exercises for Unit I 60
 
UNIT II MATHEMATICS OF THE TIME VALUE OF MONEY 63
 
Introduction 65
 
1 Simple Interest 67
 
1.1. Total Interest, 67
 
1.2. Rate of Interest, 67
 
1.3. Term of Maturity, 68
 
1.4. Current Value, 68
 
1.5. Future Value, 69
 
1.6. Finding n and r When the Current and Future Values are Both Known, 69
 
1.7. Simple Discount, 70
 
1.8. Calculating the Term in Days, 72
 
1.9. Ordinary Interest and Exact Interest, 73
 
1.10. Obtaining Ordinary Interest and Exact Interest in Terms of Each Other, 73
 
1.11. Focal Date and Equation of Value, 75
 
1.12. Equivalent Time: Finding an Average due Date, 78
 
1.13. Partial Payments, 80
 
1.14. Finding the Simple Interest Rate by the Dollar-Weighted Method, 81
 
2 Bank Discount 83
 
2.1. Finding FV Using the Discount Formula, 84
 
2.2. Finding the Discount Term and the Discount Rate, 84
 
2.3. Difference Between a Simple Discount and a Bank Discount, 85
 
2.4. Comparing the Discount Rate to the Interest Rate, 87
 
2.5. Discounting a Promissory Note, 88
 
2.6. Discounting a Treasury Bill, 90
 
3 Compound Interest 93
 
3.1. The Compounding Formula, 94
 
3.2. Finding the Current Value, 97
 
3.3. Discount Factor, 98
 
3.4. Finding the Rate of Compound Interest, 100
 
3.5. Finding the Compounding Term, 100
 
3.6. The Rule of 72 and Other Rules, 101
 
3.7. Effective Interest Rate, 102
 
3.8. Types of Compounding, 104
 
3.9. Continuous Compounding, 105
 
3.10. Equations of Value for a Compound Interest, 106
 
3.11. Equated Time For a Compound Interest, 108
 
4 Annuities 110
 
4.1. Types of Annuities, 110
 
4.2. Future Value of an Ordinary Annuity, 111
 
4.3. Current Value of an Ordinary Annuity, 114
 
4.4. Finding the Payment of an Ordinary Annuity, 116
 
4.5. Finding the Term of an Ordina