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Mathematical Finance

Book Description

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.

    Table of Contents

    1. Coverpage
    2. Titlepage
    3. Copyright
    4. Dedication
    5. Contents
    6. Preface
    7. UNIT I MATHEMATICAL INTRODUCTION
      1. 1 Numbers, Exponents, and Logarithms
        1. 1.1. Numbers
        2. 1.2. Fractions
        3. 1.3. Decimals
        4. 1.4. Repetends
        5. 1.5. Percentages
        6. 1.6. Base Amount, Percentage Rate, and Percentage Amount
        7. 1.7. Ratios
        8. 1.8. Proportions
        9. 1.9. Aliquots
        10. 1.10. Exponents
        11. 1.11. Laws of Exponents
        12. 1.12. Exponential Function
        13. 1.13. Natural Exponential Function
        14. 1.14. Laws of Natural Exponents
        15. 1.15. Scientific Notation
        16. 1.16. Logarithms
        17. 1.17. Laws of Logarithms
        18. 1.18. Characteristic, Mantissa, and Antilogarithm
        19. 1.19. Logarithmic Function
      2. 2 Mathematical Progressions
        1. 2.1. Arithmetic Progression
        2. 2.2. Geometric Progression
        3. 2.3. Recursive Progression
        4. 2.4. Infinite Geometric Progression
        5. 2.5. Growth and Decay Curves
        6. 2.6. Growth and Decay Functions with a Natural Logarithmic Base
      3. 3 Statistical Measures
        1. 3.1. Basic Combinatorial Rules and Concepts
        2. 3.2. Permutation
        3. 3.3. Combination
        4. 3.4. Probability
        5. 3.5. Mathematical Expectation and Expected Value
        6. 3.6. Variance
        7. 3.7. Standard Deviation
        8. 3.8. Covariance
        9. 3.9. Correlation
        10. 3.10. Normal Distribution
      4. Unit I Summary
      5. List of Formulas
      6. Exercises for Unit I
    8. UNIT II MATHEMATICS OF THE TIME VALUE OF MONEY
      1. Introduction
      2. 1 Simple Interest
        1. 1.1. Total Interest
        2. 1.2. Rate of Interest
        3. 1.3. Term of Maturity
        4. 1.4. Current Value
        5. 1.5. Future Value
        6. 1.6. Finding n and r When the Current and Future Values are Both Known
        7. 1.7. Simple Discount
        8. 1.8. Calculating the Term in Days
        9. 1.9. Ordinary Interest and Exact Interest
        10. 1.10. Obtaining Ordinary Interest and Exact Interest in Terms of Each Other
        11. 1.11. Focal Date and Equation of Value
        12. 1.12. Equivalent Time: Finding an Average due Date
        13. 1.13. Partial Payments
        14. 1.14. Finding the Simple Interest Rate by the Dollar-Weighted Method
      3. 2 Bank Discount
        1. 2.1. Finding FV Using the Discount Formula
        2. 2.2. Finding the Discount Term and the Discount Rate
        3. 2.3. Difference Between a Simple Discount and a Bank Discount
        4. 2.4. Comparing the Discount Rate to the Interest Rate
        5. 2.5. Discounting a Promissory Note
        6. 2.6. Discounting a Treasury Bill
      4. 3 Compound Interest
        1. 3.1. The Compounding Formula
        2. 3.2. Finding the Current Value
        3. 3.3. Discount Factor
        4. 3.4. Finding the Rate of Compound Interest
        5. 3.5. Finding the Compounding Term
        6. 3.6. The Rule of 72 and Other Rules
        7. 3.7. Effective Interest Rate
        8. 3.8. Types of Compounding
        9. 3.9. Continuous Compounding
        10. 3.10. Equations of Value for a Compound Interest
        11. 3.11. Equated Time For a Compound Interest
      5. 4 Annuities
        1. 4.1. Types of Annuities
        2. 4.2. Future Value of an Ordinary Annuity
        3. 4.3. Current Value of an Ordinary Annuity
        4. 4.4. Finding the Payment of an Ordinary Annuity
        5. 4.5. Finding the Term of an Ordinary Annuity
        6. 4.6. Finding the Interest Rate of an Ordinary Annuity
        7. 4.7. Annuity Due: Future and Current Values
        8. 4.8. Finding the Payment of an Annuity Due
        9. 4.9. Finding the Term of an Annuity Due
        10. 4.10. Deferred Annuity
        11. 4.11. Future and Current Values of a Deferred Annuity
        12. 4.12. Perpetuities
      6. Unit II Summary
      7. List of Formulas
      8. Exercises for Unit II
    9. UNIT III MATHEMATICS OF DEBT AND LEASING
      1. 1 Credit and Loans
        1. 1.1. Types of Debt
        2. 1.2. Dynamics of Interest–Principal Proportions
        3. 1.3. Premature Payoff
        4. 1.4. Assessing Interest and Structuring Payments
        5. 1.5. Cost of Credit
        6. 1.6. Finance Charge and Average Daily Balance
        7. 1.7. Credit Limit vs. Debt Limit
      2. 2 Mortgage Debt
        1. 2.1. Analysis of Amortization
        2. 2.2. Effects of Interest Rate, Term, and Down Payment on the Monthly Payment
        3. 2.3. Graduated Payment Mortgage
        4. 2.4. Mortgage Points and the Effective Rate
        5. 2.5. Assuming a Mortgage Loan
        6. 2.6. Prepayment Penalty on a Mortgage Loan
        7. 2.7. Refinancing a Mortgage Loan
        8. 2.8. Wraparound and Balloon Payment Loans
        9. 2.9. Sinking Funds
        10. 2.10. Comparing Amortization to Sinking Fund Methods
      3. 3 Leasing
        1. 3.1. For the Lessee
        2. 3.2. For the Lessor
      4. Unit III Summary
      5. List of Formulas
      6. Exercises for Unit III
    10. UNIT IV MATHEMATICS OF CAPITAL BUDGETING AND DEPRECIATION
      1. 1 Capital Budgeting
        1. 1.1. Net Present Value
        2. 1.2. Internal Rate of Return
        3. 1.3. Profitability Index
        4. 1.4. Capitalization and Capitalized Cost
        5. 1.5. Other Capital Budgeting Methods
      2. 2 Depreciation and Depletion
        1. 2.1. The Straight-Line Method
        2. 2.2. The Fixed-Proportion Method
        3. 2.3. The Sum-of-Digits Method
        4. 2.4. The Amortization Method
        5. 2.5. The Sinking Fund Method
        6. 2.6. Composite Rate and Composite Life
        7. 2.7. Depletion
      3. Unit IV Summary
      4. List of Formulas
      5. Exercises for Unit IV
    11. UNIT V MATHEMATICS OF THE BREAK-EVEN POINT AND LEVERAGE
      1. 1 Break-Even Analysis
        1. 1.1. Deriving BEQ and BER
        2. 1.2. BEQ and BER Variables
        3. 1.3. Cash Break-Even Technique
        4. 1.4. The Break-even Point and the Target Profit
        5. 1.5. Algebraic Approach to the Break-Even Point
        6. 1.6. The Break-Even Point When Borrowing
        7. 1.7. Dual Break-Even Points
        8. 1.8. Other Applications of the Break-Even Point
        9. 1.9. BEQ and BER Sensitivity to their Variables
        10. 1.10. Uses and Limitations of Break-Even Analysis
      2. 2 Leverage
        1. 2.1. Operating Leverage
        2. 2.2. Operating Leverage, Fixed Cost, and Business Risk
        3. 2.3. Financial Leverage
        4. 2.4. Total or Combined Leverage
      3. Unit V Summary
      4. List of Formulas
      5. Exercises for Unit V
    12. UNIT VI MATHEMATICS OF INVESTMENT
      1. 1 Stocks
        1. 1.1. Buying and Selling Stocks
        2. 1.2. Common Stock Valuation
        3. 1.3. Cost of New Issues of Common Stock
        4. 1.4. Stock Value with Two-Stage Dividend Growth
        5. 1.5. Cost of Stock Through the CAPM
        6. 1.6. Other Methods for Common Stock Valuation
        7. 1.7. Valuation of Preferred Stock
        8. 1.8. Cost of Preferred Stock
      2. 2 Bonds
        1. 2.1. Bond Valuation
        2. 2.2. Premium and Discount Prices
        3. 2.3. Premium Amortization
        4. 2.4. Discount Accumulation
        5. 2.5. Bond Purchase Price Between Interest Days
        6. 2.6. Estimating the Yield Rate
        7. 2.7. Duration
      3. 3 Mutual Funds
        1. 3.1. Fund Evaluation
        2. 3.2. Loads
        3. 3.3. Performance Measures
        4. 3.4. The Effect of Systematic Risk (β)
        5. 3.5. Dollar-Cost Averaging
      4. 4 Options
        1. 4.1. Dynamics of Making Profits With Options
        2. 4.2. Intrinsic Value of Calls and Puts
        3. 4.3. Time Value of Calls and Puts
        4. 4.4. The Delta Ratio
        5. 4.5. Determinants of Option Value
        6. 4.6. Option Valuation
        7. 4.7. Combined Intrinsic Values of Options
      5. 5 Cost of Capital and Ratio Analysis
        1. 5.1. Before- and After-Tax Cost of Capital
        2. 5.2. Weighted-Average Cost of Capital
        3. 5.3. Ratio Analysis
        4. 5.4. The DuPont Model
        5. 5.5. A Final Word About Ratios
      6. Unit VI Summary
      7. List of Formulas
      8. Exercises for Unit VI
    13. UNIT VII MATHEMATICS OF RETURN AND RISK
      1. 1 Measuring Return and Risk
        1. 1.1. Expected Rate of Return
        2. 1.2. Measuring the Risk
        3. 1.3. Risk Aversion and Risk Premium
        4. 1.4. Return and Risk at the Portfolio Level
        5. 1.5. Markowitz’s Two-Asset Portfolio
        6. 1.6. Lending and Borrowing at a Risk-Free Rate of Return
        7. 1.7. Types of Risk
      2. 2 The Capital Asset Pricing Model (CAPM)
        1. 2.1. The Financial Beta (β)
        2. 2.2. The CAPM Equation
        3. 2.3. The Security Market Line
        4. 2.4. SML Swing by Risk Aversion
      3. Unit VII Summary
      4. List of Formulas
      5. Exercises for Unit VII
    14. UNIT VIII MATHEMATICS OF INSURANCE
      1. 1 Life Annuities
        1. 1.1. Mortality Table
        2. 1.2. Commutation Terms
        3. 1.3. Pure Endowment
        4. 1.4. Types of Life Annuities
      2. 2 Life Insurance
        1. 2.1. Whole Life Insurance Policy
        2. 2.2. Annual Premium: Whole Life Basis
        3. 2.3. Annual Premium: m-Payment Basis
        4. 2.4. Deferred Whole Life Policy
        5. 2.5. Deferred Annual Premium: Whole Life Basis
        6. 2.6. Deferred Annual Premium: m-Payment Basis
        7. 2.7. Term Life Insurance Policy
        8. 2.8. Endowment Insurance Policy
        9. 2.9. Annual Premium for the Endowment Policy
        10. 2.10. Less than Annual Premiums
        11. 2.11. Natural Premium vs. the Level Premium
        12. 2.12. Reserve and Terminal Reserve Funds
        13. 2.13. Benefits of the Terminal Reserve
        14. 2.14. How Much Life Insurance Should You Buy?
      3. 3 Property and Casualty Insurance
        1. 3.1. Deductibles and Co-Insurance
        2. 3.2. Health Care Insurance
        3. 3.3. Policy Limit
      4. Unit VIII Summary
      5. List of Formulas
      6. Exercises for Unit VIII
    15. References
    16. Appendix
    17. Index