Elementary Mathematics Review

Increasing sophistication in financial markets, design of financial securities and computational technology has intensified the business practitionerís need for understanding of mathematics. In order to understand and participate in financial decision making, students and practitioners of finance need to be familiar with the basic tools and techniques used in the formal analysis of financial problems. Mathematics is among the most important of these  tools. In addition to being crucial to the analysis of financial problems, mathematics is a language that is most useful to communication of financial concepts and results. Yet many of us approach mathematics very tentatively, feeling overwhelmed by its apparent complexity and insecure about our own preparation to understand and apply the variety of useful tools available to us. In many cases, our preparation for handling financial analysis merely requires a review of mathematics that we may have been quite comfortable with in the past; in others, we may need to learn new tools and techniques in mathematics. The reviews presented through this web page endeavors to facilitate review and presentation of more elementary mathematics concepts sufficient to understand elementary finance and investments. Many students of finance, particularly at the introductory level, will require at least some review of mathematics tools, and most of these will be surprised at how quickly they can be grasped with only a little dedication and effort.

The mathematics review material for Dr. Teall's finance courses is prepared in chapters outlined below. Exercises and problems with solutions follow each chapter. Tables and figures for each chapter are usually located just prior to the exercises and problems. The review appendices also contain introductory information on the use of spreadsheets in finance. The chapters are provided in *.pdf format which can be downloaded by selecting the appropriate links below. After downloading, you will need to access this material with Adobe Acrobat Reader . If you do not have this program, it can be downloaded without cost by linking to Adobe's site , selecting the "Get Acrobat Reader link" and following instructions. Feel free to report any difficulties to or obtain any needed assistance from  John Teall .

Students enrolled in all of Dr. Teall's finance courses should be comfortable working with all of the material in Chapters 2 and 3 before the term starts. Sections 2.J, 2.K, 2.L and 3.C will be less important for introductory finance courses, though are still considered prerequisite material and may still be useful. A link to all exercise solutions follows the chapters listed below. Students in all of Dr. Teall's finance courses beyond the introductory level should be comfortable with all of the material presented in Chapters 2, 3, 4, 5, 6, 7 and 8.A to 8.D. Students enrolled in courses beyond the introductory finance course should also be familiar with all material normally presented in an introductory finance course. It is strongly recommended that reviews of relevant material be completed before the second week of classes. Students who are not comfortable with the appropriate prerequisite quantitative material will almost certainly experience difficulties in their courses.

Students who need help with spreadsheets and spreadsheet modelling can link to Professor Teall's Spreadsheet Assistance Site . This site will provide assistance with setting up a spreadsheet and building models. It will also assist with data retreival and student projects and provides a brief introduction to VBA, Visual Basic for Applications.

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2   Review of Elementary Mathematics: Functions and Operations
    2.A  Introduction 
    2.B  Variables, Equations and Inequalities 
    2.C  Exponents 
          Application 2.1  Interest and Future Value 
    2.D  Order of Arithmetic Operations and Rules of Algebra 
          Application 2.2  Initial Deposit Amounts 
    2.E The Number e 
    2.F  Logarithms 
          Application 2.3  Time Needed to Double Oneís Money 
    2.G  Subscripts 
    2.H  Summations 
          Application 2.4  Mean Values 
    2.I Double Summations 
    2.J   Products 
          Application 2.5  Geometric Mean Return 
          Application 2.6  The Term Structure of Interest Rates 
    2.K  Factorial Products 
          Application 2.7  Deriving the Number e 
    2.L  Permutations and Combinations 
 Appendix 2.A An Introduction to the Excel Spreadsheet 

3   Review of Elementary Mathematics: Algebra and Solving Equations  
    3.A  Algebraic Manipulations 
          Application 3.1  Purchase Power Parity 
          Application 3.2  Finding Break-Even Production Levels 
          Application 3.3  Spot and Forward Interest Rates 
    3.B The Quadratic Formula 
          Application 3.4  Finding Breakeven Production Levels 
          Application 3.5  Finding the Perfectly Hedged Portfolio 
    3.C  Solving Systems of Equations Containing Multiple Variables 
          Application 3.6  Pricing Factors 
          Application 3.7  External Financing Needs 
    3.D  Geometric Expansions 
          Application 3.8  Money Multipliers 
    3.E  Functions and Graphs 
          Application 3.9  Utility of Wealth 
 Appendix 3.A  Solving Systems of Equations on a Spreadsheet 

4   Time Value of Money  
    4.A  Introduction and Future Value 
    4.B  Simple Interest 
    4.C  Compound Interest 
    4.D  Fractional Period Compounding of Interest 
    Application 4.1 APY and Bank Account Comparisons
    4.E  Continuous Compounding of Interest 
    4.F Annuity Future Values
    Application 4.2  Planning for Retirement
    4.G  Discounting and Present Value
    4.H   Present Value of a Series of Cash Flows 
    4.I  Annuity Present Values
    Application 4.3  Planning for Retirement, Part II
    Application 4.4 Valuing a Bond
    4.J Amortization
    Application 4.5  Determining the Mortgage Payment
    4.K  Perpetuity Models
    4.L Single Stage Growth Models 
    Application 4.6  Stock Valuation Models
     4.M Multiple Stage Growth Models
    Appendix 4.A Time Value Spreadsheet Applications 

5   Return, Risk and Co-movement  
    5.A  Return on Investment 
          Application 5.1  Fund Performance         
    5.B  Geometric Mean Return on Investment 
    Application 5.2  Fund Performance, Part II
    5.C  Internal Rate of Return 
    5.D  Bond Yields 
    5.E  Introduction to Risk 
    5.F  Expected Return
    5.G Variance and Standard Deviation
    5.H Historical Variance and Standard Deviation
    5.I   Covariance
    5.J  Coefficient of Correlation and Coefficient of Determination
    Appendix 5.A Return and Risk Spreadsheet Applications
6   Elementary Portfolio Mathematics   
    6.A  Introduction to Portfolio Analysis 
    6.B  Portfolio Return 
    6.C  Portfolio Variance 
    6.D Diversification and Efficiency
    6.E The Market Portfolio and Beta
    6.F  Deriving the Portfolio Variance Expression

7   Elements of Matrix Mathematics    
    7.A Introduction to Matrices 
          Application 7.1  Portfolio Mathematics 
    7.B  Matrix Arithmetic 
          Application 7.2  Portfolio Mathematics: Part II 
          Application 7.3  Put-Call Parity 
    7.C  Inverting Matrices 
    7.D  Solving Systems of Linear Equations 
          Application 7.4  External Funding Requirements 
          Application 7.5  Coupon Bonds and Deriving Yield Curves 
          Application 7.6  Arbitrage with Riskless Bonds 
          Application 7.7  Fixed Income Portfolio Dedication 
          Application 7.8  Binomial Option Pricing 
    7.E  Spanning the State Space 
          Application 7.9 Using Options to Span the State Space 
 Appendix 7.A  Matrix Mathematics on a Spreadsheet 

8   Differential Calculus
    8.A  Functions and Limits 
          Application 8.1  The Natural Log 
    8.B  Slopes, Derivatives, Maxima and Minima 
    8.C  Derivatives of Polynomials 
          Application 8.2   Marginal Utility 
          Application 8.3  Duration and Immunization 
          Application 8.4  Portfolio Risk and Diversification 
    8.D  Partial and Total Derivatives 
    8.E  The Chain Rule, Product Rule and Quotient Rule 
          Application 8.5  Plotting the Capital Market Line 
    8.G  Taylor Series Expansions 
          Application 8.6  Convexity and Immunization 
    8.H  The Method of LaGrange Multipliers 
          Application 8.7  Optimal Portfolio Selection 
          Appendix 8.A  Derivatives of Polynomials 
          Appendix 8.B  Rules for Finding Derivatives 
          Appendix 8.C  Portfolio Risk Minimization on a Spreadsheet

9   Integral Calculus   
    9.A  Antidifferentiation and the Indefinite Integral 
    9.B Riemann Sums 
    9.C  Definite Integrals and Areas 
          Application 9.1  Cumulative Densities 
          Application 9.2  Expected Value and Variance 
          Application 9.3  Valuing Continuous Dividend Payments 
          Application 9.4  Expected Option Values 
    9.D  Differential Equations 
          Application 9.5  Security Returns in Continuous Time 
          Application 9.6  Annuities and Growing Annuities 
          Appendix 9.A  Rules for Finding Integrals 
          Appendix 9.B  Riemann Sums on a Spreadsheet   

Exercise Solutions  

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  updated 08/22/07