Quantitative Methods for Finance and

Quantitative Methods for Finance and Investments ensures that readers will gain a reasonable degree of comfort and proficiency in applying elementary mathematics to financial analysis in a variety of areas. All of the methodology in this book is geared toward the development, implementation, and analysis of financial models to solve problems encountered by both finance students and practitioners. The book: bull; bull;analyzes theoretical and practitioner-oriented models long with the mathematics required to construct them bull;presents the most essential mathematical techniques and their applications to financial analysis bull;provides dozens of practical applications, examples, and end-of-chapter exercises with detailed solutions bull; bull;demonstrates key spreadsheet applications of the mathematical models in chapter appendices bull;emphasizes practical applications of modeling technique.

John L. Teall is Professor of Finance at Pace University. He has published numerous articles in scholarly journals and has served on university faculties around the world. Dr Teall is a former member of the American Stock Exchange and has done consulting work for many of the world's leading financial institutions.

Iftekhar Hasan is Professor of Finance at the New Jersey Institute of Technology. He has published numerous articles in academic journals and has been associated with several universities and regulatory organizations in Europe. He is the co-editor of Research in Banking and Finance.

Preface

xii

Acknowledgments

xiv

Introduction and Overview

1

(6)

The importance of mathematics in finance

1

(1)

Mathematical and computer modeling in finance

2

(1)

Money, securities, and markets

3

(2)

Time value, risk, arbitrage, and pricing

5

(1)

The organization of this book

6

(1)

A Review of Elementary Mathematics: Functions and Operations

7

(18)

Introduction

7

(1)

Variables, equations, and inequalities

7

(1)

Exponents

8

(2)

Application 2.1: Interest and future value

9

(1)

The order of arithmetic operations and the rules of algebra

10

(1)

Application 2.2: Initial deposit amounts

11

(1)

The number e

11

(1)

Logarithms

12

(2)

Application 2.3: The time needed to double your money

13

(1)

Subscripts

14

(1)

Summations

14

(2)

Application 2.4: Mean values

15

(1)

Double summations

16

(1)

Products

17

(2)

Application 2.5: Geometric means

17

(1)

Application 2.6: The term structure of interest rates

18

(1)

Factorial products

19

(1)

Application 2.7: Deriving the number e

19

(1)

Permutations and combinations

20

(5)

Exercises

21

(2)

Appendix 2.A An introduction to the Excel™ spreadsheet

23

(2)

A Review of Elementary Mathematics: Algebra and Solving Equations

25

(26)

Algebraic manipulations

25

(4)

Application 3.1: Purchase power parity

27

(1)

Application 3.2: Finding break-even production levels

28

(1)

Application 3.3: Solving for spot and forward interest rates

29

(1)

The quadratic formula

29

(3)

Application 3.4: Finding break-even production levels

30

(1)

Application 3.5: Finding the perfectly hedged portfolio

31

(1)

Solving systems of equations that contain multiple variables

32

(6)

Application 3.6: Pricing factors

35

(1)

Application 3.7: External financing needs

35

(3)

Geometric expansions

38

(3)

Application 3.8: Money multipliers

40

(1)

Functions and graphs

41

(10)

Application 3.9: Utility of wealth

43

(1)

Exercises

44

(4)

Appendix 3.A Solving systems of equations on a spreadsheet

48

(3)

The Time Value of Money

51

(28)

Introduction and future value

51

(1)

Simple interest

51

(1)

Compound interest

52

(1)

Fractional period compounding of interest

53

(3)

Application 4.1: APY and bank account comparisons

55

(1)

Continuous compounding of interest

56

(1)

Annuity future values

57

(3)

Application 4.2: Planning for retirement

59

(1)

Discounting and present value

60

(1)

The present value of a series of cash flows

61

(1)

Annuity present values

62

(3)

Application 4.3: Planning for retirement, part II

64

(1)

Application 4.4: Valuing a bond

64

(1)

Amortization

65

(2)

Application 4.5: Determining the mortgage payment

66

(1)

Perpetuity models

67

(1)

Single-stage growth models

68

(4)

Application 4.6: Stock valuation models

70

(2)

Multiple-stage growth models

72

(7)

Exercises

73

(4)

Appendix 4.A Time value spreadsheet applications

77

(2)

Return, Risk, and Co-movement

79

(24)

Return on investment

79

(3)

Application 5.1: Fund performance

81

(1)

Geometric mean return on investment

82

(2)

Application 5.2: Fund performance, part II

83

(1)

Internal rate of return

84

(3)

Bond yields

87

(1)

An introduction to risk

88

(1)

Expected return

88

(1)

Variance and standard deviation

89

(2)

Historical variance and standard deviation

91

(2)

Covariance

93

(1)

The coefficient of correlation and the coefficient of determination

94

(9)

Exercises

95

(4)

Appendix 5.A Return and risk spreadsheet applications

99

(4)

Elementary Portfolio Mathematics

103

(12)

An introduction to portfolio analysis

103

(1)

Portfolio return

103

(1)

Portfolio variance

104

(2)

Diversification and efficiency

106

(4)

The market portfolio and beta

110

(1)

Deriving the portfolio variance expression

111

(4)

Exercises

113

(2)

Elements of Matrix Mathematics

115

(30)

An introduction to matrices

115

(2)

Application 7.1: Portfolio mathematics

116

(1)

Matrix arithmetic

117

(6)

Application 7.2: Portfolio mathematics, part II

120

(1)

Application 7.3: Put-call parity

121

(2)

Inverting matrices

123

(2)

Solving systems of equations

125

(8)

Application 7.4: External funding requirements

126

(1)

Application 7.5: Coupon bonds and deriving yield curves

127

(3)

Application 7.6: Arbitrage with riskless bonds

130

(1)

Application 7.7: Fixed income portfolio dedication

131

(1)

Application 7.8: Binomial option pricing

132

(1)

Spanning the state space

133

(12)

Application 7.9: Using options to span the state space

136

(1)

Exercises

137

(5)

Appendix 7.A Matrix mathematics on a spreadsheet

142

(3)

Differential Calculus

145

(35)

Functions and limits

145

(2)

Application 8.1: The natural log

146

(1)

Slopes, derivatives, maxima, and minima

147

(2)

Derivatives of polynomials

149

(8)

Application 8.2: Marginal utility

151

(2)

Application 8.3: Duration and immunization

153

(3)

Application 8.4: Portfolio risk and diversification

156

(1)

Partial and total derivatives

157

(1)

The chain rule, product rule, and quotient rule

158

(7)

Application 8.5: Plotting the Capital Market Line

159

(6)

Logarithmic and exponential functions

165

(1)

Taylor series expansions

166

(2)

Application 8.6: Convexity and immunization

167

(1)

The method of Lagrange multipliers

168

(12)

Application 8.7: Optimal portfolio selection

170

(2)

Exercises

172

(4)

Appendix 8.A Derivatives of polynomials

176

(1)

Appendix 8.B A table of rules for finding derivatives

177

(1)

Appendix 8.C Portfolio risk minimization on a spreadsheet

178

(2)

Integral Calculus

180

(23)

Antidifferentiation and the indefinite integral

180

(1)

Riemann sums

181

(4)

Definite integrals and areas

185

(6)

Application 9.1: Cumulative densities

186

(2)

Application 9.2: Expected value and variance

188

(1)

Application 9.3: Valuing continuous dividend payments

189

(2)

Application 9.4: Expected option values

191

(1)

Differntial equations

191

(12)

Application 9.5: Security returns in continuous time

193

(1)

Application 9.6: Annuities and growing annuities

194

(1)

Exercises

195

(3)

Appendix 9.A Rules for finding integrals

198

(1)

Appendix 9.B Riemann sums on a spreadsheet

199

(4)

Elements of Options Mathematics

203

(19)

An introduction to stock options

203

(2)

Binomial option pricing: one time period

205

(2)

Binomial option pricing: multiple time periods

207

(3)

The Black-Scholes option pricing model

210

(2)

Puts and valuation

212

(1)

Black-Scholes model sensitivities

213

(2)

Estimating implied volatilities

215

(7)

Exercises

219

(3)

References

222

(2)

Appendix A Solutions to Exercises

224

(42)

Appendix B The z-Table

266

(1)

Appendix C Notation

267

(3)

Appendix D Glossary

270

(4)

Index

274

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Quantitative Methods for Finance and Investments

Summary

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