# Introduction to Actuarial and Financial Mathematical Methods

## 1st Edition

**Authors:**Stephen Garrett

**eBook ISBN:**9780128004913

**Hardcover ISBN:**9780128001561

**Imprint:**Academic Press

**Published Date:**7th April 2015

**Page Count:**624

## Description

This self-contained module for independent study covers the subjects most often needed by non-mathematics graduates, such as fundamental calculus, linear algebra, probability, and basic numerical methods. The easily-understandable text of *Introduction to Actuarial and Mathematical Methods* features examples, motivations, and lots of practice from a large number of end-of-chapter questions. For readers with diverse backgrounds entering programs of the Institute and Faculty of Actuaries, the Society of Actuaries, and the CFA Institute, *Introduction to Actuarial and Mathematical Methods* can provide a consistency of mathematical knowledge from the outset.

## Key Features

- Presents a self-study mathematics refresher course for the first two years of an actuarial program
- Features examples, motivations, and practice problems from a large number of end-of-chapter questions designed to promote independent thinking and the application of mathematical ideas
- Practitioner friendly rather than academic
- Ideal for self-study and as a reference source for readers with diverse backgrounds entering programs of the Institute and Faculty of Actuaries, the Society of Actuaries, and the CFA Institute

## Readership

Actuarial and finance students worldwide who need to learn or revisit fundamental applied mathematical tools and techniques

## Table of Contents

- Dedication
- Preface
- Part 1: Fundamental Mathematics
- Chapter 1: Mathematical Language
- Abstract
- 1.1 Common mathematical notation
- 1.2 More advanced notation
- 1.3 Algebraic expressions
- 1.4 Questions

- Chapter 2: Exploring Functions
- Abstract
- 2.1 General Properties and Methods
- 2.2 Combining Functions
- 2.3 Common Classes of Functions
- 2.4 Inverse Functions
- 2.5 Actuarial Application: The Time Value of Money
- 2.6 Questions

- Chapter 3: Differential Calculus
- Abstract
- 3.1 Continuity
- 3.2 Derivatives
- 3.3 Derivatives of More Complicated Functions
- 3.4 Algebraic Derivatives on Your Computer
- 3.5 Actuarial Application: The Force of Interest
- 3.6 Questions

- Chapter 4: Differential Calculus II
- Abstract
- 4.1 An Introduction to Smoothness
- 4.2 Higher-Order Derivatives
- 4.3 Stationary and Turning Points
- 4.4 Higher-Order Derivatives and Stationary Points on Your Computer
- 4.5 Actuarial Application: Approximating Price Sensitivities
- 4.6 Questions

- Chapter 5: Sequences and Series
- Abstract
- 5.1 Sequences
- 5.2 Series and Summations
- 5.3 Evaluating Summations
- 5.4 Taylor and Maclaurin Series
- 5.5 Series and Summations on Your Computer
- 5.6 Actuarial Application: Annuities
- 5.7 Questions

- Chapter 6: Integral Calculus I
- Abstract
- 6.1 Indefinite Integrals of Basic Functions
- 6.2 Change of Variables Approach
- 6.3 Indefinite Integrals of Products of Functions
- 6.4 Indefinite Integrals of Rational Functions
- 6.5 Indefinite Integrals on Your Computer
- 6.6 Questions

- Chapter 7: Integral Calculus II
- Abstract
- 7.1 Definite Integrals
- 7.2 Integration Strategies
- 7.3 Area Between Curves
- 7.4 Definite Integrals on Your Computer
- 7.5 Actuarial Application: The Force of Interest as a Function of Time
- 7.6 Questions

- Chapter 1: Mathematical Language
- Part II: Further Mathematics
- Chapter 8: Complex Numbers
- Abstract
- 8.1 Imaginary and Complex Numbers
- 8.2 Simple Operations on Complex Numbers
- 8.3 Complex Roots of Real Polynomial Functions
- 8.4 Argand Diagrams and the Polar Form
- 8.5 A Simplified Polar Form
- 8.6 Complex Numbers on Your Computer
- 8.7 Questions

- Chapter 9: Probability Theory
- Abstract
- 9.1 Fundamental Concepts
- 9.2 Combinations and Permutations
- 9.3 Introductory Formal Probability Theory
- 9.4 Conditional Probabilities
- 9.5 Probability on Your Computer
- 9.6 Actuarial Application: Mortality
- 9.7 Questions

- Chapter 10: Introductory Linear Algebra
- Abstract
- 10.1 Basic Matrix Algebra
- 10.2 Matrix Multiplication
- 10.3 Square Matrices
- 10.4 Solving Matrix Equations
- 10.5 Solving Systems of Linear Simultaneous Equations
- 10.6 Matrix Algebra on Your Computer
- 10.7 Actuarial Application: Markov Chains
- 10.8 Questions

- Chapter 11: Implicit Functions and ODEs
- Abstract
- 11.1 Implicit Functions
- 11.2 Ordinary Differential Equations
- 11.3 Algebraic Solution of First-Order Boundary Value Problems
- 11.4 Implicit Functions and ODEs on Your Computer
- 11.5 Questions

- Chapter 12: Multivariate Calculus
- Abstract
- 12.1 Partial Derivatives and Their Uses
- 12.2 Critical Points of Bivariate Functions
- 12.3 The Method of Lagrange Multipliers
- 12.4 Bivariate Integral Calculus
- 12.5 Multivariate Calculus on Your Computer
- 12.6 Questions

- Chapter 13: Introductory Numerical Methods
- Abstract
- 13.1 Root Finding
- 13.2 Numerical Differentiation
- 13.3 Numerical Integration
- 13.4 Actuarial Application: Continuous Probability Distributions
- 13.5 Questions

- Chapter 8: Complex Numbers
- Part III: Worked Solutions to Questions
- Chapter 1 Solutions
- Chapter 2 Solutions
- Chapter 3 Solutions
- Chapter 4 Solutions
- Chapter 5 Solutions
- Chapter 6 Solutions
- Chapter 7 Solutions
- Chapter 8 Solutions
- Chapter 9 Solutions
- Chapter 10 Solutions
- Chapter 11 Solutions
- Chapter 12 Solutions
- Chapter 13 Solutions

- Appendix A: Mathematical Identities
- A.1 Trigonometric Identities
- A.2 Derivatives of Standard Functions

- Appendix B: Long Division of Polynomials
- B.1 Motivation
- B.2 Performing the Long Division
- B.3 Questions
- B.4 Solutions

- Bibliography
- Index

## Details

- No. of pages:
- 624

- Language:
- English

- Copyright:
- © Academic Press 2015

- Published:
- 7th April 2015

- Imprint:
- Academic Press

- eBook ISBN:
- 9780128004913

- Hardcover ISBN:
- 9780128001561

## About the Author

### Stephen Garrett

Prof. Stephen Garrett is Professor of Mathematical Sciences at the University of Leicester in the UK. He is currently Head of Actuarial Science in the Department of Mathematics, and also Head of the Thermofluids Research Group in the Department of Engineering. These two distinct responsibilities reflect his background and achievements in both actuarial science education and fluid mechanics research. Stephen is a Fellow of the Royal Aeronautical Society, the highest grade attainable in the world's foremost aerospace institution.

### Affiliations and Expertise

Professor of Mathematical Sciences, University of Leicester, UK

## Reviews

"This book is an ideal introduction to the mathematical background required for students who wish to embark on an actuarial or financial career. It will be especially useful to students who have not taken a mathematical degree. The extensive examples throughout the book show how the mathematics can be used in practice and enable students to gain a thorough understanding of the material." --**John Millett, University of Kent**