Introduction to Actuarial and Financial Mathematical Methods (eBook)

Stephen Garrett (Autor)

eBook Download: PDF | EPUB

2015 | 1. Auflage
624 Seiten
Elsevier Science (Verlag)
978-0-12-800491-3 (ISBN)

Infos

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. - 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

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.

Textauszug

Chapter 2

Exploring Functions

Abstract

In this chapter, we give a detailed discussion of functions of a single independent variable. Functions are a hugely important concept in mathematics and will be used extensively in all that follows in this book. We begin with a general discussion of functions in the broad sense, and then proceed to discuss particular fundamental classes of functions: polynomial, rational, exponential, logarithmic, and circular (trigonometric) functions. These fundamental classes of functions will form the “building blocks” for the expressions that arise in the actuarial and financial context.

Keywords

Mappings

Functions

Roots

Plotting

Composite functions

Polynomials

Rational

Exponential

Logarithmic

Circular

Trigonometric

Inverse

Time value of money

Contents

2.1 General Properties and Methods   24

2.1.1 Mappings   24

2.1.2 Functions   26

Functions vs. mappings   26

Odd and even functions   27

Roots   28

2.1.3 Root finding on your computer   29

2.1.4 Plotting functions   30

2.2 Combining Functions   33

2.2.1 Simple combinations   33

2.2.2 Composite functions   35

2.3 Common Classes of Functions   37

2.3.1 Polynomial functions   37

2.3.2 Rational functions   43

2.3.3 Exponential functions   50

2.3.4 Logarithmic functions   56

2.3.5 Circular (trigonometric) functions   61

2.4 Inverse Functions   67

2.5 Actuarial Application: The Time Value of Money   68

2.6 Questions   77

• Chapter 1 • Inverse trigonometric operations

• Identify and work with examples of • mappings • functions • composite functions • inverse functions • Recall properties of the standard classes of functions (polynomial, rational, exponential, logarithmic, and circular/trigonometric), including • continuity and smoothness • limits • asymptotes • singularities • Ability to explore properties of more complicated functions using algebraic and visual methods

In this chapter, we give a detailed discussion of functions of a single independent variable. Functions are a hugely important concept in mathematics and will be used extensively in all that follows in this book. We begin with a general discussion of functions in the broad sense, and then proceed to discuss particular fundamental classes of functions: polynomial, rational, exponential, logarithmic, and circular (trigonometric) functions. These fundamental classes of functions will form the “building blocks” for the expressions that arise in the actuarial and financial context.

A crucial aim of this chapter is to encourage you to think of functions as mathematical objects and to be able to explore their properties. You should assume that we are working with the real numbers in all that follows.

2.1 General Properties and Methods

2.1.1 Mappings

Chapter 1 ended with a discussion of the expression

x+4)2

  (2.1)

After studying the previous chapter, it should be clear that Eq. (2.1) is not an example of an equation or an identity; that is, it does not have an equal sign separating a LHS and RHS. Nor is it an inequality. In fact, the expression is an example of a function. As we shall see, functions have a precise meaning and not all expressions are necessarily functions. We now build toward an understanding of what is meant by the term “function.”

For the moment, you should think of Eq. (2.1) as a “rule” that returns an output for a given value of x as an input. For example, particular input and output values are stated in Table 2.1.

Table 2.1

Example input and associated output values of (x+4)2

x

→

(x+4)2

− 5

→

1

− 4

→

0

− 3

→

1

− 2

→

4

− 1

→

9

0

→

16

1

→

25

2

→

36

3

→

49

4

→

64

5

→

81

With the results of Table 2.1 in mind, it makes sense to think of Eq. (2.1) as a mapping. For example, in this particular case, − 4 has been mapped to 0, 1 has been mapped to 25, and 4 has been mapped to 64. Alternatively, using the “goes to” notation listed in Table 1.1, we can write these particular mappings as

4→(x+4)20,1→(x+4)225,4→(x+4)264

Rather than always writing the full expression that defines the mapping, it is convenient to label it with a single letter. For example, we could denote Eq. (2.1) by the letter f and write

:x→(x+4)2

In fact, the mapping is fully defined by the statement

:x→(x+4)2,∀x∈R

  (2.2)

which is read as

f is such that x goes to (x+4)2 for all x in the set of real numbers.

It is often convenient to write this simply as

(x)=(x+4)2

which states the mapping’s label, f, and the symbol used to define the independent variable, x. Of course, the shorthand notation f(x) gives no information about the properties of the independent variable, for example, that this f is defined for all real numbers. However, properties of the input will typically be known from the context of the problem.

There is nothing special about using the letter f to denote a mapping and x to denote the independent variable. For example, the following statement defines exactly the same mapping as in Eq. (2.2)

:y→(y+4)2,∀y∈R

and can be summarized as g(y) = (y+4)2.

The set of possible values of the independent variable on which the mapping is defined is called the domain. For example, the domain of mapping (2.2) is the set of real numbers. However, the mapping

:z→(z+2)3,∀z∈[−2,2]

  (2.3)

has a domain formed from the closed interval of real numbers between − 2 and 2 (including the end points).

If we understand that the domain defines the extent of the input of a mapping, the range defines the extent of the output of the domain under that mapping. For example, the range of mapping (2.2) is +∪{0}, equivalently 0,∞), and the range of function (2.3) is h(z) ∈ [0,64]. We can interpret this as it being impossible to find an input within the domain that gives an output outside of the range. We return to question of how to determine the mapping’s range later...

Produkt-Details

• Erscheint lt. Verlag: 2.5.2015
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Recht / Steuern ► Wirtschaftsrecht
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Finanzierung
• Themenwelt: Betriebswirtschaft / Management ► Spezielle Betriebswirtschaftslehre ► Bankbetriebslehre
• ISBN-10: 0-12-800491-6 / 0128004916
• ISBN-13: 978-0-12-800491-3 / 9780128004913