Introduction to Actuarial and Financial Mathematical Methods

By Stephen Garrett

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

• Language: English
• Publisher: Academic Press
• Release date: May 2, 2015
• ISBN: 9780128004913

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.

Book preview

2015

Part 1

Fundamental Mathematics

Chapter 1

Mathematical Language

Abstract

In this chapter, we state and illustrate the use of common mathematical notation that will be used without further comment throughout this book. It is assumed that much of this section will have been familiar to you at some point of your education and is included as an aide-mémoire. Of course, given that the book will explore many areas of the application of mathematics, the material presented here may well prove to be incomplete. It should therefore be considered as an illustration of the level of mathematics that will be assumed as prerequisite, rather than a definitive list.

Keywords

Number systems

Mathematical symbols

Set notation

Interval notation

Quantifiers

Equations

Identities

Inequalities

Contents

1.1 Common Mathematical Notation   3

1.1.1 Number systems   3

1.1.2 Mathematical symbols   6

1.2 More Advanced Notation   8

1.2.1 Set notation   8

1.2.2 Interval notation   12

1.2.3 Quantifiers and statements   13

1.3 Algebraic Expressions   14

1.3.1 Equations and identities   14

1.3.2 An introduction to mathematics on your computer   17

1.3.3 Inequalities   18

1.4 Questions   20

In this chapter, we state and illustrate the use of common mathematical notation that will be used without further comment throughout this book. It is assumed that much of this section will have been familiar to you at some point of your education and is included as an aide-mémoire. Of course, given that the book will explore many areas of the application of mathematics, the material presented here may well prove to be incomplete. It should therefore be considered as an illustration of the level of mathematics that will be assumed as prerequisite, rather than a definitive list.

1.1 Common mathematical notation

1.1.1 Number systems

We begin by summarizing the types of numbers that exist. As this book in concerned with the practical application of mathematics, it should be unsurprising that the set of real numbers forms the building blocks of most (but not quite all, see Chapter 8) of what we will study.

A real number is a value that represents a position along a continuous number line. For example, numbers 5 and 6 have clear positions on the number line in , is therefore seen as the fundamental collection of numbers that we might want to work with in real-world applications.

Figure 1.1 The real number line.

has many subsets, each with an infinite number of members. Such subsets include

The meaning of the terms positive real numbers and negative real numbers should be clear, although note that 0 is technically neither. You may however need to be reminded that the integers are the subset of real numbers that are whole. For example, 0, − 10, and 34 are integers, but − 10.1 and 34.8 are not.

The natural numbers are easily understood as the positive integers and zero.as the nonzero natural numbers.

In addition to the sets of whole and natural numbers, a rational number is any real number that can be expressed as the fraction of two integers. It should be clear that the set of integers are also rational numbers, for example, 32 = 32/1 and − 7 = −7/1, but so are numbers like 45/2 and − 98,736/345,298.

In contrast, irrational numbers are those which cannot be represented as a fraction of two integers. Irrational numbers are numbers which have an infinite number of decimal places, for example, π. Irrational numbers cannot therefore be integers or natural numbers.

The relationship between the different sets of real numbers is summarized in Figure 1.2. From this it is clear that the sum of the sets of rational and irrational numbers form the broader set of real numbers. The set of rational numbers can be further subdivided into integers and nonintegers; the set of integers contains the natural numbers.

Figure 1.2 Venn diagram of the real number systems.

Example 1.1

Where would 0 appear in the Venn diagram of Figure 1.2?

Solution

.

Example 1.2

Give three examples for each of the following number systems.

Solution

is the set of positive real numbers. Examples could be 0.0001, 3.2, and 100.

is the set of integers. Examples could be − 10, 0, and 35.

is the set of natural numbers. Examples could be 1, 7, and 92.

is the set of rational numbers. Examples could be − 2, 5/6, and 9,883/3.

is the set of irrational numbers.

1.1.2 Mathematical symbols

In addition to the symbols used to denote the different number systems, mathematics is full of notation. At this stage, it is useful to list the most common items of notation that you should be able to identify, this done in Table 1.1.

Table 1.1

Basic mathematical notation

Note that a strike through indicates its negation, e.g., ≠ denotes is not equal to.

Note that the list of basic notation in , and this prompts discussion of our first mathematical subtly: The symbol ≈ is commonly used to reflect that in practical situations we are often forced to report approximate values of exact values. For example, the mathematical constant e is an irrational number and so has a numerical value with an infinite number of decimal places

The practical use of e therefore requires one to truncate this to a manageable number of decimal places, say three or four. This truncation is an approximation of the actual value and we write

Similarly, the mathematical constant π is an irrational number with value

In practice one might use

is used when a mathematical method is used explicitly to generate an approximation. For example, as we shall see in Chapter 5, it is often possible to develop a method to approximate the value of an equation. In this case, one would acknowledge that the method is not intended to deliver the exact . The practical use of this symbol can be seen in Chapters 5 and 13, and in particular Eq. (5.9), for example.

The other items of notation in Table 1.1 are assumed to be self explanatory.

Example 1.3

.

a. y > 5.4

b. z ≤ 10

c. x + 2 > 4

d. y = x and y = z

f. |q| = 7

Solution

a. y is greater than 5.4. For example, y = 5.41 or y = 6.

b. z is less than or equal to 10. For example, z = 10 or z = 9.6.

c. x + 2 is greater than 4. For example, x = 2.1 or x = 3.

d. y = x and y = z implies and is implied by x = z. Any identical values of x, y, and z are examples of this.

e. x . For example, x = 0.503866281774 or x = 0.4986827.

f. The modulus of q is 7. For example, q = 7 or q = −7, that is q = ±7.

1.2 More advanced notation

1.2.1 Set notation

Table 1.2 lists the basic items of set notation. We have loosely used the term set when discussing the number systems, for example, we have discussed subsets of the set of real numbers, without a proper definition of what a set actually is.

Table 1.2

Basic set notation

For all intents and purposes in this book, a set is simply understood as a collection of distinct , is interpreted as the collection of all possible real numbers. Any particular real number is a member is read as 4.56 is a member of the set of real numbers. Any set formed from a collection of particular real numbers is considered to be a subset. From Figure 1.2 it should be clear that

Consider the sets

   (1.1)

It is clear that A and C are subsets of the sets of real numbers, rational numbers, and integers, and B is a subset of real numbers, rational numbers, integers, and natural numbers. None is a subset of the irrational numbers. The mathematical shorthand for these statements would be

Of course, set theory is not limited to discussing number systems and we can work with sets of any objects. Where appropriate in this book we will use capital letters to refer to sets and lower case letters to refer to a particular member, that is element, of a set.

Example 1.4

Give all possible values of x that would satisfy the following statements concerning the sets in Eq. (1.1).

a. x ∈ A

b. x ∈ A and x ∈ B

c. x ∈ B and − x ∈ C

Solution

, or 2.

b. x ∈ A

c. x ∈ B and 3.

It is possible to form larger sets by adding two sets using the union . For example,

The union operation forms a new set that consists of all members of the original two sets. Note that 2 ∈ A and 2 ∈ B but it is only listed once in the resulting union of A and B. This is because a set is a list of distinct elements. The idea of a union can be extended to three or more sets in the obvious way.

Furthermore, we can form the set that consists of the common elements of two sets using the intersection . For example,

A set with no elements is called an empty set for obvious reasons, and is denoted by Ø. For example, since A and C have no common elements

, has an obvious meaning.

In terms of the number system, we can write the following statements with the union and intersection notation

The complement of a set can be understood in broad terms as the set of items outside of the set. However, in order to define the items outside of a set, we need to define the space of items that the set exists in. For example, the complement of the set of irrational numbers is the set of all items that are not irrational numbers; without somehow specifying that we actually meant the complement of the irrational numbers within the set of real numbers, there is nothing stopping us listing cats, dogs, and apples alongside the set of rational numbers as members of the complement! For this reason it is useful to define the absolute complement of a set within some broad space of all possible elements Ω, and the relative complement of two sets that are both within Ω.

. Now that Ω is defined, we can consider the absolute complement of subsets of Ω. The absolute complement of set A or A.

, we define the relative complement of A in B as

Using Tables 1.1 and 1.2, we can translate this to words as "the relative complement of A in B are those things, x, in B such that are not in A." Even more simply, it is what remains of set B after having removed those items also in A. The analogue to the subtraction B − A should be clear.

Example 1.5

Using the sets in Eq. (1.1), determine the relative complement of A in B.

Solution

The relative complement of A = {−1,1,2} in B = {0,2,3} is all the elements in B that are not in A. Therefore,

Back to our motivating example of number systems, we can broaden our space Ω to include both the real and imaginary number systems (see Chapter 8), and define the relative complement of the irrational numbers in the real numbers,

Example 1.6

.

f. B \ B = ∅

Solution

a. x is a real number. For example, x = 1.53.

b. y is an integer. For example, y = 9.

c. z is a member of set formed from the union of the two sets {0,1,2,3} and {5,6}, i.e., z is from {0,1,2,3,5,6}. For example, z = 2.

d. y is a member of the set formed from the intersection (i.e., overlap) of the integers and positive real numbers. For example, y = 892.

e. The intersection of the set of the union positive real numbers and zero with the set of integers is the set of natural numbers. For example, 3 is a positive real number (one can label it on the positive half of the number line), it is an integer, and is also a natural number.

f. The complement of set B within itself is the empty set. That is, there are no elements outside of B than are simultaneously also in B.

The basic set operations discussed here are summarized visually in Figure 1.3.

Figure 1.3 Basic set operations illustrated with Venn diagrams.

We now leave aside explicit mention of set theory for the while and return to this in Chapter 9 on probability theory. Unless otherwise stated, you should assume that all mathematical quantities represent real numbers in all that follows.

1.2.2 Interval notation

Throughout this book we will make extensive use of interval bracket notation. In particular, we will use the following bracket notation

• [a,b] denotes the interval {x : a ≤ x ≤ b}

• [a,b) denotes the interval {x : a ≤ x < b}

• (a,b] denotes the interval {x : a < x ≤ b}

• (a,b) denotes the interval {x : a < x < b}

where the term interval can be interpreted as subset of the real number line. Using Tables 1.1 and 1.2 to translate these statements into words, it should be clear that the interval [a,b] is read as "the set of numbers x such that x is between and including a and b." In contrast, the interval (a,b) is read as "the set of numbers x such that x is between but not including a and b." The interpretation of the intervals [a,b) and (a,b] follows in a similar manner. The key point, of course, is that a square bracket denotes an inclusive endpoint of the interval, and a rounded bracket does not.

We refer to an interval that does not include its endpoints as an open interval. For example, (1,5) consists of all numbers x such that 1 < x < 5 and is open. A closed interval, however, does include its endpoints. For example, [10,102] consists of all numbers x such that 10 ≤ x ≤ 102 and is closed.

is not a number that one can draw on a number line, rather it represents that we can keep on using more and more of the number line without imposing any bound.

Example 1.7

Interpret the following mathematical statements in words and give two examples in each case.

b. y ∈ (0,10]

c. p ∈ [0,1]

d. z ∈ (−9.9,−9.8)

Solution

a. x . For example, x = 100 or x = 564.3.

b. y is such that 0 < y ≤ 10. For example, y = 0.1 or y = 10.

c. p is such that 0 ≤ p ≤ 1. For example, p = 0 or p = 1.

d. z is such that − 9.9 < z < −9.8. For example, z = −9.87 or z = −9.82.

1.2.3 Quantifiers and statements

There are two mathematical quantifiers which, when combined with the notation described previously, form a powerful means of writing a wide variety of mathematical statements in a concise way. These are

• ∀, read as for all

• ∃, read as there exists

The quantifier ∀ is often referred to as the universal quantifier, and ∃ as the existential quantifier. The meaning of both should be immediately apparent, although their power might not be. To hint at the power of the two quantifiers in simplifying statements, we begin with an example:

Example 1.8

Demonstrate the intuitive fact that it is possible to find a rational number that approximates the value of π to any finite level of accuracy. Use concise mathematical notation to express that this is true for all real numbers.

Solution

We list the approximations to the value of the irrational number π to an increasing number of decimal places, expressed as a rational number:

That this is true for all real numbers (not just the irrational π) is expressed by

The mathematical statement given in the solution to Example 1.8 is translated to words as

for all x in the set of real numbers and for all in the set of positive real numbers, there exists r in the set of rational numbers such that the absolute value of the difference between x and r is smaller than the value of

Some thought should convince you that this statement is a reflection of our process for approximating π. However, aside from that this is an interesting mathematical fact, the benefits of using the concise mathematical statement formed from the two quantifiers should be immediately apparent.

Example 1.9

Translate the following mathematical statements into words. Give a numerical example in each case.

and z is odd.

Solution

a. For all x and y in the set of real numbers, the product of x and y .

b. For all p and q in the set of negative real numbers, the product of p and q .

c. There exists z in the set of integers such that z is less then 7 and is odd. For example, z = 5.

d. For all p in the set of rational numbers, there exists q in the set of rational numbers such that 3p = q. For example, p = 4/3 and q = 4.

1.3 Algebraic expressions

As we shall see throughout this book, mathematical methods require the manipulation of mathematical expressions. At this stage, it is important to define what we mean by the distinct types of mathematical expressions: equations, identities, inequalities, and functions. The distinction between these terms is the topic of this section, and functions will be considered in detail in Chapter 2.

1.3.1 Equations and identities

The key distinction between an equation and an identity is the number of values of the independent variable (x in Eqs. 1.2 and 1.3) for which the expression is true. An identity is true for all values of the independent variable, but this is not true of an equation.

Put another way, for an identity, it is possible to show that the expression on the left-hand side (LHS) of the equal sign is algebraically equal to that on the right-hand side (RHS). This is not true of an equation and one might be required to find the particular values of the independent variable for which the equality between the LHS and RHS holds. In general, an equation could have a finite or infinite number of values of the independent variable for which the equality holds; typically we might refer to these values of the independent variable as the solutions of the equation.

Consider the following expression:

   (1.2)

It should be immediately clear that Eq. (1.2) is not an identity. In particular, we might note that the LHS is a quadratic expression, that is the highest power of x is 2, and the RHS is linear, that is the highest power of x is 1. For this reason the behavior of the LHS and the RHS will be very different as x takes different values. We return to a discussion of polynomials in Chapter 2.

It is natural to enquire which values of x satisfy Eq. (1.2); that is, for which values of x does LHS = RHS? It is assumed that you will be familiar with the algebraic manipulations required to solve such equations, however, for completeness, we detail the process below.

and so, by the standard quadratic formula,

Equation (1.2) is therefore shown to be true for x = −6 and x = −1 only. The reader is invited to confirm that this is true.

The solution of general equations is not always as simple in practice and you are likely to be familiar with some analytical approaches to finding the solution to polynomial expressions, for example, the quadratic formula or factorization. However, in many practical instances it may not be possible to find an analytical solution. With this in mind, various numerical approaches to solving equations are discussed in Chapter 13.

Consider now the expression

   (1.3)

In this case, it should be immediately clear that the RHS is algebraically identical to the LHS, and the mathematical statement is true for all values of x. Expression (1.3) is therefore an example of an identity, and, using the notation of Table 1.1, it would be correct to write

It is not always possible to confirm an identity by simple observation and we should consider alternative methods. The most obvious approach is the algebraic manipulation of both the LHS and RHS to confirm that they are indeed identical.

Example 1.10

Using an algebraic approach, classify the following expression as an equation or an identity:

  

(1.4)

Solution

The answer is not immediately obvious. One should attempt to cast the LHS in terms of a partial fraction of the following form:

with A,B,C, and D unknown constants to be determined. Some manipulation leads to A = 1, B = −6, C = −3, and D = 2 which confirms the original expression as an identity.

One might consider the algebraic manipulations of complicated expressions to be time consuming and prone to error. In practical situations, an alternative to the algebraic approach could be to plot both sides of an expression over some particular interval of the independent variable that we deem appropriate. In the case of Eq. (1.4), it might be quicker to plot the RHS and the LHS for a reasonable interval in y and compare the result, and this is particularly true if you have access to a computer. The result of such a graphical approach is given in Figure 1.4 and it does appear to be an identity, at least over the values of y considered. We should of course also explore the comparison over different intervals of y to convince ourselves that it is actually an identity, that is, it is true for all y. However, the graphical approach can never be as rigorous as an algebraic approach and Example 1.11 is given as a warning against its use without proper consideration of the ranges used. Before considering that example, let us first discuss how it is possible to use computers in our studies.

Figure 1.4 Computational plot LHS and RHS of Eq. ( 1.4 ). LHS (–) over y ∈ [−1.3,1.3] and RHS (…) over y ∈ [−2,2].

1.3.2 An introduction to mathematics on your computer

Computers are extremely useful tools in mathematics, as will be reflected throughout this book. The advent of relatively cheap and powerful computers has led to the parallel development of programming languages and specialist software that can be used to perform highly complicated mathematical operations at great speed and accuracy. Indeed many commercially available packages are at the very center of the modern financial and scientific industries. However, rather than learning how to do the mathematics on a computer, in this book we will use computers to complement our understanding of the mathematical techniques developed. Specialist software packages are therefore over and above our needs.

A particularly powerful and free online tool is Wolfram Alpha, available at the url http://www.wolframalpha.com. Wolfram Alpha is an extremely useful computational engine that can be used for checking or exploring much of the mathematics we will study in this book and we will make repeated mention of it. The great advantage of Wolfram Alpha is that it does not require knowledge of a specialist computational language, indeed it is usually possible to write the mathematical request in plain English.

For example, mathematical expressions can be very quickly plotted on Wolfram Alpha. If, say, we would like to plot the expression x² − 4x + 4 over x ∈ [−5,5] we would navigate to the Web site and simply enter the instruction

plot xˆ2-4x+4 between x=-5 and 5

The engine will then report the required plot. Two or more expressions can be plotted simultaneously using, for example, the input

plot xˆ2-4x+4, (x-2)ˆ2, xˆ2-2x+2 between x=-5 and 5

The online tool is therefore a useful means for visually comparing and exploring mathematical expressions.

Wolfram Alpha can also be used for algebraic manipulations. For example, the expression in Example 1.10 can very quickly be confirmed as an identity using the instruction

express (yˆ2+1)/(y(2yˆ2-1)(y-1)) as a partial fraction

It is likely that you have access to Excel on your computer. While Excel is not as easy to use as Wolfram Alpha and is not aimed at doing mathematics, it is an extremely powerful numerical tool that is used very widely in business. You should therefore spend some time familiarizing yourself with both Excel and Wolfram Alpha if you are not already familiar with them.

Producing plots in Excel is slightly more cumbersome than with Wolfram Alpha. In particular, one would need to specify the values of x, calculate the associated value of the expression at each of these values, and produce a scatter plot from these data points. It is assumed that you are familiar enough with Excel to do this.

We will make further reference to both Wolfram Alpha and Excel throughout this book, however, the focus will always be as a complement to the concepts under discussion. It is extremely important to realize that computers should not be used a replacement for mathematical knowledge. While much of the mathematics presented in this book can be performed using Wolfram Alpha and similar tools, it is still crucial that you understand the operations and mathematical concepts behind the results. This is the aim of this book.

Example 1.11

Use Wolfram Alpha to investigate whether

is an equation or an identity.

Solution

This example is given as a warning against the graphical approach for confirming identities. The output from the following command demonstrates the reason for this warning,

plot (x-2)ˆ4 and xˆ4-8xˆ3+24xˆ2-32x+15

Without specifying the range for x, the engine chooses two ranges, a narrow range, say, x ∈ [0.5,3.5] and a broad range, say, x ∈ [−20,20]. The plot over the narrow range clearly shows that the LHS and RHS are not equal. However, had we seen only the plot over the broader range, we might have falsely concluded the curves were identical without further experimentation.

A much better approach would be to base the decision on the algebraic expansion of (x − 2)⁴. This could be done using the command

expand (x-2)ˆ4

which yields x⁴ − 8x³ + 24x² − 32x + 16. The RHS and LHS are therefore seen to differ and the expression is correctly classified as an equation.

1.3.3 Inequalities

To complete this section on the types of mathematical expressions, consider

   (1.5)

The use of > makes it clear that this expression is neither an equation nor an identity; it is in fact an example of an inequality. In general, an inequality will be true within particular intervals of the independent variable on the real line, or possibly not true over any interval. An inequality is therefore distinct from identities and more akin to equations. As with equations, the appearance of an inequality will usually trigger the need to find the particular ranges of the independent variable for which it is true. It is assumed that the reader is familiar with handling inequalities algebraically and the straightforward solution to Eq. (1.5) is detailed for completeness only.

That is, Eq. (. Inequalities that arise in practice are unlikely to be so simple and one must be prepared to resort to more involved algebraic manipulations or graphical techniques to solve them. We will return to techniques for solving equations and inequalities at many points throughout this book, including Section 2.1.2 in the next chapter. However the following examples are given to illustrate the process that we will be required to