Mathematical Foundations of Risk Measurement

By Terry Watsham and Keith Parramore

Every financial risk manager should be able to assess risks and understand the core mathematics behind these risks. With nuances to the practice of financial risk management that go beyond the quantitative, the chapters in this book are accessible for individuals with little experience in quantitative financial risk management.

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• Language: English
• Publisher: Professional Risk Managers' International Ass
• Release date: Oct 6, 2022
• ISBN: 9798986164687

Book preview

Mathematical Foundations of Risk Measurement

This textbook contains material aligned with the fourth major domain of the PRM Designation syllabus. The syllabus section is listed here for your reference.

Visit the PRM Resources and Case Studies page at www.prmia.org/PRM for a current list of reading materials aligned to the PRM Desigation program.

IV. MATHEMATICAL FOUNDATIONS OF RISK MEASUREMENT

A. Algebraic Methods

1. Solve equations using algebraic methods (e.g., linear and quadratic equations).

B. Calculus Methods Related to Risk Management

1. Apply calculus methods (e.g., exponential and integration, approximation, differentiation, stochastic, etc.) to risk management.

C. Basic Statistics Related to Risk Management

1. Compute and interpret basic statistical measures relating to risk management (e.g., mean, standard deviation, kurtosis, correlation, etc.).

2. Understand the application of extreme value theory.

D. Numerical Methods

1. Discuss, calculate, and interpret various optimization and numerical methods (e.g., LaGrange, Newton-Raphson, Monte Carlo simulation, Multi- state Markov model, etc.).

E. Matrix Algebra

1. Understand and apply matrix algebra as it relates to risk management (e.g., Cholesky decomposition, etc.).

2. Calculate and interpret principal components analysis (PCA).

3. Solve linear simultaneous equations using matrix algebra.

F. Probability Theory in Finance

1. Understand probability theory including Bayesian theory.

2. Calculate, apply, and interpret probability distributions (e.g., normal, lognormal, Poisson, Copula, probit, and logit models).

G. Regression Analysis in Finance

1. Understand and interpret time series, simple, and multiple linear regression.

2. Apply confidence intervals and hypothesis testing.

H. Compounding Methods

1. Use compounding methods (continuous and discrete) and describe the differences between the two.

Chapter 1: Foundations for Algebraic Methods

Our mathematical backgrounds accumulate from birth, sometimes naturally, sometimes forcibly. We cannot hope to cover even a small part of that ground in this introductory section. We can only try to nudge the memory and fill a few relevant gaps to lay the groundwork for the sections that follow.

Symbols and Rules

Expressions, Functions, Graphs, Equations, and Greek

We begin with a brief examination of various notations and their uses. Letters (x, y, etc.) are often used in mathematics or statistics to represent numbers. These substitute values are either variables or constants. Variables are wildcard characters that can take variable values. Constants represent fixed values. For example, in the expression 3x + 5 (x∈R – see endnote¹), x is a variable. There are ranges of values it can take. In this case, x∈R tells us that x can be any real number (x is in the domain of real numbers). In most cases, the context determines the values a variable can take. The expression is shorthand for (3 × x) + 5, so that if x = –3, the value of the expression is (3 × (–3)) + 5 = –4.

In the expression α + 3x is the constant and x the variable. So, if we make α = 5 and x = –3, we have the answer (3 × (–3)) + 5 = –4 again.

We call 3x + 5 an expression that represents a function of x, designated f(x) so in this instance, f(x) = 3x + 5 defines a function f. We also need to know the domain (i.e., the set of possible x values such as x∈R) before the function is defined, but this is often omitted. A function has the property such that for any value of the input(s), there is a unique value output. Note the possible plurality of inputs; we often operate with functions of more than one variable.

Yet another variant on the theme is y = 3x + 5. Expressed this way, x is the independent variable and y, computed from a given value of x, is the dependent variable. Strictly, y = 3x + 5 is the graph of the function f(x) = 3x + 5. It is a graph² because for every value given to the variable x, a value for the variable y can be computed. Thus, it represents a set of (x, y) pairs such as (–3, –4), all of which have the property that the y coordinate is 3 times the x coordinate plus 5. We can plot all of these (x, y) pairs. Together, they form a line (Figure IV.1).

Figure IV.1: Graph of a Straight Line

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Of course, the choice of 3 and 5 in the expression/function/graph was arbitrary. We could just as well have chosen 2x – 4, or –5x + 0.5, and so on. We can express this thought by writing mx + c, or f(x) = mx + c, or y = mx + c. The letters m and c indicate constants, but constants that we can change. They are called parameters.

In the graph y = mx + c, m is the gradient of the line and c is the intercept (i.e., the value at which the line crosses the y-axis). The gradient might not appear to be correct if the scales on the axes are different, but careful counting resolves the issue. For example, in Figure IV.1, the line passes through (–1⅔, 0) and (0, 5). In doing so, it climbs 5 while moving across 1⅔, and 5 divided by 1⅔ is 3. To find the point (–1⅔, 0) we determined where the line crosses the x-axis. This happens when y = 0, so the x value can be found by solving 3x + 5 = 0. This is an equation.

In financial contexts, we often fit straight lines to data in a process called regression. In that context, it is more common to use a and b for the parameters, with a being the intercept and b being the slope, so that y = a + bx

Example IV.1

Suppose the return on a stock, Rs, relates to the return on a related index, Ri, by equation Rs = –0.0064 + 1.19Ri. The return to the index is the independent variable, and the return to the stock is the dependent variable. Based on this relationship, the expected returns on the stock can now be computed for various market returns. For Ri = 0.05, we compute Rs = –0.0064 + 1.19×0.05 = 0.0531. Similarly, Ri = 0.10 gives Rs = 0.1126 and Ri = 0.15 gives Rs = 0.1721. The parameter 1.19 represents the slope or gradient (i.e., b) in the function y = a + bx; it is the coefficient of Ri. In this example, it measures the sensitivity between stock market returns and stock or portfolio returns

Greek letters are often used to label parameters. Appendix A shows common uses in finance. Appendix B gives some of the symbols frequently met in financial math, together with their meaning.

The Algebra of Number

Algebra is a branch of mathematics that describes relationships and expresses those relationships using letters and symbols to represent unknown values of the elements of the relationship. Thus, when studying finance, you will come across many algebraic expressions that have been created to express the relationship between the financial variables.

There are rules for manipulating algebraic expressions. We will not be too formal in what follows, but an outline should help jog those memories laid down during the years of formal education.

Types of Number

Numbers are classified into six main groups: Integers, Natural, Rational, Irrational, Real, and Complex Numbers. There is a hierarchy of number type. The hierarchy is both historical and logical. The historical is the order of their invention, and the logical is the level of complexity; the naturals lie within the integers, the integers lie within the rationals, etc.

We start with the Integers. These are labeled Z = {…, –3, –2, –1, 0, 1, 2, 3, …}. They are whole numbers and can have negative or positive values from minus infinity to plus infinity. Thus, –308, –250, 1, 10, 208, 10956 are all integers, but 208.6 or √5 are not.

Natural numbers {1, 2, 3, 4, 5, …} are labeled N.³ This group of numbers consists only of the positive integers (i.e., those ranging from 1 to infinity). There is some debate as to whether zero should be included. If it is included, Natural Numbers are all the non-negative integers. If zero is excluded, Natural Numbers are all the positive integers.

After the Naturals come the Rationals (fractions), labeled Q. They are fractions that each consist of a pair of integers, the numerator and denominator. The denominator cannot be zero, and there is an extra element of structure that defines 1/2, 2/4, 3/6, etc. as being identical. Since the denominator can be one, integers are also Rationals.

If the fraction cannot be expressed as a ratio of integers, the number is an Irrational number. Examples of irrationals include π, e, and numbers such as 0.101001000100001….

Next, we have the Real numbers, denoted R. These consist of Rational plus Irrational numbers.

Finally, we have complex numbers, C. These do not concern us here.

Operations in Natural Numbers

We are used to dealing with addition, subtraction, multiplication, and division. In fact, subtraction is defined in terms of addition. For instance, 6 – 2 is the number that needs to be added on to 2 to give 6. Similarly, division is defined in terms of multiplication. For example, 6/2 is the number that when multiplied by 2 gives 6. Thus, we only need to deal with rules related to addition and multiplication. Furthermore, multiplication within the natural numbers is defined in terms of repeated addition.

Having defined the operations within N, we need to extend the definitions to sets further up the hierarchy.

Operations on Rational Numbers (Fractions)

Much time is spent in school dealing with operations on fractions. We summarize the algorithms below.

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Rules

The rule defining how the operations interact with numbers are drilled into us throughout our formative years. Here are some of them:

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From these rules, we can deduce such results as a – (–b) = a + b and (–a) × (–b) = a × b.

The Order of Operations

The rules of algebra define the order in which operations are applied. The importance of this can be seen by considering expressions such as 12 ÷ 4 ÷ 3. Is this to mean 12 ÷ (4 ÷ 3) = 9 or (12 ÷ 4) ÷ 3 = 1?

The solution is to use brackets. However, when complex expressions are written, it is common practice to follow conventions that limit the number of brackets needed. This makes the expressions easier to read, but one must first know the conventions. Thus, it is common practice to write a linear expression in the form 2x + 3 (understanding that 2x is itself shorthand for 2 × x). It is an accepted convention that this is shorthand for (2x) + 3, rather than 2(x + 3). In other words, the multiplication is to be done before the addition. So, if the expression is to be evaluated when x = 5, the answer is 10 + 3 = 13, not 2 × 8 = 16. If the latter order of operations is required, brackets must be used, i.e., 2(x + 3).

These conventions are summarized in the mnemonic BIDMAS, which gives the order in which to apply operations, unless brackets indicate otherwise: Brackets, Indices (powers), Division, Multiplication, Addition, Subtraction.

Thus, to evaluate (2x + 3)² + 40 / (5 – x/2)+5 when x = 6, proceed as:

B: Evaluate the brackets (i.e., work within them). With x = 6, we have 15² + 40 / 2 + 5

I: Indices. In this example, do the squaring. We therefore get 225 + 40 / 2 + 5

D: Division. We get 225 + 20 + 5

M: Multiplication. In this example, there is none to do.

A: Addition. In this example, 225 + 20 + 5 =250

S: Subtraction. In this example, there is none to do.

Sequences and Series

Sequences

In mathematics, a sequence is an ordered set of mathematical terms. In other words, it is a list. The fact that the terms are ordered means that there is a specific defined list (i.e., the order in which the various elements are placed is important). For example, the sequence of the first 10 odd integers above zero is {1, 3, 5, 7, 9, 11, 13,…,19}. Another example of a sequence is the Fibonacci sequence, where after the first two numbers, each number is the sum of the previous two {0, 1, 1, 2, 3, 5, 8, 13, 21…}. Some sequences can be defined by a formula or rule.

In finance, a frequently encountered sequence is the future cash flows from a bond. For example, the cash flows due under a five-year bond with principal of 100, paying annual coupons at 5%, results in a finite sequence as {5, 5, 5, 5, 105}.

Some sequences are infinite. Such an example in finance would be the cash flows due under a perpetual bond. A perpetual bond paying 5 per annum results in an infinite sequence (5, 5, 5 …). The three dots indicate that the series goes on without bound. Other sequences are finite as in the cases of the first 10 odd integers above, or the bond cash flows.

The Geometric Sequence

One sequence that is often employed in finance is the geometric sequence. This has a first term, usually denoted by a. Subsequent terms are derived by repeatedly multiplying by a fixed number x, known as the common ratio:

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A very common example of a geometric sequence in finance results from the compounding of interest. To illustrate, assume an annual rate of interest of 2.5% and a common ratio of 1+ the rate of interest; in this case, 1.025. This is x in the example above. Using compound interest, the common ratio is called the compounding factor.

Assume an initial deposit of 1000 is placed for 20 years. The initial deposit is the first term of the geometric progression; a in the notation above. The second term is 1000(1.025) = 1025. The third term is 1025(1.025)² = 1050.625, and the 21st term is 1000×1.02520 ≈ 1638.62. So, a sequence representing the periodic compounded value of a deposit over its life is a geometric sequence.

Series

A series is the accumulated sum of a sequence, and the accumulated sum of a geometric sequence is a geometric series. When a sequence has many terms, a short cut to summing the terms is useful. Two such short cuts pertaining to the geometric series are explained below. Their use in finance primarily has to do with the valuing of annuities or determining the amortization schedules of fixed-term, fixed-interest loans. However, another application is employed in the pricing of a non-redeemable bond or an infinite annuity.

The Sum to n Terms of a Geometric Series

In what follows, x stands for the ratio of successive terms. In much work in financial math, r stands for the rate of interest, so we often see (1 + r), i.e., the compounding factor, appearing as the ratio of successive terms.

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Thus, Sn – xSn = a − axn, giving:

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(1)

Of course, Sn = na if x = 1.

We illustrate equation (1) later in this chapter in the section on the Time Value of Money, where we calculate the amortization payment of a fixed-term loan.

For a geometric sequence, if −1 < x < 1 (i.e. if |x| < 1 ), Sn does tend to a limit as n tends to infinity. This is because the xn term tends to zero, so that:

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(2)

We also illustrate equation (2) in the section about the Time Value of Money by pricing a Perpetual Bond.

Exponentiation and Logarithms

Exponentiation

Exponentiation involves raising a base, a, to a power, n. It is written an. The power is also called the exponent. Common examples include a², which is commonly called a-squared, and a³, which is known as a-cubed. Exponentiation is the equivalent of repeated multiplication. For example, a² = a × a, and a³ = a × a × a.

The fundamental rule of exponents (or powers) is abstracted from simple observations such as 2³ × 2² = 2⁵, since 2³ = 2 × 2 × 2 = 8, 2² = 2 × 2 = 4, and 2⁵ = 2 × 2 × 2× 2× 2 = 32.

The generalization of this is known as the first rule of exponents: For a > 0,

1. DESCRIBE HERE (This may appear obvious, but it applies to all numbers x and y, not just to natural numbers.)

There are two further rules of exponents:

2. DESCRIBE HERE

3. DESCRIBE HERE (3)

The consequences of these rules are:

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Here we see that an exponent can have three parts. It has a sign, positive or negative, it has a numerator (e.g., 0 or ±1 in the examples above), and it can have a denominator (e.g., the 2 in the last example above).

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These three operations can be done in any order. In this case, it is easiest to take the fourth root first since 81 = 3⁴. So, 3 is the 4th root. The negative sign indicates that the answer should be 1 over the fourth root (i.e., 1/3). The 3 in the numerator indicates that we should cube 1/3 (i.e., 1/3³). So, 81-3/4 = 1/3³ = 1 / 27.

Logarithms

A logarithm is the inverse of an exponent. For example:

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This is encapsulated in the following defining relationship:

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(4)

It says that loga(x) is that power to which a must be raised to give the answer x.

Corresponding to the three rules of exponents are three rules of logarithms. They are:

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(5)

We demonstrate the correspondence in the case of rule 1, showing how to use equation (4):

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Suppose we needed to find log2(6). Since 2² = 4 and 2³ = 8, it must be between 2 and 3. However, we can again use equation (4). Let x = log2(6). Then, 2x = 6. Taking logs (base 10) of both sides gives:

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Note that in Microsoft Excel, you can find log2(6) directly from the formula =log(6,2).

‘The Exponential Function’ and Natural Logarithms

In the section above on exponentiation, we introduced exponential functions generally. However, one exponential function is used so widely, it is simply called the exponential function. This function has base e. This is an irrational number, approximately equal to 2.7182818285. The logarithm to the base e is known at the Natural Logarithm.

The obvious question is "What is natural about e?" To answer this, consider Figure IV.2, which shows graphs of functions y = 2x and y = 3x.

Figure IV.2: Graphs of 2x and 3x

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