Essential Maths 1 The information in this document is the minimum assumed knowledge for students undertaking the Macquarie University Masters of Applied Finance, Graduate Diploma of Applied Finance, and Graduate Certificate of Finance. 1 These notes were prepared y Anne Cooper and Catriona March. Macquarie University MAFC_Essential_Maths Page 1 of 20
Contents 1 Introduction... 3 1.1 Foundations... 3 1.2 Excel... 3 1.3 References... 3 2 Numers and Notation... 4 2.1 Integers and Real Numers... 4 2.2 e... 4 2.3 Pi ( π )... 5 2.4 Commonly used Greek letters and their uses... 6 3 Solving Equations... 7 4 Functions... 8 4.1 Linear functions... 8 4.2 Quadratic functions... 11 4.3 Powers... 13 4.4 Exponential function... 14 4.5 Logarithms... 14 4.6 Natural Logarithm... 16 5 Derivatives... 17 5.1 Rates of Change... 17 5.2 Rules of differentiation... 19 Macquarie University MAFC_Essential_Maths Page 2 of 20
1 Introduction The information in this document is the minimum assumed knowledge for students undertaking the Macquarie University Masters of Applied Finance, Graduate Diploma of Applied Finance, and Graduate Certificate of Finance. 1.1 Foundations As a prerequisite to the topics discussed elow, students are assumed to have a asic understanding of: Numers and arithmetic operations: addition, multiplication, division, order of operations, etc. The use of symols to represent numers and relationships etween numers. Simple equations and inequalities, and how to rearrange them. For a refresher on such concepts, see the references listed elow, in particular Swift & Piff [1], Chapter EM (Essential Maths), or Alexander & Sheedy [2], Section II.A. 1.2 Excel Students are also assumed to have access to and the aility to use Microsoft Excel. This will e used for data analysis and other exercises throughout many courses in the Applied Finance programs. 1.3 References The following texts and wesites are referred to elow as sources for additional information: [1] Louise Swift and Sally Piff, Quantitative Methods for Business, Management and Finance, Fourth Edition. Palgrave Macmillan 2014. Companion wesite is: https://www.palgrave.com/companion/swift-quantitative-methods-4/ [2] Carol Alexander and Elizaeth Sheedy (Editors), The Professional Risk Managers Handook, Volume II, PRMIA, 2011. [3] Wolfram MathWorld, a free we resource y Wolfram Research: http://mathworld.wolfram.com Macquarie University MAFC_Essential_Maths Page 3 of 20
2 Numers and Notation 2.1 Integers and Real Numers Positive whole numers or integers {1, 2, 3, 4, } are also called the natural numers and are usually denoted y N. The set of all integers, oth positive and negative, as well as zero, { -2, -1, 0, 1, 2, }, is denoted y Z. Rational numers can e expressed as a ratio of two integers, for instance 2 3 The set of all rational numers is denoted y Q. or 22 7. Some numers cannot e represented as a ratio of two integers ut can e expressed as the limit of a sequence of rational numers. These are called irrational numers. Particularly important examples of irrational numers for our purposes are e and π. The set of all numers, oth rational and irrational, is called the real numers and is denoted y R. 2.2 e The irrational numer e is one of the most important numers in mathematics. It occurs in many contexts in mathematics and many applications in the physical sciences. Being irrational, e cannot e exactly expressed as a decimal ut it can e evaluated to any required degree of accuracy, for example: e = 2.7182818284 5904523536 0287471352 6624977572 The exponential function (Section 4.4) is ased on the constant e. It is also the ase used for the natural logarithm (Section 4.6 elow). The numer e occurs naturally in finance in the context of compound interest. For an annual interest rate of r compounding n times per year, the accrual factor is: (1 + r n n ) As the frequency of compounding increases, in the limit as n goes to infinity: lim (1 + r n n n ) = e r This leads to the concept of continuously compounding interest rates. Macquarie University MAFC_Essential_Maths Page 4 of 20
For a continuously compounding interest rate of r C, the future value of an investment of A after T years is A e r C T. For more information aout e, see: http://mathworld.wolfram.com/e.html 2.3 Pi ( π ) The ratio of any circle s circumference C to its diameter d is a constant, the irrational numer π. C d Like e, the numer π cannot e exactly expressed as a rational numer ut it can e evaluated to any required degree of accuracy: π = 3.1415926535 8979323846 2643383279 5028841971 While its formal definition is the geometrical relationship for a circle given aove, it occurs in many other contexts, including financial applications. The special numers e and π are connected in the relationship: e x2 /2 = 2π This relationship leads to the definition of the proaility density function of the Standard Normal variale, which is given y the formula: φ(x) = 1 2π e x2 /2 The Normal variale is widely used for modelling in the physical sciences and in finance. Applications include the modelling of asset returns and option pricing models. For more information aout π, see: http://mathworld.wolfram.com/pi.html Macquarie University MAFC_Essential_Maths Page 5 of 20
2.4 Commonly used Greek letters and their uses The tale elow lists letters from the Greek alphaet which are commonly used as symols in finance applications. Lower Case Upper Case Pronunciation Examples of use in finance α Α Alpha Regression intercept β Β Beta Regression slope, systemic risk γ Γ Gamma Options sensitivity measure (change in delta) δ Δ Delta Price sensitivity measure for options ε Ε Epsilon Random error in regression analysis θ Θ Theta Time sensitivity measure for options κ Κ Kappa Kurtosis μ Μ Mu Mean of distriution; expected return ν Ν Nu Integer constant, e.g. degrees of freedom π Π Pi π - circle constant Π symol for a product ρ Ρ Rho Correlation coefficient Option interest rate sensitivity σ Σ Sigma σ standard deviation or volatility Σ symol for a sum τ Τ Tau Time to maturity φ Φ Phi Normal distriution density function χ Χ Chi Statistical distriution for testing fit Source: Alexander & Sheedy [2], Section II.A.1.1 Macquarie University MAFC_Essential_Maths Page 6 of 20
3 Solving Equations Solving simple equations requires an understanding of asic algera. That is, how to expand terms, factorise terms and power rules. If you are unfamiliar with these concepts and cannot solve the exercises elow, it is recommended that you work through the readings listed. Example Answer a) Expand (x 5) 2 x 2 10x + 25 ) Expand (x + 2)(2x 2 4x + 5) 2x 3 3x + 10 c) Factorise 2x 2 5x 3 (2x + 1)(x 3) d) Simplify: x x + 1 = 2 x = 2 e) If x 3 = 4096, what is the value of x? x = (4096) 1 3 = 16 f) Simplify: x 2 (1 + y) 3 (1 + y) 2 x 4 1 + y x 2 g) Solve y factorising the quadratic: x 2 2x 15 = 0 x 2 2x 15 = (x + 3)(x 5) so x = 3 or x = 5 Further Reading For further explanation and more examples, see Swift & Piff [1], Chapter EM, (Essential Maths), in particular: Section 5, Expanding Brackets, pages 43-48 Section 6, Factorising, pages 48-52 Section 7, Powers, pages 53-69 Macquarie University MAFC_Essential_Maths Page 7 of 20
4 Functions A function descries a relationship etween variales. Functions are often represented y letters such as f. Whereas the symol f refers to the function itself, we write f(x) for the value taken y the function evaluated at the point x. The set of all possile values for x that the function acts on is known as the domain of the function. For each value of x in the domain, there is a unique value of f(x). The set of all values f(x) taken y the function is known as the range. It is also common to write y = f(x). When expressed in this way, x is referred to as the independent variale and y is referred to as the dependent variale. 4.1 Linear functions A linear function is a function of the form: y = mx + where m and are constants. Since the constants m and take different values for different linear functions, they are sometimes referred to as parameters. When graphed in the (x, y) plane, the function y = mx + is a straight line. When x = 0, the value of y is, so the straight line intercepts the y-axis at. For example, the linear function y = 4x + 5 represents a straight line which intercepts the y-axis at 5. Macquarie University MAFC_Essential_Maths Page 8 of 20
When x = 3, the function value is y = 4 3 + 5 = 17. This is highlighted in the chart elow. The slope or gradient of the line is the rate of change in y for a unit change in x. Since a linear function represents a straight line, its graph has the same slope at every point. This slope is m, the coefficient of x in the formula for the line. For example, the linear function y = 4x + 5 has slope 4. This means that y changes 4 units for each 1 unit change in x. 4 unit change in y 1 unit change in x Since the linear equation aove makes sense for any real numer x, the domain of a linear function is R. The linear equation can e rearranged as: x = 1 m y m This means that for any real numer y, there is a value x for which y = mx + and so the range of the linear function is also R. Macquarie University MAFC_Essential_Maths Page 9 of 20
Sometimes linear functions are written as: a x + c y = k This can e rearranged into the format aove: y = a c x + k c That is, a straight line with y-intercept = k c and gradient m = a c. Many relationships in usiness and finance are naturally linear. In other situations, linear functions are used as approximations ecause they are simple to use and easy to understand. Examples include: It is usually assumed in economics that the demand Q for a product will decrease as its price P increases. The simplest possile model for this is a linear function with negative slope, that is, for constants a and, where a is negative: Q = a P + In financial markets theory, the return on an individual asset R A is modelled as a linear function of the market return R M : R A = α + β R M The slope in this case is referred to as the eta (β) of the asset; the y-intercept is the return on the asset, independent to the market return. This is related to the Capital Assets Pricing Model (CAPM). A udget constraint might e modelled in a simple way as a linear function expressed as: a x + c y = k where in this case x and y are the numers of two different goods eing manufactured; the constants a and represent the fixed cost per unit of manufacturing goods of type x and y respectively; and k is the total funds availale for this purpose. Further Reading For more explanation, see Swift & Piff [1], Chapter EM, (Essential Maths), Section 4, Modelling Using Straight Lines. Macquarie University MAFC_Essential_Maths Page 10 of 20
4.2 Quadratic functions A quadratic function is a function of the form y = ax 2 + x + c where a, and c are constants and where a is not equal to 0. For example, with a = 1, = 5 and c = 4, the quadratic function is y = x 2 5x + 4 The roots of the quadratic occur when y = 0. Sometimes these can e found directly y factorising the quadratic function. For example, we can rewrite the quadratic aove as: y = x 2 + 5x + 4 = (x 1) (x 4) This shows that the roots of this quadratic are x = 1 and x = 4. More generally, the roots of any quadratic function can e found using the following formula, which is known as the quadratic formula: x 2 4ac 2a The numer of roots depends on the term inside the square root in the quadratic formula: When 2 4ac > 0, there are two roots, given y the formula aove. When 2 4ac = 0, there is exactly one solution, x = 2a. When 2 4ac < 0, the quadratic has no real roots. 2 Graphically, a quadratic function represents a paraola and its roots are where the paraola hits the x-axis. The turning point of the quadratic occurs at x = 2a. When a > 0, the paraola is concave up with minimum value at the turning point, while when a < 0, the paraola is concave down with maximum value at the turning point. The quadratic function is symmetric aout the vertical line x = 2a. 2 The roots in this case can e expressed as complex numers ut this is eyond the scope of this document and is not needed for the Applied Finance programs. Macquarie University MAFC_Essential_Maths Page 11 of 20
The graph of the quadratic function y = x 2 5x + 4 is shown elow. The curve cuts the x-axis at the roots of this quadratic, x = 1 and x = 4. The minimum value occurs at the turning point, when x = ( 5) 2 = 2.5, and the curve is symmetric aout x = 2.5. Notice that in this example 2 4ac = 25 16 = 9 > 0. y = x 2-5x + 8 y = x 2-5x + 6.25 y = x 2-5x + 4 For the quadratic, y = x 2 5x + 8, 2 4ac = 25 32 = 7 < 0. As can e seen in the chart aove, this curve does not hit the x-axis so there are no real roots. The graph of the quadratic function y = x 2 5x + 6.25 is also shown in the chart aove. In this case 2 4ac = 25 25 = 0 and there is a single real root at x = 2.5. An example of use of the quadratic function in economics is modelling the profit P that a firm makes on the production of a particular product, as a quadratic function of the numer of units Q produced: P = aq 2 + Q + c To calculate the numer of units that needs to e produced for the firm to reak even, that is to make zero profit, we solve the quadratic equation: aq 2 + Q + c = 0 Thus the reak-even production level is given y the roots of the quadratic equation. Further Reading For more explanation, see Swift & Piff [1], Chapter MM, (More Maths), Section 1, Some Special Equations, and Section 2, Modelling Using Curves. Macquarie University MAFC_Essential_Maths Page 12 of 20
4.3 Powers In general, a power or exponent can consist of 3 operations that can e done in any order. For example, to evaluate the expression 16 5 4 the 1 4 indicates to take the 4th root; the 5 indicates to raise to the 5 th power; the negative sign indicates to invert the result. 16 5 4 4 = [( 16) 5 ] 1 = [(2) 5 ] 1 = 32 1 = 1 32 The fundamental rules for manipulating exponents for any positive real numer and real numers x and y are listed elow: 1 = 0 = 1 x y = x+y x / y = x y x y x y 1 1 x can never equal 0 For any positive numer and for every real numer x the power function y = x is always positive. This means that the domain of the function y = x is all real numers, while its range is the positive real numers. Power functions occur naturally in the context of compounding interest. Suppose you invest an amount A at a compounding interest rate of r% per period. Assuming reinvestment of interest, after m periods your investment has grown to: A (1 + r) m You will learn more aout interest rate conventions and compounding interest in later units in your Applied Finance degree. Macquarie University MAFC_Essential_Maths Page 13 of 20
4.4 Exponential function Raising the special numer e to power x for different real numers x gives the exponential function y = e x. This function is graphed in the diagram elow. Notice that the function only takes positive values, and although it gets aritrarily close to the x-axis as x goes to, it never reaches it. The exponential function can e calculated in Excel using the EXP( ) function. Financial calculators should also contain a key which enales you to calculate e x. The point highlighted in the chart elow is e 2 7.389056. 4.5 Logarithms For a fixed positive numer, if x can e written as x = y for some real numer y, then we say y is the logarithm of x for the ase and we write: y = log (x) Equivalently, the logarithm the answer x: x log is that power to which must e raised to give x = log (x) This means that the logarithm function function of the power function y = x. x y log for the ase is the inverse Macquarie University MAFC_Essential_Maths Page 14 of 20
For any positive numer and for every possile x the power x will always e positive. It follows that, the domain of its range is all real numers. x y log is the positive real numers, while The logarithm for the ase can e calculated in Excel using the function LOG(x, ). For example, the graph of the logarithm of x for the ase 2 is shown in the chart elow. Because 2 0 = 1, 2 1 = 2, 2 2 = 4 and 2 3 = 8, the graph of x y log 2 crosses the x-axis at 1 and passes through the points (2, 1), (4, 2) and (8, 3). It can e seen that the curve gets aritrarily close to the x-axis, ut never meets it. Corresponding to each of the rules aove for powers given in Section 4.3, are the following rules for logarithms: 1 = log 1 0 = 1 log 1 0 x y = x+y log x y log x log y x / y = x y log x / y log x log y x y x y y log x y log x 1 1 x can never equal 0 1 log log x log 0 x is not defined Macquarie University MAFC_Essential_Maths Page 15 of 20
For an amount of A invested at a rate of r% per period, logarithms can e used to solve for the numer of periods it will take for the investment to reach a given amount. For instance, suppose the interest rate is 2% per period. To calculate the numer of periods m for an investment to doule in size, we solve for m in: A (1.02) m = 2A Taking logs of oth sides of the equation, for instance to ase 10 (ut any ase could e used) and manipulating using the rules aove: Log 10(1.02) m = Log 10(2) m Log 10(1.02) = Log 10(2) m = Log 10(2) Log 10(1.02) = 35 periods 4.6 Natural Logarithm The natural logarithm of a positive numer x is the logarithm of x to the ase e, the special irrational numer defined aove in Section 2.2. The natural logarithm is usually written ln(x). The graph of the function y = ln(x) is shown in the diagram elow. As for all logarithms, since e 0 = 1, the function y = ln(x) crosses the x-axis at 1, and since e 1 = e, log e (e) = 1. Since x = e ln(x) for all x > 0, the natural logarithm function y ln x is the inverse function of the exponential function y = e x, and vice versa. 5 y = EXP(x) 4 y = x 3 2 y = LN(x) 1 0-5 -4-3 -2-1 -1 0 1 2 3 4 5-2 -3-4 Macquarie University MAFC_Essential_Maths Page 16 of 20
Further Reading For more information on logarithms, see: Swift & Piff [1], Chapter MM, (More Maths), Section 4, Some Special Equations, pages 123 to 132. http://mathworld.wolfram.com/logarithm.html http://mathworld.wolfram.com/naturallogarithm.html 5 Derivatives 5.1 Rates of Change Rates of change are extremely important in economics and finance. Example include: The rate at which revenue changes as output (the quantity produced) changes is the marginal revenue. The marginal product of laour is the rate of change of output as the hours of laour changes. The price elasticity of demand is the rate of change of the demand for a good as the price of the good changes. Delta is the change in the value of an option as the price of the underlying asset changes. It is important for hedging option positions. The rate of change is referred to as the derivative of the function. For a function expressed as y = f(x), the derivative of the function is commonly denoted: dy dx or f (x) In other words, the change in the function y = f(x) for a very small change in x, is the derivative of y with respect to x. Graphically, the rate of change of the function is the slope of the tangent to the curve representing the function. We have seen aove that for a linear function, this is constant for all values of x. For non-linear functions, the slope of the tangent changes as we move around the function. The slope of the tangent to the curve is given y the ratio of the change in y, to the change in x, as the change in x ecomes aritrarily small. Macquarie University MAFC_Essential_Maths Page 17 of 20
Tangent at x=3 Δy = change in y Δx = change in x We can, however, find the rate of change of a function without graphing it. To determine the rate of change of a function we determine its derivative, for instance, y using the rules for differentiation given in the next section. The derivative of a function is itself a function. The rate of change of the function at a point is the value of the derivative function at that point. For example, the quadratic displayed in the chart aove is: y = f(x) = x 2 2x + 1 The derivative of this function is: dy dx = f (x) = 2x 2 This tells us that the slope of the function y = x 2 2x + 1 at x = 0 is 2 0 2 = 2, and the slope of the function at x = 3 is 2 3 2 = 4, etc. At the point x = 1, the derivative is 2 1 2 = 0. This means that the curve is flat at this point. It can e seen in the chart aove that this is the turning point and minimum value of the quadratic y = x 2 2x + 1. In general, setting the derivative equal to zero and solving for x is useful in finding maximums and minimums of functions. Further Reading More explanation and examples can e found in Swift & Piff [1], Chapter MM, (More Maths), Section 3, Rates of Change. Macquarie University MAFC_Essential_Maths Page 18 of 20
5.2 Rules of differentiation A numer of rules to differentiate functions are demonstrated in the tale elow. Rule / Example Function Derivative Power rule y = x n dy dx = n xn 1 Example y = x 3 dy dx = 3 x2 Multiply y a constant y = a g(x) dy dx = a dg dx Example y = 5x 3 dy dx = 5 d dx (x3 ) = 5 3 x 2 = 15 x 2 Addition Rule y = g(x) + h(x) dy dx = dg dx + dh dx Example y = 6x 3 + 3x 2 dy dx = 18 x2 + 6 x Product Rule y = g(x) h(x) dy dx = dg dx h(x) + g(x) dh dx Example y = 6x 3 (3 + x 2 ) dy dx = 18 x2 (3 + x 2 ) + 6x 3 (2x) = 6 x 2 (9 + 15x 4 ) Quotient Rule y = g(x) h(x) dy dx = 1 h(x) 2 ( dg dx h(x) g(x) dh dx ) Example y = 3x2 + 3 2x dy dx = (6x) 2x (3x2 + 3) 2 (2x) 2 = 3x2 3 2x 2 Macquarie University MAFC_Essential_Maths Page 19 of 20
Rule / Example Function Derivative Chain Rule y = g(h(x)) dy dx = dg dh(x) dh dx Example y = (x 4 + 1) 3 dy dx = 3 (x4 + 1) 2 (4x 3 ) = 12 x 3 (x 4 + 1) 2 Exponential Rule y = e x dy dx = ex Logarithmic Rule y = ln(x) dy dx = 1 x Note that the exponential function is the only function whose derivative is itself. At every point the rate of change of the exponential function is the same as the value of the function. Macquarie University MAFC_Essential_Maths Page 20 of 20