Chapter 2

Mathematical and Statistical Foundations

• A function is a mapping or relationship between an input or set of inputs and an output
• Functions can be linear, where the relationship can be expressed on a straight line, or non-linear, where it would be expressed graphically as a curve.
• A linear function can be expressed as y = a + bx, where y and x are variables and a and b are parameters
• a is the intercept and b is the slope or gradient
• The intercept is the point at which the line crosses the y-axis.

Straight Lines

• Example: Modeling the relationship between a student's average mark (y, in percent) and the number of hours studied per year (x).
• Example equation: y = 25 + 0.05x
• Intercept (a) = 25
• Slope (b) = 0.05
• This means that with no study (x = 0), the student could expect to earn a mark of 25%.
• For every hour of study, the grade would on average improve by 0.05%, and 100 hours of study would lead to a 5% increase in the mark.

Roots

• The root of an equation is the point where the line crosses the x-axis.
• A straight line has one root unless it is horizontal (e.g., y = 4).
• To find the root, set y = 0 and rearrange the equation.
• Example: for the equation y = 25 + 0.05x, setting y to zero yields x = -500

Quadratic Functions

• A linear function is often not flexible enough to accurately describe the relationship between two series.
• A quadratic function can be a better fit.
• The general form of a quadratic function is y = a + bx + cx².
• a, b, and c are parameters describing the function's shape.
• A linear function is a special case of a quadratic where c = 0.
• If c is positive, the function is U-shaped, and if it's negative, it's inverted U-shaped.

The Roots of Quadratic Functions

• A quadratic equation has two roots.
• Roots can be distinct, repeated, real, or complex.
• Roots can be found by factoring the equation, completing the square, or using the quadratic formula.

The Roots of Quadratic Functions (Cont'd)

• If b² > 4ac, the function has two unique roots and crosses the x-axis in two separate places.
• If b² = 4ac, the function has two equal roots and crosses the x-axis in one place.
• If b² < 4ac, the function has no real roots and does not cross the x-axis.

Calculating the roots of quadratics - examples

• Examples of quadratic equations are provided to determine their roots.

Calculating the roots of quadratics - solutions

• The equations are solved by setting them to zero and then finding possible factors
• Quadratic equations always have two roots.
• The examples demonstrate the use of factoring and sometimes the quadratic equation is required to find the roots.

Powers of Number or of Variables

• Raising a number or variable to a power is a way of writing repeated multiplication.
• x² = x × x, x³ = x × x × x
• The number being raised to a power is called the index.

Manipulating Powers and their Indices

• Any number or variable raised to the power one is simply that number or variable.
• Raising to the power zero yields one, except zero to the power zero, which is undefined.
• A negative index means reciprocal, e.g., xˉ³ = 1/x³.
• Multiplying terms with the same base involves adding indices, e.g., x² x x³ = x⁵.
• Raising a power to another power involves multiplying the indices e.g., (x²)³ = x⁶.
• Dividing terms with the same base involves subtracting indices, e.g., x⁵ / x² = x³.
• Non-integer powers are also possible, but harder to calculate by hand (e.g. √x, x⁰.⁷⁶).

The Exponential Function, e

• Exponential functions describe relationships where a variable grows or declines proportionally to its current value.
• The equation is expressed as y = eˣ
• e is a mathematical constant approximately equal to 2.71828.
• Exponential functions are useful in finance for representing compound interest.
• The exponential function is always positive and approaches zero as x becomes very negative.

Logarithms

• Logarithms simplify complex calculations, converting multiplication and division to addition and subtraction.
• Logarithms can help rescale data, making variance more consistent (addressing heteroscedasticity).
• Logarithms can transform a skewed distribution to a more normal distribution.
• Logarithms transform a non-linear multiplicative relationship into a linear additive relationship.

How do Logs Work?

• Logarithms are the power to which a base (e.g., 2) must be raised to equal a given number (e.g., log₂8 = 3).
• log functions usually intersect the x-axis at one
• as x increases y increases at a slower rate, which is opposite to exponential functions.

A Graph of a Log Function

• A graph showing a typical log function.

How do Logs Work?(Cont'd)

• Natural logarithms are logs to the base e (often denoted as ln(x) or logₑ(x)).
• Taking a natural logarithm is the inverse operation of exponentiation (so the exponential function is sometimes called the antilog).
• The log of a number less than one will be negative.
• You cannot take the log of a negative number.

The Laws of Logs

• Rules for manipulating logarithms. ln(xy) = ln(x) + ln(y), ln(x/y) = ln(x) - ln(y), ln(y^c) = c ln(y)., ln(1)=0

Sigma Notation

• Sigma notation is used to denote sums of numbers or observations.
• Σ is used to denote a sum, and implies summing over a set of values. Σᵢ=₁ ⁴ xᵢ means to add up the values of xᵢ from i=1 to i=4.

Properties of the Sigma Operator

• Rules for manipulating sigma notation (summation)

Pi Notation

• Similar to summation, pi (Π) notation implies multiplying over a set of values. Пᵢ=₁⁴ xᵢ means to multiply the values of xᵢ from i=1 to i=4

Differential Calculus

• The rate of change of one variable with respect to another is measured by a derivative.
• If a relationship between the two variables can be represented by a curve, the gradient of the curve is the rate of change.
  • y = f(x)
• dy/dx is notation for the derivative.

Differentiation: The Basics

• The derivative of a constant is zero.
• The derivative of a linear function is its slope.
• Non-linear functions have differing gradients at various points along the curve.
• The tangent line's slope represents the gradient of a curve at a particular point.
• Turning points are points on a curve where the tangent slope is zero (change in direction).

The Tangent to a Curve

• Diagram of a tangent to a curve.

The Derivative of a Power Function or of a Sum

• The derivative of a power function y = cxⁿ is dy/dx = cnxⁿ⁻¹.
• The derivative of a sum is equal to the sum of individual derivatives.
• The derivative of a difference is equal to the difference of individual derivatives.

The Derivatives of Logs and Exponentials

• The derivative of log x is 1/x.
• The derivative of a log of a function is f'(x) / f(x)
• The derivative of eˣ is eˣ.
• The derivative of e^(f(x)) is f'(x). e^(f(x)).

Higher Order Derivatives

• Higher-order derivatives represent the rate of change of the previous derivatives.
• The second-order derivative represents the rate of change of the gradient.
• Second derivatives are used to determine if a turning point is a maximum or minimum.

Maxima and Minima of Functions

• To find maxima and minima of functions, set the first-order derivative to zero and solve for the x-values.
• Positive or negative second derivatives indicate minima or maxima, respectively.

Partial Differentiation

• Partial differentiation calculates the rate of change of a function of multiple variables with respect to one variable, holding the others constant.
• Notation for partial derivative of 𝑦 with respect to 𝑥₁ is (∂y/∂x₁).

How to do Partial Differentiation (Cont'd)

• Partial derivatives are calculated by treating other variables as constants.
• Ordinary least squares (OLS) estimates use partial differentiation to find parameters that minimize the residual sum of squares.

Integration

• Integration is the reverse process of differentiation.
• It calculates the area under a curve.

Matrices - Background

• A scalar is a single number (e.g. 3, -5, 0.5).
• A vector is a one-dimensional array of numbers.
• A matrix is a two-dimensional array or collection of numbers.
• Matrices are useful for organizing and manipulating data in econometrics and finance.

Working with Matrices

• Matrix dimensions are quoted as R x C (rows x columns)
• Elements of a matrix are identified with sub-scripts (e.g. m23).
• A matrix with one row is a row vector, and one column is a column vector.
• A matrix with equal rows and columns is a square matrix.
• A matrix of all zeros is a zero matrix.

Working with Matrices 2

• A symmetric matrix is symmetrical about the leading diagonal (mᵢⱼ = mⱼᵢ).
• A diagonal matrix has non-zero terms only on the main diagonal.
• The identity matrix is a diagonal matrix with 1s on the main diagonal, and zeros elsewhere.

Working with Matrices 3

• The identity matrix is the matrix equivalent of the number 1.
• Multiplication by the identity matrix leaves a matrix unchanged.
• Matrices must be conformable to perform operations.

Matrix Addition or Subtraction

• Matrices must be of the same order to add or subtract.
• Addition and subtraction are performed element by element.

Matrix Multiplication

• The number of columns in the first matrix must equal the number of rows in the second matrix to multiply them.
• Matrix multiplication is not commutative (AB ≠ BA).
• Matrices don't have a standard division operation

Matrix Multiplication Example

• Illustrative example of matrix multiplication.

The Transpose of a Matrix

• The transpose of a matrix is obtained by switching its rows and columns.
• If a matrix is R x C, its transpose is C x R.

The Rank of a Matrix

• The Rank of a matrix is the maximum number of linearly independent rows (or columns)
• A matrix that has a rank equal to its dimension is of full rank, otherwise it’s of reduced rank.
• The rank of a Matrix is equal to the rank of its transpose.

The Inverse of a Matrix

• The inverse of a matrix, denoted A⁻¹, satisfies AA⁻¹ = I.
• The inverse only exists for square and non-singular matrices.
• Properties of the inverse of a matrix (A⁻¹) are provided.

Calculating Inverse of a 2×2 Matrix

• The formula for the inverse of a 2x2 matrix is given.
• The determinant is part of the formula to find the inverse of a 2x2 matrix.

The Trace of a Matrix

• The trace of a square matrix is the sum of the elements on its main diagonal.
• Properties of the trace of a matrix (Tr(A), Tr(cA), and others) are provided

The Eigenvalues of a Matrix

• Eigenvalues of a matrix are a set of values that, when used, can satisfy an equation to isolate the set of scalars (λ).
• |Π - λIp| = 0, where Ip is an identity matrix, must be zero for the characteristic root (λ) set of values to exist.

Calculating Eigenvalues: An Example

• Illustrative example of calculating eigenvalues of a 2x2 matrix.

Portfolio Theory and Matrix Algebra - Basics

• Portfolio allocation problems use matrix algebra in Finance, especially when analyzing weights for investment and the expected returns of various options.

The Variance-Covariance Matrix

• The variance-covariance matrix includes variances on the main diagonal and covariances elsewhere.
• σᵢⱼ is used to denote the covariance between returns on stock i and stock j.

Constructing the Variance-Covariance Matrix

• Variance-covariance matrices are constructed from observations about returns (after mean subtraction).

The Variance of Portfolio Returns

• The variance of a portfolio can be calculated using the variance-covariance matrix and portfolio weights.

The Correlation between Returns Series

• Correlation matrices relate all pairs of returns;
• Correlation matrices are symmetrical around the main diagonal.
• Portfolio variance can be calculated from a correlation matrix and diagonal matrix of standard deviations of the returns.

Selecting Weights for the Minimum Variance Portfolio

• Finding the portfolio weights (min w'Vw) that minimizes the portfolio variance, subject to a restriction of total wealth (w'1N = 1)

Selecting Optimal Portfolio Weights

• Finding optimal portfolio weights for the mean-variance efficient frontier involves maximizing the portfolio's expected return subject to a maximum variance constraint.