Chapter 2

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Questions and Answers

What does the slope (b) in the linear equation $y = a + bx$ represent?

The maximum value of y

The rate of change of y with respect to x (correct)

The initial value of y when x is zero

The value of x at which y is at its minimum

In the equation $y = 25 + 0.05x$, what does the value 25 represent?

The maximum possible grade

The number of hours studied

The expected mark with no study hours (correct)

The slope of the linear relationship

Which of the following statements about a straight line in a linear function is correct?

The slope remains constant throughout the line. (correct)

The intercept can be negative but the slope cannot.

The slope is always positive in linear relationships.

The slope changes at different points along the line.

If a student studies for 100 hours, what would their expected increase in marks be in the context of the equation $y = 25 + 0.05x$?

5% Signup and view all the answers

What does a negative slope in a linear function indicate about the relationship between x and y?

As x increases, y decreases. Signup and view all the answers

What is the derivative of the function y = 4x^3?

12x^2 Signup and view all the answers

What is the second derivative of the function y = 4x^5 + 3x^3 + 2x + 6?

60x^4 + 18x Signup and view all the answers

What does a negative second derivative indicate at a turning point?

It's a maximum point. Signup and view all the answers

What is the derivative of the function y = e^(3x^2)?

6xe^(3x^2) Signup and view all the answers

If y = log(x^3 + 2x - 1), what is the derivative dy/dx?

(3x^2 + 2)/(x^3 + 2x - 1) Signup and view all the answers

What is the general rule for the derivative of a sum of two functions?

The derivative is equal to the sum of their derivatives. Signup and view all the answers

What is the derivative of log(f(x)) in terms of its function f?

f'(x)/f(x) Signup and view all the answers

What is the first derivative of the function y = 3/x?

-3/x^2 Signup and view all the answers

What does the trace of a square matrix represent?

The sum of the terms on its leading diagonal Signup and view all the answers

What is one of the properties of the trace of a matrix?

Tr(A') = Tr(A) Signup and view all the answers

What is the condition for the matrix (Π − λIp) to have a non-zero solution?

The matrix must be singular Signup and view all the answers

Given a 2 × 2 matrix Π, what characteristic roots are identified in the example?

λ = 6 and λ = 3 Signup and view all the answers

What mathematical operation provides the expected return on a portfolio?

E(r)'w Signup and view all the answers

In the equation Πc = λc, what does λ represent?

An eigenvalue or characteristic root Signup and view all the answers

What does the matrix I_p represent in the context of eigenvalues?

An identity matrix of size p × p Signup and view all the answers

Which of the following statements about eigenvectors is true?

They are solutions to the equation Πc = λc. Signup and view all the answers

What does the derivative measure in the context of two variables?

The instantaneous rate of change of one variable with respect to another. Signup and view all the answers

What is the derivative of a constant function such as y = 10?

0 Signup and view all the answers

If y = 3x + 2, what does dy/dx equal?

3 Signup and view all the answers

What occurs at a turning point on a curve?

The gradient is zero. Signup and view all the answers

Which statement about non-linear functions is true?

The gradient will vary at different points along the curve. Signup and view all the answers

If y = x^2, what is the derivative dy/dx?

2x Signup and view all the answers

What does dy/dx represent?

The change in y for a unit change in x. Signup and view all the answers

How is the gradient of a curve represented?

By the slope of the tangent at a specific point. Signup and view all the answers

What condition indicates that a quadratic function will have two distinct real roots?

b² > 4ac Signup and view all the answers

What are the roots of the quadratic equation y = x² - 4x?

2 and 2 Signup and view all the answers

Which scenario indicates that a quadratic function will not intersect the x-axis?

b² < 4ac Signup and view all the answers

How can the roots of a quadratic equation be found if it cannot be factored easily?

By applying the quadratic formula Signup and view all the answers

In the equation y = 9x² + 6x + 1, what type of roots does it have?

One repeated real root Signup and view all the answers

What does the term 'repeated roots' mean in the context of quadratic functions?

The roots are both the same number Signup and view all the answers

For which of the following equations will the roots be complex?

y = x² + 4 Signup and view all the answers

What does the leading diagonal of the variance-covariance matrix represent?

The variances of the component portfolio returns Signup and view all the answers

How is the variance-covariance matrix V calculated?

By subtracting the mean returns and then using R'R/(T-1) Signup and view all the answers

What is the correct representation of the portfolio variance VP?

VP = w′Vw Signup and view all the answers

What can be deduced about the correlation matrix C?

It is symmetrical about the leading diagonal Signup and view all the answers

In the formula VP = w′SCSw, what does S represent?

The diagonal matrix of standard deviations Signup and view all the answers

What does σ12 represent in the variance-covariance matrix?

The covariance between stock one and stock two Signup and view all the answers

Why must the mean returns be subtracted from the actual returns when constructing the variance-covariance matrix?

To normalize the data for easier comparison Signup and view all the answers

What dimension does the portfolio variance VP have as a result of the matrix operations?

It is a scalar value (1 × 1) Signup and view all the answers

Study Notes

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.

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Test your understanding of linear functions and their properties. This quiz covers key concepts such as slope interpretation, intercept values, and the implications of positive and negative slopes in linear equations. Perfect for students studying algebra and mathematics.