Notes, summaries, assignments, exams, and exercises for Mathematics for Primary education

Market Value Performance Measures

• Market Value Added (MVA): Market Capitalization (Number of shares × Price per share) - Equity.
• Market-to-Book Ratio: Measures value added per dollar invested by shareholders. Calculated as: Market Value of Equity / Book Value of Equity (Total Shareholders' Equity).

Measuring Profitability

• Economic Value Added (EVA): Profit after deducting all costs, including the cost of capital. Formula: [After-tax operating income] - [Cost of Capital × Total Capitalization at start of year (Long-term debt + Total equity)].
• Return on Equity (ROE): Net Income / Equity (start of year).
• Return on Assets (ROA): [(1 - Tax Rate) × Interest Expense + Net Income] / Total Assets (start of year).
• Return on Capital (ROC): [(1 - Tax Rate)

Capital Budgeting and Cost of Capital Analysis

13) The company cost of capital may be an inappropriate discount rate for a capital budgeting proposal if the proposal has a different degree of risk. (ch13)

Calculating Weighted Average Cost of Capital (WACC)

1) The company cost of capital for a firm with a 65/35 debt/equity split, an 8% before-tax cost of debt, a 15% cost of equity, and a 35% tax rate would be:

Calculation: 0.65 × (1 - 35%) × 8% + 0.35 × 15% = 8.63%

Determining After-Tax Cost of Debt

2) The firm has 7,500 bonds outstanding with a face value of $1,000 per bond. The bonds carry a 7% coupon, pay interest semiannually, and mature in 8 years. The bonds are selling at 98% of face value. The company's tax rate is 34%. What is the firm'...

Revenue and Cost Formulas

• Revenue = Selling price per unit × Number of units sold
• Variable costs = Variable cost per unit × Number of units sold
• Total costs = Fixed costs + Variable costs
• Profit = Total revenue − Total costs or Total contribution − Fixed costs

Market and Investment Metrics

• Market capitalisation = Number of issued shares × Current share price
• Expected value (A & B) = (Pay-off of A × Probability of A) + (Pay-off of B × Probability of B)
• Net gain = Expected value − Initial cost of decision
• Market growth (%) = (Change in market size / Original market size) × 100
• Market share (%) = (Sales of one product or business / Total sales in the market) × 100
• Added value = Sales revenue − Cost of bought-in goods and services

Operational

Two-State Actuarial Model

The Two-State Model (also known as the Dead-Alive or Binary Model) is a fundamental actuarial framework used to represent processes that exist in one of two possible states, such as Alive/Dead or Working/Retired. It is widely utilized in life insurance and pension modeling to estimate the probability of transition between states. The model assumes that at any given time, an individual occupies only one state, allowing actuaries to calculate premiums, reserves, and expected present values by simplifying complex uncertainties into binary outcomes.

Core Assumptions

• Binary States: The system exists in only one of two states at any time.
• Markov Property: Transitions depend solely on the current state.
• Constant Probabilities:

What is a Shapefile?

A shapefile is an Esri vector data storage format for storing the location, shape, and attributes of geographic features. It is stored as a set of related files and contains one feature class. Google Earth, an Online Satellite Imagery (OSI) Technique, is a tool for generating shapefiles.

Geometric Analysis Tools in Projects

Example 1: Population Maps of To Kwa Wan using ArcGIS

We used ArcGIS to create population maps of To Kwa Wan.

1. Selected population and working population data from the C&SD department.
2. Opened TPUs with the population data in ArcGIS.
3. Calculated the area of the TPUs based on Hong Kong 1980 Grid, determining population densities.
4. Created maps with different population densities, using graduated colors to represent

1) If a toy company is planning to introduce 3 new toys next Christmas. The toys sales can be a success or a failure, what is the probability that at least 2 of the toys are a success?

FFF FFS FSF FSS 4/8 = ½ = 0.5

S SFF SFS SSF SSS

For problems 2 and 3: a guitar manufacturer has an average daily production of 60 units, with a standard deviation of 7 units per day. If the guitar production distribution is mound shape:

2) What percentage of days in a month will have a daily production between 53 and 74 units?

53

3) What percentage of days in a month will have a daily production of more than 67 units?

x>67 = 16%

4) A combination lock system has three ten-digit (0 to 9) number wheels side by side. How many four-digit opening numbers are possible if...

Slope, Inclination, and Lines

PENDING

Slope

Inclination and Tilt

This refers to the straight tilt or angle of a line.

Parallel Lines Definition

Two lines are parallel if their slopes are equal: L₁ || L₂ if m₁ = m₂.

To find the angle (theta), the slope is used, often in conjunction with trigonometric tables or functions.

Perpendicular Lines Definition

Two lines are perpendicular if they form an angle of 90⁰. This occurs when the product of their slopes is -1:

(m₁)(m₂) = -1

General Equation of a Line

To formulate the general equation of a line, two components are typically needed:

• The slope (m)
• A specific point on the line (x₁, y₁)

The variables (x, y) in the equation represent any point on that line.

Midpoint of a Line Segment

The coordinates of the midpoint...

Question 1: Decimal Representation of a Fraction

Question: Consider the fraction 6/7. The decimal representation of this fraction is:

Answer: 6 ÷ 7 = 0.857142857... (repeating)

Question 2: Vaccinated to Unvaccinated Ratio

Question: If 60% of a population is vaccinated, what is the ratio of vaccinated to unvaccinated individuals?

Answer: 60% vaccinated → 60 : 40 → Simplified = 3 : 2

Question 3: Property Tax Calculation

Question: A property has been assessed at $225,000. The mill rate is 14.5. To find the property tax, you would multiply the assessed value by:

Answer: The mill rate of 14.5 means $14.50 per $1,000 of assessed value. To convert this to a decimal factor, divide by 1,000:

• 14.5 ÷ 1,000 = 0.0145
• Property tax = $225,000 × 0.0145 = $3,262.