## Topic Overview

Financial Mathematics involves using mathematical methods to solve problems related to finance, such as calculating interest, annuities, and investments. Understanding these concepts is crucial for making informed financial decisions in real life.

### Key Learning Objectives

- Understand the difference between simple and compound interest.
- Calculate future and present values of investments.
- Work with annuities and amortization schedules.
- Apply financial mathematics to real-world problems.

## Key Terms and Definitions

**Interest**: The cost of borrowing money or the earnings from investing money.**Principal (P)**: The initial amount of money borrowed or invested.**Simple Interest (SI)**: Interest calculated on the principal only.**Compound Interest (CI)**: Interest calculated on the principal and also on the accumulated interest of previous periods.**Future Value (FV)**: The value of an investment at a specific date in the future.**Present Value (PV)**: The current value of a future amount of money.**Annuity**: A series of equal payments made at regular intervals.**Amortization**: The process of spreading out a loan into a series of fixed payments over time.

## Main Content Sections

### Simple Interest

**Formula**: ( \text{SI} = P \times r \times t )

– **P**: Principal amount

– **r**: Interest rate per period

– **t**: Time period

**Example**: If you invest R5000 at an annual simple interest rate of 5% for 3 years, the interest is calculated as follows:

[

\text{SI} = 5000 \times 0.05 \times 3 = 750

]

The future value (FV) will be:

[

\text{FV} = P + SI = 5000 + 750 = 5750

]

### Compound Interest

**Formula**: ( \text{FV} = P \times (1 + \frac{r}{n})^{n \times t} )

– **P**: Principal amount

– **r**: Annual interest rate

– **n**: Number of times interest is compounded per year

– **t**: Time in years

**Example**: If you invest R5000 at an annual compound interest rate of 5% for 3 years, compounded annually:

[

\text{FV} = 5000 \times (1 + \frac{0.05}{1})^{1 \times 3} = 5000 \times (1.05)^3 \approx 5788.13

]

### Present Value

**Formula**: ( \text{PV} = \frac{\text{FV}}{(1 + \frac{r}{n})^{n \times t}} )

**Example**: To find the present value of R5788.13 needed in 3 years at an annual compound interest rate of 5%, compounded annually:

[

\text{PV} = \frac{5788.13}{(1 + 0.05)^3} \approx 5000

]

### Annuities

**Future Value of an Annuity**:

[

\text{FV}_\text{annuity} = PMT \times \left( \frac{(1 + r)^n – 1}{r} \right)

]

– **PMT**: Payment amount per period

– **r**: Interest rate per period

– **n**: Number of payments

**Present Value of an Annuity**:

[

\text{PV}_\text{annuity} = PMT \times \left(1 – \frac{1}{(1 + r)^n}\right) \div r

]

**Example**: To calculate the future value of monthly payments of R1000 for 3 years at an annual interest rate of 6%, compounded monthly:

[

r = \frac{0.06}{12} = 0.005 \quad \text{and} \quad n = 3 \times 12 = 36

]

[

\text{FV}_\text{annuity} = 1000 \times \left( \frac{(1 + 0.005)^{36} – 1}{0.005} \right) \approx 1000 \times 38.993 \approx 38993

]

### Amortization

Amortization schedules are used to pay off loans with regular payments. Each payment covers interest and principal repayment.

## Example Problems or Case Studies

**Problem 1**: Calculate the compound interest on an investment of R10000 for 5 years at 8% interest compounded quarterly.- Solution: ( r = \frac{8}{4} = 0.02 ) and ( n = 5 \times 4 = 20 )

[

\text{FV} = 10000 \times (1 + 0.02)^{20} \approx 10000 \times 1.485947 \approx 14859.47

] **Case Study**: Calculate the monthly mortgage payment for a home loan of R500000 over 20 years at an annual interest rate of 10%, compounded monthly.- Use the Present Value of an Annuity formula to find the monthly payments.

## Summary or Review Section

- Simple Interest is calculated on the principal alone.
- Compound Interest considers both the principal and previously earned interest.
- Future Value and Present Value are essential for understanding investment growth.
- Annuities involve regular payments and can be used for calculating retirement savings or loan payments.
- Amortization schedules help manage loans with fixed repayment structures.

## Self-Assessment Questions

- Calculate the simple interest on R7000 invested for 4 years at an annual interest rate of 6%.
- What is the future value of R2000 invested for 3 years at an annual compound interest rate of 5%, compounded semi-annually?
- If you need R50000 in 5 years, how much should you invest now at 7% annual interest, compounded quarterly?
- Find the future value of an annuity with monthly payments of R1500 for 10 years at an 8% annual interest, compounded monthly.
- What will be the monthly repayment for a car loan of R200000 over 6 years at an annual interest rate of 12%, compounded monthly?

## Connections to Other Topics/Subjects

**Algebra**: Understanding formulas and solving equations.**Economics**: Applying interest calculations for loans, investments, and savings.**Business Studies**: Financial planning and management.

Remember to review and practice these concepts regularly, and seek help when needed to ensure you have a firm grasp of financial mathematics. Happy studying!