Simple and Compound Interest Basics

Study Notes

Simple Interest Formula

Simple interest is calculated on the principal amount only.
The formula for simple interest is: Interest = Principal × Rate × Time
- Where:
  - Principal: The initial amount of money
  - Rate: The interest rate (expressed as a decimal)
  - Time: The duration of the loan or investment (often in years)
Example: If you borrow $1,000 at an annual interest rate of 5% for 2 years, the simple interest would be calculated as follows:
- Interest = $1,000 × 0.05 × 2 = $100
Total amount repayable = Principal + Interest

Compound Interest Calculation

Compound interest is calculated on the principal amount plus any accumulated interest.
It's calculated on the interest earned in previous periods.
The formula for compound interest is more complex and depends on whether interest is compounded annually, semi-annually, quarterly or monthly.
- Generally, the formula for compound interest is:
  - A = P (1 + r/n)^(nt) where:
    - A = the future value of the investment/loan, including interest
    - P = the principal investment amount (the initial deposit or loan amount)
    - r = the annual interest rate (decimal)
    - n = the number of times that interest is compounded per year
    - t = the number of years the money is invested or borrowed for
Example: If you deposit $1000 in a savings account that pays an annual interest rate of 5% compounded annually for 2 years, the future value would be:
- A = 1000(1+0.05/1)^(1*2) ≈ $1102.50
To show that compounding increases the return, note that the simple interest would be $100 for this example - a significant difference
Important Considerations for Compound Interest Calculation
- Frequency of compounding matters: The more frequent the compounding (e.g., daily, monthly), the higher the future value will be.
- Number of compounding periods: The calculation considers the number of times per year interest is compounded (e.g., annual compounding (n = 1), semi-annual compounding (n = 2)).
- Calculating Compound Interest for Different Compounding Periods
  - If interest is compounded semi-annually, use 0.05/2 as r and 2n as the compounding frequency, etc
Example with semi-annual compounding: Consider a principal of $1000, an annual interest rate of 5%, and a time period of 2 years, compounded semi-annually.
- A = $1000(1 + 0.05/2)^(2*2) = 1000(1.025)^4 ≈ $1103.81
Key Differences between Simple and Compound Interest
- Simple interest only calculates interest on the principal amount.
- Compound interest calculates interest on the principal and on the accumulated interest of previous periods.
- Compound interest generally yields a higher return than simple interest over the long term, especially with longer periods.

Sample Question - Compound Interest:

A principal of $5,000 is invested at an annual interest rate of 6% compounded quarterly for 3 years.
- Calculate the future value of the investment at the end of the 3-year period.
Solution:
- P = 5000
- r = 0.06
- n = 4 (quarterly compounding)
- t = 3
- Future value A = 5000 (1 + 0.06/4)^(4*3)
- A = 5000 * (1.015)^12
- A ≈ $6000.00

Sample Question - Simple Interest

A borrower takes out a loan of $2,000 at an annual interest rate of 4% for a period of 5 years. Calculate the total amount repayable in simple interest.
Solution:
- Principal = $2000
- Rate = 4% (0.04)
- Time = 5 years
- Interest = PRT = 20000.045 = $400
- Total amount repayable = $2,000 + $400 = $2,400

Description

Explore the fundamental concepts of simple and compound interest with clear formulas and practical examples. This quiz covers the calculations for both types, helping you understand how interest is accrued over time. Perfect for students learning finance principles.