Lecture on Compounded Interest

Introduction

• Most people have checking or savings accounts that pay interest.
• Banks typically use compounded interest to calculate the interest paid each month.

Compounded Interest Formula

• Principal (P): Initial investment amount.
• Interest Rate (I): Must be expressed as a decimal (e.g., 8% becomes 0.08).
  • Convert percentage to decimal by dividing by 100.
• Number of Compounds per Year (N):
  • Quarterly: N = 4
  • Monthly: N = 12
• Time (T): Expressed in years (e.g., 18 months = 1.5 years).
• Amount (A): Total amount after the given time.

Example 1: Quarterly Compounding

• Scenario: Invest $1,000 at 8% interest compounded quarterly for 3 years.
  • Principal, P = $1,000
  • Interest rate, I = 0.08
  • Compounded quarterly, N = 4
  • Time, T = 3 years
• Calculation Process:
  • Convert annual interest to quarterly: 0.08 / 4
  • Total number of compounding periods: 4 * 3 = 12 quarters
  • Formula setup: 1000 * (1 + 0.08/4)^(4*3)
• Result: $1,268.24 after 3 years*

Example 2: Daily Compounding

• Scenario: Change compounding to daily with same conditions.
  • Principal, P = $1,000
  • Interest rate, I = 0.08
  • Compounded daily, N = 365
  • Time, T = 3 years
• Calculation Process:
  • Convert annual interest to daily: 0.08 / 365
  • Total number of compounding periods: 365 * 3 = 1,095 days
  • Formula setup: 1000 * (1 + 0.08/365)^(365*3)
• Result: $1,271.22 after 3 years
• Comparison: Daily compounding results in a higher amount due to more frequent interest application.*

Conclusion

• More frequent compounding periods result in higher total amounts due to more frequent interest application.
• Though differences may seem small on a small scale, they are significant with larger principal amounts (millions or billions).
• Understanding compounded interest is crucial for managing investment returns.