Compound interest is the growth of an investment or loan where the interest earned over time is reinvested or added to the principal, so that in future periods interest is earned on both the initial amount and already-accrued interest. This compounding effect makes money grow faster than simple interest, especially with higher rates, longer time horizons, and more frequent compounding. Key concepts

• Formula: A = P(1 + r/n)^(nt)
  • A = ending amount
  • P = principal (initial amount)
  • r = annual interest rate (as a decimal)
  • n = number of times interest is compounded per year
  • t = time in years
  • Interest earned = A − P
  • The same formula can be rearranged to solve for any unknown (e.g., P, r, t) given the others.
• Compounding frequency (n)
  • The more often interest is compounded (daily, monthly, quarterly, etc.), the higher the ending balance, assuming the rate stays the same.
• Effective annual rate (EAR)
  • EAR accounts for compounding within the year and is typically higher than the nominal rate when interest is compounded more than annually.
• Practical implications
  • Small improvements in rate, extra contributions, or longer time horizons can dramatically increase final results due to compounding.
  • Compound interest benefits savers over investors, whereas borrowers pay more over time when compounding is involved.

Quick examples

• Example 1: Annual compounding
  • P = 1,000, r = 5% (0.05), n = 1, t = 3
  • A = 1000(1 + 0.05/1)^(1*3) = 1000(1.05)^3 ≈ 1,157.63
  • Interest earned ≈ 157.63
• Example 2: Monthly compounding
  • P = 1,000, r = 5% (0.05), n = 12, t = 3
  • A = 1000(1 + 0.05/12)^(12*3) ≈ 1,161.62
  • Interest earned ≈ 161.62

Common pitfalls

• Assuming simple interest when compounding occurs more frequently than yearly.
• Neglecting contributions or withdrawals that occur during the period, which alter the effective compounding base.
• Ignoring fees or taxes that reduce the net growth of the account.

If you’d like, provide: principal P, annual rate r (as a percentage), compounding frequency n, and time t (in years), and the ending amount A and interest earned can be calculated for you.