Learning goals
- Distinguish simple interest from compound interest
- Apply the compound interest formula for a lump sum
- See how compounding frequency and time change the final amount
Simple vs compound interest
Simple interest pays only on the original principal. If you invest $1,000.00 at 5% simple interest for 3 years, you earn $50.00 each year — $150.00 total — and end with $1,150.00.
Compound interest pays on the growing balance. Year 1 interest is added to principal; year 2 interest is calculated on a larger amount. Over long periods, compounding is the engine of savings and investment growth.
Banks, savings accounts, and many investments quote annual rates but may compound monthly or daily. Always match the formula to the compounding schedule.
Compound amount (lump sum)
A = P(1 + r/n)^(n t)
P is principal, r is nominal annual rate (decimal), n is compounds per year, t is years, and A is the future amount. Interest earned = A − P.
Annual compounding shortcut
A = P(1 + r)^t
When interest compounds once per year, n = 1 and the formula simplifies. Useful for back-of-envelope estimates.
Example 1 — Annual compounding
Given
$1,000.00 at 5% per year, compounded annually, for 3 years.
Steps
- Year 1: 1,000 × 1.05 = 1,050.
- Year 2: 1,050 × 1.05 = 1,102.50.
- Year 3: 1,102.50 × 1.05 = 1,157.625 ≈ $1,157.63.
- Or one shot: A = 1000 × (1.05)^3 = 1,157.625.
Answer
About $1,157.63 — roughly $7.63 more than simple interest over the same period.
Example 2 — Monthly compounding
Given
$1,000.00 at 5% APR, compounded monthly, for 3 years.
Steps
- r = 0.05, n = 12, t = 3 → n t = 36 periods.
- Periodic rate = 0.05/12 ≈ 0.0041667.
- A = 1000 × (1 + 0.05/12)^36 ≈ 1,161.47.
Answer
About $1,161.47 — slightly more than annual compounding at the same nominal rate.
Tip: More frequent compounding (with the same nominal APR) usually yields a bit more, unless the bank quotes an already-effective rate.
Example 3 — Time matters more than you feel
Given
Same $1,000.00 at 5% annual compounding for 10 years vs 3 years.
Steps
- 3 years: 1000 × (1.05)^3 ≈ 1,157.63.
- 10 years: 1000 × (1.05)^10 ≈ 1,628.89.
- Growth from year 3 to 10 is larger than the first three years alone because interest stacks on a bigger base.
Answer
Leaving money invested longer lets compounding do more of the work.
Confusing APR with APY
Common mistake
Assuming a 5% APR compounded monthly grows exactly 5% in one year.
Better approach
Effective annual yield is (1 + r/n)^n − 1. Monthly 5% APR ≈ 5.12% effective. Compare APY when shopping accounts.
Ignoring contributions
Common mistake
Using only the lump-sum formula when you add monthly deposits.
Better approach
Regular contributions need a savings / future-value-of-annuity model (or Quantarizm’s savings and compound interest tools with deposits).
Check your understanding
1.What is $500.00 at 4% annual compounding after 2 years?
Answer: 500 × (1.04)^2 = 540.80.
2.In A = P(1 + r/n)^(n t), what is n for quarterly compounding?
Answer: n = 4.
3.If A = $1,200.00 and P = $1,000.00 how much interest was earned?
Answer: A − P = $200.00.
Key takeaways
- Compound interest earns interest on prior interest; balances grow faster over long periods.
- A = P(1 + r/n)^(n t) for lump sums; match n to the compounding schedule.
- More frequent compounding usually increases effective yield for the same nominal APR.
- Time in the market often matters as much as a small rate difference.