Interest on interest
With simple interest, you earn interest only on the principal. With compound interest, interest is added to the balance, and then that interest earns interest too. Over time, the difference is huge.
The formula
A = P(1 + i)ⁿ
| Symbol | Meaning |
|---|---|
| A | future value |
| P | principal |
| i | interest rate per compounding period (annual rate ÷ periods per year) |
| n | total number of compounding periods |
Compounding periods
| Compounding | Periods per year |
|---|---|
| annually | 1 |
| semi-annually | 2 |
| quarterly | 4 |
| monthly | 12 |
| daily | 365 |
Example: annual compounding
$5,000 at 4% per year, compounded annually, for 10 years.
A = 5,000(1.04)¹⁰ ≈ $7,401.22
Interest earned: $2,401.22. (Simple interest would give only $2,000.)
Example: monthly compounding
$5,000 at 4% per year, compounded monthly, for 10 years.
- i = 0.04/12 ≈ 0.003333
- n = 120
- A = 5,000(1.003333)¹²⁰ ≈ $7,454.16
More frequent compounding gives slightly more.
Simple vs compound
| Year | Simple (5% on $1,000) | Compound (5% annually) |
|---|---|---|
| 0 | $1,000 | $1,000 |
| 10 | $1,500 | $1,628.89 |
| 20 | $2,000 | $2,653.30 |
| 30 | $2,500 | $4,321.94 |
Compound growth is exponential; simple growth is linear.
Present value
How much to invest now to have $10,000 in 5 years at 6% compounded annually?
P = A ÷ (1 + i)ⁿ = 10,000 ÷ 1.06⁵ ≈ $7,472.58
This is the present value.
The Rule of 72
Years to double ≈ 72 ÷ rate.
- At 6%: about 12 years.
- At 9%: about 8 years.
The Rule of 72 is a quick mental estimate.
Start early
$2,000 invested at age 20 at 6% grows to about $20,571 by 60 (40 years). The same $2,000 invested at 40 grows to about $6,414. Time is the biggest factor in compound growth.
Compound interest on debt
It works against you on loans and credit cards: unpaid interest is added to the balance, and you pay interest on it.
Worked examples
Example 1: $1,200 at 3% compounded annually for 4 years? 1,200 × 1.03⁴ ≈ $1,350.61.
Example 2: $800 at 6% compounded quarterly for 2 years? i = 0.015, n = 8 → ≈ $901.19.
Example 3: How long to double at 8%? Rule of 72 → about 9 years.
Common mistakes
- Using the annual rate without dividing by the periods per year.
- Using years for n instead of total periods.
- Confusing future value and present value.
Try it yourself
- $2,500 at 5% compounded annually for 6 years.
- $10,000 at 4.8% compounded monthly for 3 years.
- Present value of $5,000 due in 4 years at 3% compounded annually.
- Rule of 72: years to double at 4%?
Answers: 1) ≈ $3,350.24 2) ≈ $11,545.52 3) ≈ $4,442.44 4) about 18 years
Quick recap
- Compound interest: A = P(1 + i)ⁿ.
- Adjust i and n for the compounding period.
- Present value = A ÷ (1 + i)ⁿ; the Rule of 72 estimates doubling time.