How to Calculate Compound Interest by Hand
You don't need a calculator to understand compound interest — doing it by hand once makes the concept click in a way that using a tool never does. This guide walks through the calculation step by step.
The Formula
A = P × (1 + (r) / (n))^(n × t)
- A = final amount (principal + interest)
- P = principal (starting amount)
- r = annual interest rate as a decimal (e.g. 5% = 0.05)
- n = compounding periods per year
- t = time in years
Step-by-Step: Annual Compounding
Example: $2,000 at 6% annual interest for 3 years, compounded annually (n=1).
Step 1: Write down the values.
- P = 2000, r = 0.06, n = 1, t = 3
Step 2: Simplify the formula for annual compounding. When n = 1, the formula becomes: A = P × (1 + r)^t
Step 3: Calculate (1 + r). 1 + 0.06 = 1.06
Step 4: Raise to the power of t. 1.06^3 = 1.06 × 1.06 × 1.06
Do this in steps:
- 1.06 × 1.06 = 1.1236
- 1.1236 × 1.06 = 1.191016
Step 5: Multiply by the principal. A = 2000 × 1.191016 = $2,382.03
Interest earned = $2,382.03 − $2,000 = $382.03
Year-by-Year Breakdown
You can also track it year by year — same result, more insight:
| Year | Opening Balance | Interest (6%) | Closing Balance |
|------|----------------|---------------|-----------------|
| 1 | $2,000.00 | $120.00 | $2,120.00 |
| 2 | $2,120.00 | $127.20 | $2,247.20 |
| 3 | $2,247.20 | $134.83 | $2,382.03 |
Notice: year 2 earns $7.20 more than year 1, and year 3 earns $7.63 more than year 2. That's compounding — interest on interest.
Monthly Compounding (n = 12)
Same example: $2,000 at 6% for 3 years, now compounded monthly.
Step 1: Calculate the monthly rate. r/n = 0.06/12 = 0.005
Step 2: Calculate total compounding periods. n × t = 12 × 3 = 36
Step 3: Calculate (1 + r/n). 1 + 0.005 = 1.005
Step 4: Raise to the power 36. 1.005^36 — this is harder to do by hand. Use logarithms:
ln(1.005^36) = 36 × ln(1.005) = 36 × 0.004988 = 0.17957
e^0.17957 ≈ 1.1967
Step 5: Multiply. A = 2000 × 1.1967 = $2,393.40
Monthly compounding earns $11.37 more than annual — the difference grows with time and rate.
The Shortcut: Rule of 72
For rough mental estimates, divide 72 by the annual interest rate to find years to double:
- 6% → 72/6 = 12 years to double
- 8% → 72/8 = 9 years to double
- 10% → 72/10 = 7.2 years to double
This works because of how exponential growth relates to the natural logarithm of 2 (≈0.693). The rule slightly overestimates for high rates and is very accurate for 5–10%.
Finding Interest Only
If you only need the interest amount (not the total):
I = P × [(1 + (r) / (n))^(n × t) - 1]
Example: $5,000 at 4% monthly for 5 years.
- Monthly rate = 0.04/12 = 0.003333
- Periods = 60
- (1.003333)^60 ≈ 1.2210
- I = 5000 × (1.2210 − 1) = 5000 × 0.2210 = $1,105
Verify with Simple Interest
Always sanity-check against simple interest (I = Prt):
- Simple: I = 5000 × 0.04 × 5 = $1,000
- Compound: I = $1,105
Compound earns $105 more over 5 years — sensible, not dramatic. Over 30 years the gap becomes enormous.
Use the Calculator
For quick calculations with multiple scenarios — different rates, terms, compounding frequencies — our compound interest calculator shows you the full year-by-year breakdown instantly.