Compound interest formula, step by step
Understand every variable in the compound interest formula and apply it to annual, monthly and recurring-contribution examples.
The compound interest formula
The standard formula for the future value of an initial principal is:
FV = P x (1 + r / n) ^ (n x t)| Variable | Meaning |
|---|---|
FV | future value |
P | initial principal |
r | annual rate as a decimal |
n | compounding periods per year |
t | time in years |
For annual compounding, n = 1; quarterly compounding uses n = 4; monthly uses n = 12.
Convert the rate to a decimal
Divide a percentage by 100 before placing it in the formula:
8% / 100 = 0.08Using 8 instead of 0.08 is one of the most consequential mistakes in this calculation. Keep the units consistent as well: do not combine a monthly rate with a term expressed in years unless the formula converts one of them.
Annual-compounding example
Suppose you invest $10,000 at 8% for five years, compounded annually:
P = 10000
r = 0.08
n = 1
t = 5
FV = 10000 x (1 + 0.08 / 1) ^ (1 x 5)
FV = 14693.28The estimated ending balance is $14,693.28: $10,000 of principal and $4,693.28 of interest.
Monthly-compounding example
With the same principal, rate and term but monthly compounding, n = 12:
FV = 10000 x (1 + 0.08 / 12) ^ (12 x 5)
FV = 14898.46| Compounding | Ending balance |
|---|---|
| Annual | $14,693.28 |
| Monthly | $14,898.46 |
Monthly compounding produces $205.18 more because interest is added to the balance more frequently. This is a model with a constant rate, not a guaranteed investment outcome.
Formula with recurring contributions
If a contribution is made at the end of every compounding period, a reference formula is:
FV = P x (1 + i) ^ m + C x (((1 + i) ^ m - 1) / i)Here, i = r / n, m = n x t, and C is the contribution per period. For contributions at the beginning of each period, multiply the contribution portion by (1 + i).
At a 0% rate, avoid dividing by i; future value is simply the initial principal plus total contributions. For irregular deposits or mismatched contribution and compounding schedules, calculate period by period or use a tool designed for those inputs.
See compound interest with monthly contributions for a worked savings example.
Common mistakes
- Entering a percentage instead of its decimal form.
- Mixing months and years in the same equation.
- Using the wrong compounding frequency.
- Treating a nominal rate as an effective rate.
- Rounding intermediate values too early.
If two products compound at different frequencies, first review nominal vs. effective interest rates.
Calculate a complete scenario
The compound interest calculator handles the initial balance, periodic contributions, term and compounding frequency together. It also separates total contributions from the earnings produced.
Frequently asked questions
What does n mean?
It is the number of times interest is compounded each year: 1 annually, 4 quarterly or 12 monthly.
What happens if the rate is 0%?
The original balance earns no interest. With contributions, the ending balance equals the initial principal plus all deposits.
Does the basic formula include monthly contributions?
No. Use the annuity component shown above or calculate each contribution separately.
Should I use an annual or monthly rate?
Use a rate that matches the periods in the equation. The standard formula starts with an annual rate and divides it by the number of compounding periods per year.