The Compound Interest Formula Explained
The compound interest formula A = P(1 + r/n)nt calculates the future value of a deposit when interest is repeatedly added to the balance and begins earning interest itself. It's the single equation behind savings accounts, index-fund projections, and every calculator on this site.
The Compound Interest Formula
- A
- = final amount (future value)
- P
- = principal (initial deposit)
- r
- = annual interest rate (decimal)
- n
- = number of times interest is compounded per year
- t
- = time in years
Understanding the Variables
Each variable in the formula has a precise meaning; get any one wrong (most often r, which must be a decimal) and the result is off by orders of magnitude.
| Icon | Symbol | Name | Definition |
|---|---|---|---|
| P | Principal (Initial Deposit) | The initial amount of money you invest or deposit. Example: $10,000 | |
| r | Annual Interest Rate | The expected annual rate of return expressed as a decimal. Example: 7% = 0.07 | |
| n | Compounding Frequency | How many times interest is calculated and added to the balance each year. Example: 1 (annually), 12 (monthly), 365 (daily) | |
| t | Time (Years) | The total number of years your money is invested. Example: 30 | |
| A | Final Amount (Future Value) | The total your investment grows to, including principal and all interest. Example: $81,164.97 — see the worked example below |
How Compound Interest Works
Compound interest works by adding each period's interest to your balance so the next period's interest is calculated on a larger amount.
- You invest money (the principal).
- You earn interest on your principal at the periodic rate r/n.
- That interest is added to your balance.
- The next period, you earn interest on the new, larger balance.
- This repeats n × t times, so growth accelerates — that snowball is compounding.
Compounding Frequency (n)
The more often interest compounds, the more you earn — with quickly diminishing returns, as the same $10,000 at 7% for 30 years shows:
| Frequency | n (times per year) | Example | $10,000 at 7%, 30 yrs |
|---|---|---|---|
| Annually | 1 | Once per year | $76,122.55 |
| Semi-annually | 2 | Twice per year | $78,780.91 |
| Quarterly | 4 | Every 3 months | $80,191.83 |
| Monthly | 12 | Once per month | $81,164.97 |
| Daily | 365 | Every day | $81,645.26 |
| Continuously | ∞ | At every instant | $81,661.70 |
Special Cases
If interest is compounded once per year (n = 1):
With annual compounding, r/n is simply r.
If no compounding (simple interest):
Interest accrues on the principal only, so growth is linear.
Continuous Compounding
When interest compounds at every instant, we use the formula:
Where e ≈ 2.71828 (Euler's number)
For $10,000 at 7% over 30 years, continuous compounding gives $81,661.70 — about $496.73 more than monthly compounding.
Learn more about compounding frequency →Example Calculation
Suppose you invest $10,000 at an annual interest rate of 7%, compounded monthly (n = 12), for 30 years. Substituting into the formula:
After 30 years, your investment grows to $81,164.97.
Compounding frequency matters: the same $10,000 at 7% for 30 years compounded only annually (n = 1) grows to $76,122.55 — $5,042.42 less than with monthly compounding.
See more examples →Excel Formula
You can calculate compound interest in Excel using:
=FV(rate_per_period, nper, -pmt, -pv, type)Where:
rate_per_period= r / nnper= n × tpmt= periodic payments (use 0 if none)pv= principal (use a negative value)type= 0 for end of period, 1 for beginning
Questions the formula tends to raise — APR vs. APY, how long money takes to double, what changes with monthly contributions — are answered in the compound interest FAQ.
