General Compound Interest Formula and Examples

The general compound interest formula is A = P(1 + r/n)^nt, where A is the ending balance, P is the starting principal, r is the annual interest rate as a decimal, n is the number of compounding periods per year, and t is the time in years. To find only the interest earned or charged, subtract the principal: I = A − P.

Formula

For a fixed annual nominal rate compounded at regular intervals, the ending balance is:

A = P(1 + r/n)^nt

• A is the ending balance, in dollars in this calculator.
• P is the initial principal, in dollars.
• r is the annual interest rate written as a decimal. For example, 5% is 0.05.
• n is the number of compounding periods each year, such as 12 for monthly or 4 for quarterly compounding.
• t is the elapsed time in years.

The calculator accepts the annual rate as a percentage, so its equivalent expression is A = P(1 + apr/(100n))^nt. Interest alone is I = A − P, provided there are no additional deposits, withdrawals, fees, or other balance changes. The calculator rounds displayed dollar outputs to cents; calculations use the unrounded balance until the final display.

Worked Example

Using the calculator defaults—$1,000 principal, 5% annual interest, 10 years, and monthly compounding—convert the rate to a decimal: r = 5/100 = 0.05. Monthly compounding means n = 12.

1. Find the rate per period: 0.05 / 12 = 0.0041666667.
2. Find the total number of periods: 12 × 10 = 120.
3. Substitute into the formula: A = 1,000(1 + 0.05/12)¹²⁰.
4. Evaluate: A ≈ $1,647.01.
5. Subtract the starting principal: I = $1,647.01 − $1,000 = $647.01.

Thus, under these assumptions, the ending balance is $1,647.01 and the interest portion is $647.01. A different compounding frequency can produce a different result even when the stated annual rate and time remain unchanged.

Assumptions

• The annual rate remains fixed throughout the period.
• Interest is credited at regular intervals, and no deposits or withdrawals occur between them.
• The entered annual percentage is treated as a nominal rate divided by the number of compounding periods per year.
• No fees, taxes, contributions, or other balance changes are included.

How Compounding Works

With simple interest, each period’s interest is calculated only on the original principal. With compound interest, previously credited interest becomes part of the balance on which later interest is calculated. This is often described as earning “interest on interest.” The effect becomes more noticeable over longer periods because each new calculation applies to a balance that may already have grown.

The factor 1 + r/n represents the balance multiplier for one compounding period. Raising that factor to nt applies it once for every period in the full term. For example, monthly compounding over ten years means 120 applications of the monthly multiplier—not ten.

Choosing the Compounding Frequency

Set n to match the schedule specified for the account or loan: commonly, 1 for annual, 2 for semiannual, 4 for quarterly, or 12 for monthly compounding. Daily compounding is sometimes represented with 365 periods, but an actual contract or account may define day-count conventions differently. Use the stated method rather than assuming a frequency.

For a fixed nominal annual rate and the same principal and term, more frequent compounding generally results in a larger ending balance when interest is earned, because interest is added to the balance sooner. For interest owed, it can likewise increase the calculated amount. The formula does not, by itself, tell you the effective annual rate: that rate is (1 + r/n)ⁿ − 1, expressed as a decimal. It helps compare the annual effect of different compounding schedules when the nominal rate is known.

Common Questions and Misconceptions

A frequent error is entering a percentage as though it were already a decimal. If the rate is 5%, use 0.05 in the textbook formula—not 5. In this calculator, enter 5 because the input is explicitly labeled as a percentage. Another mistake is mixing time units: if the formula uses years, enter years, and calculate the number of periods as years multiplied by periods per year.

Another important distinction is between a stated nominal rate and an effective annual rate. The standard formula above assumes the input is a nominal annual rate divided into regular periods. If a quoted rate is already an effective annual rate, applying the formula as if it were nominal may count compounding twice. Check how the rate is defined before using it.

This formula also assumes a constant rate and no cash flows other than interest. Regular contributions, irregular deposits, withdrawals, changing rates, and fees require additional calculations. A balance subject to those changes cannot generally be represented by this one-term formula alone.

Frequently Asked Questions

What is the general compound interest formula?
The standard formula is A = P(1 + r/n)^nt. It calculates an ending balance from the starting principal, annual rate, compounding frequency, and time.

How do I calculate compound interest only?
First calculate the ending balance using the compound interest formula, then subtract the original principal: I = A − P. This gives interest alone when no other transactions or charges affect the balance.

What does n mean in the formula?
n is the number of times interest compounds per year. Use 12 for monthly compounding or 4 for quarterly compounding when those schedules apply.

Can I use a monthly interest rate?
Yes, but use a formula adjusted for periods: if i is the monthly rate as a decimal and m is the number of months, then A = P(1 + i)^m. Do not divide a monthly rate by 12 again.

What happens if interest compounds continuously?
Continuous compounding uses a different formula, A = Pe^rt, where e is the mathematical constant and r is the annual rate as a decimal. It is a limiting model as compounding periods become increasingly frequent, not the same schedule as monthly or daily compounding.

The result is an estimate; an account statement or lender disclosure gives the exact figures applicable to the account.

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in p | Principal | $ | 1000 | 0 | 100000000
in apr | Annual interest rate | % | 5 | 0.01 | 100
in yrs | Time | years | 10 | 0 | 100
in freq | Compounding periods per year | periods/year | 12 | 1 | 365
out amt | Ending balance | $ | p * (1 + apr / (100 * freq)) ^ (freq * yrs) | 2
out int | Interest earned | $ | amt - p | 2