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 = P\left(1 + \frac{r}{n}\right)^{nt}

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.

The five variables of the compound interest formula, with definitions and examples
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. The returns diminish quickly, though, as the same $10,000 at 7% for 30 years shows:

Compounding frequency values and their effect on $10,000 at 7% for 30 years
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):

A = P(1 + r)^t

With annual compounding, r/n is simply r.

If no compounding (simple interest):

A = P(1 + rt)

Interest accrues on the principal only, so growth is linear.

Continuous Compounding

When interest compounds at every instant, we use the formula:

A = Pe^{rt}

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:

A = 10{,}000\left(1 + \frac{0.07}{12}\right)^{12 \times 30} = 10{,}000 \times 8.116497

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, which is $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 / n
nper = n × t
pmt = 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.

The Compound Interest Formula Explained