The Compound Interest Formula: A = P(1 + r/n)^(nt), Explained
The compound interest formula is A = P(1 + r/n)ⁿᵗ, where P is the starting principal, r is the annual rate as a decimal, n is the number of times interest compounds per year, t is the number of years, and A is the final amount. This guide defines each variable, works a full example, and shows how compounding frequency changes the result.
The compound interest formula is
A = P(1 + r/n)^(nt). Feed it your starting balance, your annual rate as a decimal, how often interest compounds each year, and the number of years, and it returns the final balance with interest included. To find the interest alone, subtract the principal:A − P.
Want the answer without the algebra? Our compound interest calculator takes the same four inputs and returns A instantly. The rest of this page explains what the formula is doing.
What each variable means
Every letter in the formula has one job. The principal is the amount you start with, the base everything grows from. The rate goes in as a decimal, not a whole-number percentage, which is the single most common mistake. The frequency counts how many times a year interest is calculated and added back. Time is always measured in years. The result is the full ending balance.
The exponent is the engine. It counts the total number of compounding events, and each event multiplies the balance by a small factor. That same factor applied many times is what makes the balance curve upward instead of climbing in a straight line.
A full worked example
Suppose you deposit a starting balance at a fixed annual rate, compounded monthly, over a decade. Identify the variables first, then substitute them into the formula and work it in order: the rate per period, then the exponent, then the final power.
rate per period = 0.05 ÷ 12 = 0.0041667 growth factor = 1 + 0.0041667 = 1.0041667 total events = 12 × 10 = 120 A = 1000 × (1.0041667)^120 A = 1000 × 1.647009 A ≈ $1,647.01
The balance grows past the starting amount, and the interest earned is whatever sits on top of the principal. The original deposit earned almost two-thirds of itself again, without a single extra contribution.
How compounding frequency changes the result
Hold the principal, rate, and time steady and change only the frequency. More frequent compounding produces a higher final balance, because interest starts earning its own interest sooner. Every step up in frequency raises the result, but the gains shrink fast.
Two things stand out. Daily beats annually over the decade, but the jumps get smaller at every step. Going from annual to quarterly adds far more than going from monthly to daily. There is a ceiling: as the frequency grows without bound, the result approaches the continuous-compounding limit, which lands barely above daily.
annual → quarterly = +$14.73
monthly → daily = +$1.65
daily → annually gap = $19.77
continuous limit ≈ $1,648.72Compound vs. simple interest
Simple interest skips the compounding step entirely. It pays a fixed amount each period based only on the original principal, using I = P · r · t. Earned interest never gets to earn anything itself, so over a long term the gap between the two methods is real. If you need that calculation on its own, use our simple interest calculator.
The Rule of seventy-two for doubling time
For a quick estimate of how long money takes to double, divide seventy-two by the interest rate written as a whole-number percentage. It is an approximation, not the exact figure the formula gives, but it holds up well for mental math at moderate rates. Run the full formula when you need the precise number.
years to double ≈ 72 ÷ rate
at 5%: 72 ÷ 5 = 14.4 yearsA higher rate moves the needle more than any change in how often interest compounds. Get the rate and term right first, then treat frequency as a minor tuning knob.
Common questions
It means interest is calculated and added to the balance twelve times a year. Each month the applied rate is the annual rate split into twelve parts, and the next month's interest is figured on the slightly larger balance.
Subtract the principal from the final amount. The formula returns the full balance, and the interest is whatever sits on top of the principal.
Yes, but with sharply diminishing returns. Each increase in frequency raises the result, but the increases get smaller and converge on the continuous-compounding limit A = Peʳᵗ. Past daily compounding the difference is usually fractions of a percent, so the rate matters far more than the frequency.
