Compound Interest: Formula, Frequency and Examples
Compound interest adds earned or charged interest to principal so later periods calculate interest on a larger balance; the standard lump-sum formula shows how rate, frequency and time interact.
Timeline
- Define inputs: Identify principal P, annual decimal rate r, compounding periods per year n and years t.
- Calculate: For a fixed lump sum with no transactions, use A = P(1 + r/n)^(nt).
- Reality-check: Add fees, taxes, changing rates, contributions or payments, and inflation before treating the result as a forecast.
Compound interest means that interest already added to a balance can itself earn or incur interest in later periods. For a single starting amount with a constant nominal annual rate and no deposits or withdrawals, a common formula is A = P(1 + r/n)^(nt). Here A is the ending amount, P the principal, r the annual rate written as a decimal, n the number of compounding periods per year and t the number of years. [1][2]
Suppose $1,000 earns a 5% nominal annual rate compounded monthly for 10 years. The inputs are P = 1,000, r = 0.05, n = 12 and t = 10. The calculation is 1,000 × (1 + 0.05/12)^120, which is about $1,647.01 before tax or fees. The growth above $1,000 includes interest earned on earlier interest, not merely 10 separate payments of $50. [1][3]
Compounding frequency describes how often accumulated interest is added for the next calculation. With the same positive nominal rate and all else equal, monthly compounding produces slightly more than annual compounding, and daily slightly more than monthly. The extra gain becomes smaller as frequency increases. Crediting frequency and compounding frequency can differ, so the disclosure, rather than an account's marketing shorthand, controls the actual calculation. [1][4]
For U.S. deposit accounts, annual percentage yield is designed to help comparison by reflecting the interest rate and compounding frequency over a 365-day period. A nominal interest rate by itself does not include the effect of compounding. Compare APY with APY for accounts with similar conditions, then account for minimum-balance tiers, fees, rate changes and withdrawal rules. A variable-rate account cannot guarantee that today's APY will persist. [4][5]
Regular contributions require a different calculation from the simple lump-sum formula because each contribution compounds for a different length of time. The timing of deposits—beginning or end of a period—also changes the answer. Investor.gov's calculator accepts an initial amount, monthly contribution, years, estimated rate and compounding frequency, making assumptions visible. A spreadsheet should use a matching cash-flow convention rather than silently adding all contributions to the starting principal. [2]
Compounding also works against a borrower when unpaid interest is added to a debt balance. Payment timing, fees, promotional periods and the contract's day-count and compounding rules determine the result; the savings formula alone may not model a credit card, mortgage or student loan correctly. For borrowing, use the required APR disclosures, amortization schedule and agreement, and distinguish interest that accrues from interest that is actually capitalized. [1][5]
A formula is a scenario, not a promise. Taxes and fees can reduce the nominal ending balance, inflation can reduce what that balance buys, and investment returns vary and can be negative. Run several rates and time horizons, verify units and avoid rounding each intermediate period. The strongest lesson is structural: time magnifies the effects of rate, repeated contributions, costs and debt, so starting assumptions deserve as much attention as the arithmetic. [1][2][3]
Sources
- Consumer Financial Protection Bureau — How Does Compound Interest Work?
- Investor.gov — Compound Interest Calculator
- FDIC — Chapter 5: Compound Interest
- Consumer Financial Protection Bureau — Regulation DD Definitions
- Consumer Financial Protection Bureau — Truth in Savings Regulation DD