Interest is the compensation paid for the use of money. The lender gives up something (immediate access to the funds, the chance to put them to other use), and the borrower compensates them at a contractually agreed rate. Equivalently, interest is the return on capital — the rate at which invested money grows over time.
Interest is expressed as a percentage rate per interest period — the time unit over which the rate applies. The period can be annual, semi-annual, quarterly, monthly, weekly, daily, or (in the theoretical limit) continuous. The same headline rate can mean very different things at different compounding frequencies, which is why the distinction between nominal and effective rates matters.
For a single period starting with principal and rate , the future value is
The total accumulated interest .
Simple interest holds the principal flat across all periods: interest is computed each period only on the original principal. Total interest over periods is , and the future value is . Simple interest is almost never used in real-world finance; it’s mostly a teaching device or a regulatory accounting choice.
Compound interest pays interest on accumulated interest as well as on principal. Each period’s interest gets added to the principal for next period. The future value after periods is
— the same formula that powers all of time-value-of-money analysis.
Why the difference matters in practice: at 10% over 30 years, simple interest gives . Compound interest gives . The gap widens with rate and time. Long-horizon engineering decisions live entirely in the compound-interest world.
For the link between the time-value framework and broader economic principles, see Time value of money. For the role of interest in project evaluation, see Minimum acceptable rate of return.