Compound interest adds each period's interest to the balance before the next period begins. The next interest charge or credit is therefore calculated on the original principal plus earlier interest. With a fixed rate per period and no deposits or withdrawals, this repeated multiplication creates exponential growth rather than equal cash increases.
Principal, rate, periods, final amount, and interest
Let be the starting principal, the effective rate per compounding period and the number of periods. The amount is the balance after those periods. The interest earned or charged is . A percentage rate must be written as a decimal in the formula, so 6% becomes 0.06.
The rate and the period count must use the same unit. With monthly compounding, use a monthly rate and a number of months. A nominal annual rate divided by 12 is not the same thing as an effective annual rate converted to an equivalent monthly rate. The contract or problem statement must tell you which convention applies.
Worked example: 1,000 at 6% for three years
Assume 6% is an effective annual rate and interest is added once per year. Multiply the balance by 1.06 three times. The amount before rounding is 1,191.016. Rounded to cents, the balance is 1,191.02 and the compound interest is 191.02.
Rounding only at the end preserves the intended model. If an institution rounds each period, its ledger may differ by a few cents. Fees, taxes, changing rates, and day-count conventions can produce a much larger difference and require a more detailed model.
Compare simple and compound growth
Change the inputs to see equal simple-interest increments and the increasing increments produced by compounding.
Why the same percentage produces larger later increases
In the example, first-year interest is 60. The second year starts at 1,060, so its interest is 63.60. The third starts at 1,123.60 and adds 67.416. The percentage stays at 6%, but the base to which it applies grows. By contrast, simple interest would add 60 every year and reach 1,180 after three years.
The gap is small over short periods and widens over long ones. That does not make every investment forecast reliable. Constant returns are a mathematical scenario, not a promise about markets. For borrowing, the formula also does not include repayments unless the cash-flow schedule is explicitly modeled.
More frequent compounding needs a stated annual convention
If a nominal annual rate is convertible times per year, a common textbook model uses periodic rate and periods over years. The corresponding effective annual rate is . If the quoted 6% is already an effective annual rate, find an equivalent monthly rate from instead of simply dividing by 12.
Continuous compounding uses an exponential limit, but it is a separate convention and should not be assumed from the words “compound interest” alone. Always identify when interest is posted.
Regular payments create a cash-flow problem
Depositing the same amount every month produces an annuity rather than a single-principal calculation. Each deposit compounds for a different number of periods. Payments at the beginning and end of a period also give different results. A correct calculation needs the payment amount, timing, periodic rate and number of payments. The single-sum formula above remains the building block for each cash flow.
