Compound growth: what time and frequency actually change
Two examples can quote the same annual rate yet finish with different balances because returns are added at different intervals. Separate the starting amount, annual rate, duration and compounding frequency before treating the output as meaningful.
Earlier returns become part of the next calculation
Compound interest applies to the original principal and accumulated interest. Simple interest instead applies the rate to the original principal each time.
The arithmetic updates the balance and uses that new balance for the next period. Withdrawing the return instead of leaving it with the principal changes the problem; the two cases do not follow the same compound-growth path.
References: Investor.gov — Compound interest
An annual rate is not a rate for every month
Compounding frequency counts additions per year: 1 annually, 4 quarterly or 12 monthly. It is separate from the duration: 3 years of monthly compounding means 36 periods, not 12.
This model divides the annual rate across those periods, then applies the growth repeatedly. Do not use 6% for every month when you mean 6% a year. If a quoted number already represents a full year’s growth after compounding, do not automatically treat it as a pre-compounding annual rate and add that effect again.
References: Investor.gov — Compound-interest calculator and frequency
Compare the assumptions before comparing balances
For one starting amount, the final balance is principal × (1 + annual percentage ÷100 ÷periods per year), raised to the power of periods per year ×years. The calculated increase is the final balance minus the principal, not the whole final balance.
This single-amount model includes no recurring deposits, withdrawals, fees, taxes or inflation adjustment. Variable rates and new contributions require a different model. Treat the rate as an explicit assumption: a mathematical scenario is not a financial offer, forecast or investment recommendation.
Worked example: earlier returns make the difference
Assume 5000 currency units, an illustrative annual rate of 4%, annual compounding and 3 years with no deposits or withdrawals. The balances become 5200, then 5408, then 5624.32. The increase is 624.32. Simple interest on the starting amount would total 600; the extra 24.32 comes from applying the rate to earlier interest.
For a separate one-year comparison, an illustrative annual rate of 6% compounded monthly means 0.5% per month. The factor (1 +0.06 ÷12) raised to 12 is about 1.061678: roughly 6.17% growth, not 72%. This follows fixed assumptions before fees or taxes; it is not an available product rate or a guaranteed return.