The Compound Interest Formula
When contributions are made monthly alongside a starting balance, the future value is:
FV = P × (1 + mr)ⁿ + PMT × [(1 + mr)ⁿ − 1] / mr
Where P = starting principal, PMT = monthly contribution, mr = equivalent monthly interest rate, and n = total months (years × 12). The equivalent monthly rate depends on compounding frequency: monthly → r/12; quarterly → (1 + r/4)^(1/3) − 1; annually → (1 + r)^(1/12) − 1.
Example: $10,000 starting balance, $200/month, 7% annual rate, monthly compounding, 10 years. n = 120, mr ≈ 0.005833. Final balance ≈ $54,705. Total contributed: $34,000. Interest earned: $20,705 — 38% of the final balance.
Simple vs. Compound Interest
Simple interest is calculated only on the original principal — it grows linearly. Compound interest is calculated on the principal plus all previously earned interest — it grows exponentially.
Over 30 years, monthly compounding on $10,000 at 7% produces $50,165 more than simple interest. The gap compounds on itself — this is why time horizon matters more than any other variable in long-term savings.
Compounding Frequency and APY
More frequent compounding produces higher effective returns. The APY (Annual Percentage Yield) reflects the true annual return after accounting for intra-year compounding: APY = (1 + r/n)ⁿ − 1.
At 7% APR: annual compounding → APY of exactly 7.000%; quarterly compounding → 7.186%; monthly compounding → 7.229%. The difference between monthly and annual compounding on $10,000 over 10 years is approximately $425 — meaningful over a long horizon but small compared to the effect of rate or time.
The Rule of 72
The Rule of 72 gives a quick estimate for doubling time: divide 72 by the annual interest rate. At 6%, money doubles in roughly 12 years. At 9%, roughly 8 years. At 4%, roughly 18 years.
The rule is derived from ln(2) ≈ 0.693. For rates in the 6–10% range, the Rule of 72 is accurate to within a few months. It becomes less precise at extreme rates. For continuous compounding, exact doubling time = ln(2) / r.
Finding the Monthly Contribution for a Goal
The reverse calculator answers: given a target future balance, what monthly contribution is needed? The formula is the algebraic inverse of the FV equation:
PMT = [FV_goal − P × (1 + mr)ⁿ] × mr ÷ [(1 + mr)ⁿ − 1]
If your current plan already meets or exceeds the goal, the calculator reports zero additional monthly contribution needed. The "extra monthly needed" figure shows how much more to contribute on top of what you've already entered.