Compound interest and present value, step by step

Match the rate to each period

Compound interest adds each period’s interest to the balance, so later interest can earn interest too. For a fixed nominal annual rate j, compounded m times per year over t years:

• Periodic rate: i = j/m.
• Number of periods: n = mt.
• Future value: FV = PV(1 + i)ⁿ.

PV is the opening deposit. The formulas here assume a fixed rate, whole compounding periods, and no deposits, withdrawals, taxes, or fees during the term. An effective annual rate is a different rate type; do not automatically divide it by m.

Example: grow $7,500 for three years

A hypothetical deposit earns 4.8% nominal annual interest, compounded quarterly. With four quarters per year:

i = 0.048/4 = 0.012 per quarter

n = 4 × 3 = 12 quarters

FV = 7,500 × (1.012)¹² = 8,654.209681…

The future value is $8,654.21 and the interest earned is $1,154.21.

With simple interest at 4.8% for the same three years, the total would be 7,500 × (1 + 0.048 × 3) = $8,580.00. Quarterly compounding adds $74.21 to that total here because interest remains in the account and earns further interest.

Do not use 4.8% in every quarter: that would apply the annual rate four times per year. Do not use an exponent of 3 with a quarterly rate: it would describe only three quarters.

Example: find a deposit for a future goal

You want $12,000 in three years. Assume a fixed 5.4% nominal annual rate compounded monthly. This time the future amount is known, so reverse the growth calculation:

PV = FV ÷ (1 + i)ⁿ

i = 0.054/12 = 0.0045 per month

n = 12 × 3 = 36 months

PV = 12,000 ÷ (1.0045)³⁶ = 10,209.003836…

The present value rounded to the nearest cent is $10,209.00. If the task is to choose a cent-denominated deposit that reaches at least $12,000 under this exact model, round up to $10,209.01 instead.

Check using the unrounded present value: multiplying it by (1.0045)³⁶ returns $12,000. With a positive rate and no later deposits, present value should be smaller than future value.

Check the rate label and the timeline

For the first example, the effective annual yield is (1 + 0.048/4)⁴ − 1 = about 4.89%. It is above the nominal 4.8% because it includes one year of quarterly compounding. A three-year gain is not an annual yield.

If money arrives at different dates, calculate a future value for each deposit using only its own time invested, then add the results. A deposit made one year later cannot earn that missing first year of interest.

For these examples, retain the calculator’s full growth factor and round money at the end. A question that supplies a rounded interest table may produce a slightly different result; follow its stated method and precision.