Compound Interest Formula

The standard compound interest formula

For a single lump sum growing at a fixed interest rate, the most common compound interest formula is:

FV = PV × (1 + r/m)^(m×t)

PV = present value (starting amount)
FV = future value (ending balance)
r  = nominal annual interest rate (decimal, e.g. 0.05 for 5%)
m  = compounding periods per year
t  = time in years

The exponent m×t is the total number of compounding periods. Each period multiplies the balance by (1 + r/m), so after m×t periods, the factor is raised to that power.

Understanding each variable

Example: classic compound interest problem

You invest $5,000 at 5% per year, compounded monthly, for 10 years. What is the future value?

• PV = 5,000
• r = 0.05
• m = 12
• t = 10

Plug into the formula:

FV = 5000 × (1 + 0.05/12)^(12×10)
   = 5000 × (1 + 0.0041666...)^120
   ≈ 5000 × 1.647
   ≈ 8,236 (approx)

The calculator on this page uses the same formula under the hood. Adjust the inputs to see how the future value reacts.

Compound interest formula with regular contributions

If you also contribute a regular amount PMT every compounding period, the future value is:

FV = PV × (1 + r/m)^(m×t)
   + PMT × [((1 + r/m)^(m×t) − 1) ÷ (r/m)]

The first term is the grown starting lump sum. The second term is the future value of all those regular contributions. This is the formula the calculator uses whenever the contribution box is non-zero.

In many textbooks this is called the future value of an annuity combined with a single sum.

Continuous compounding formula

If interest compounds continuously, the formula changes slightly. Instead of compounding a finite number of times per year, you approach an infinite number of infinitesimally small steps. The limit is:

FV = PV × e^(r×t)

where e is the mathematical constant approximately equal to 2.71828. Continuous compounding is common in some theoretical finance contexts, but less so in day-to-day retail banking.

Effective annual rate vs nominal rate

The nominal rate r doesn’t fully describe your real growth unless you also know how often interest is compounded. The effective annual rate (EAR) converts any (r, m) pair into a single yearly growth figure:

EAR = (1 + r/m)^m − 1

For example, 5% compounded monthly has an effective rate slightly above 5% because interest earned in one month also earns interest in later months.

Common mistakes when using the compound interest formula

• Mixing up percentage and decimal. Always convert percentage rates into decimals (5% → 0.05) before using the formula.
• Using the wrong m. If you choose monthly compounding, m = 12, not 1.
• Inconsistent t units. The formula expects t in years. If you have months, divide by 12 before plugging in.
• Forgetting contributions. If you add money regularly, you need the extended formula that includes PMT.

Frequently asked questions

Is the compound interest formula only for savings?

No. The same math can describe the growth of investments or the accumulation of interest on loans. For loans you typically know the rate and payment and use rearranged formulas, but the compounding structure is the same.

Can compound interest be negative?

If the balance is shrinking (for example, a negative rate or fees larger than the interest), you effectively have a negative growth rate. The formula still works if you plug in a negative r, it just produces a smaller FV.

What happens if m = 1?

If m = 1, you are compounding once per year. The formula becomes FV = PV × (1 + r)^t, which is the classic annual compounding equation.

How accurate is the formula for real-world accounts?

The formula assumes a constant rate and regular compounding. Real accounts may have variable rates, fees or irregular top-ups. In those cases it’s an approximation, but still a very useful one for understanding the direction and scale of growth.