The Cumulative Interest Formula Explained (with Worked Examples)

Whether you're a student, a saver, or a finance professional, knowing the cumulative interest formula by heart pays dividends — literally. This guide walks through every variant of the formula with worked examples you can verify in our cumulative interest calculator.

1. Lump-sum cumulative interest formula

For a one-time deposit with no further contributions:

A = P × (1 + r/n)^(n × t)

Cumulative Interest = A − P

• P — principal (starting amount)
• r — annual interest rate as a decimal
• n — compoundings per year
• t — years
• A — final amount (future value)

Worked example

$10,000 at 7% compounded monthly for 20 years:

A = 10,000 × (1 + 0.07/12)^(12 × 20)
A = 10,000 × (1.005833...)^240
A ≈ 10,000 × 4.0387
A ≈ $40,387

Cumulative Interest = $40,387 − $10,000 = $30,387.

2. Future-value formula with recurring contributions

Most real-life situations involve monthly deposits. Combining a lump sum with an ordinary annuity:

FV = P × (1 + r/n)^(nt) + PMT × [((1 + r/n)^(nt) − 1) / (r/n)]

Cumulative Interest = FV − P − (PMT × n × t)

Worked example

$10,000 starting + $500/month for 20 years at 7% (monthly compounding):

Lump portion: $40,387 (from above)
Annuity portion: 500 × ((1.005833)^240 − 1) / 0.005833 = 500 × 520.93 ≈ $260,463

FV ≈ $300,850. Total contributions = $10,000 + ($500 × 240) = $130,000.

Cumulative Interest ≈ $170,850.

3. Annuity-due variant (contributions at the start of period)

If you deposit at the beginning of each period, every contribution earns one extra period of interest:

FV_due = FV_ordinary × (1 + r/n)

On a 30-year horizon with monthly contributions, switching from end-of-month to start-of-month typically adds 0.5–1% to the final balance.

4. Continuous compounding

As n → ∞, the discrete formula collapses to the elegant exponential form:

A = P × e^(rt)

where e ≈ 2.71828.

Worked example

$10,000 at 7% compounded continuously for 20 years:

A = 10,000 × e^(0.07 × 20)
A = 10,000 × e^1.4
A ≈ 10,000 × 4.0552
A ≈ $40,552

Slightly more than monthly compounding ($40,387) — as expected, because continuous compounding is the theoretical maximum.

5. Effective annual yield (APY)

To compare rates fairly across compounding frequencies, convert them to the equivalent annual yield:

APY = (1 + r/n)^n − 1

For a 6% nominal rate:

• Annually: 6.00%
• Quarterly: 6.136%
• Monthly: 6.168%
• Daily: 6.183%
• Continuously: 6.184%

The differences look small but compound dramatically over decades.

6. Cumulative interest on a loan

For a fixed-rate amortizing loan, the monthly payment is:

M = P × [r(1 + r)^n] / [(1 + r)^n − 1]

Where r is the monthly rate and n the number of payments.

Cumulative interest paid = (M × n) − P

Worked example

$300,000 at 6.5% over 30 years (360 monthly payments):

M = 300,000 × [0.005417 × (1.005417)^360] / [(1.005417)^360 − 1]
M ≈ $1,896.20/month

Total paid = $682,633. Cumulative interest = $382,633.

7. Inflation-adjusted cumulative interest

To convert a future value to today's purchasing power, divide by the cumulative inflation factor:

Real FV = Nominal FV / (1 + inflation)^t

Example: a nominal $300,000 in 20 years at 2.5% inflation has a real value of ≈ $183,000 today.

Putting it all together

Real-world planning combines several of these formulas at once: a lump-sum start, monthly contributions that step up annually, monthly compounding, taxes on interest, and inflation. Doing all that by hand is painful — which is why the cumulative interest calculator on our home page implements every formula above and gives you the answer instantly with charts and a year-by-year schedule.

Quick-reference cheat sheet

Situation	Formula
Lump sum, periodic compounding	A = P(1 + r/n)^(nt)
Lump sum, continuous compounding	A = Pe^(rt)
Ordinary annuity FV	FV = PMT × [((1+r/n)^(nt) − 1) / (r/n)]
Annuity due FV	FV_ord × (1 + r/n)
Effective annual rate	APY = (1 + r/n)^n − 1
Loan monthly payment	M = P × [r(1+r)^n] / [(1+r)^n − 1]
Real future value	FV / (1 + inflation)^t