# Compound interest calculator

> Calculate compound interest: the future value of a deposit plus regular contributions at any compounding frequency, including daily and continuous.

Interactive version: https://www.calcopenly.com/finance/compound-interest-calculator
Subject: Finance calculators

Compound interest adds each period's interest to the balance, so later interest is earned on earlier interest. A single deposit grows to A = P(1 + r/k)^(kt) at annual rate r compounded k times a year for t years, or P × e^(rt) with continuous compounding. Regular contributions are added as an annuity at the equivalent rate per contribution period, paid at the start or the end of each period.

With the defaults, 10,000 at 7% compounded monthly plus 200 at the end of each month grows to 54,713.58 in 10 years. Deposits total 34,000.00, so 20,713.58, or 37.9% of the final balance, is interest. The effective annual rate is 7.2290%.

The rate is held constant and no tax or fees are deducted. Investment returns vary from year to year, so for anything other than a fixed-rate account the result shows what a steady average rate would produce.

## How to calculate compound interest by hand

Take 5,000 at 6% a year compounded quarterly for 8 years, plus 100 paid in at the end of every month. The deposit and the contributions are handled separately and then added.

1. Rate per compounding period: 6% ÷ 4 = 1.5%.
2. Growth of the deposit over 32 quarters: 5,000 × 1.015^32 = 5,000 × 1.610324 = 8,051.62.
3. Rate per contribution period. Interest compounds quarterly but money arrives monthly, so find the monthly rate that compounds to 1.5% over three months: i = 1.015^(1/3) − 1 = 0.4975%.
4. Growth of 96 contributions: 100 × ((1 + i)^96 − 1) ÷ i = 100 × 122.6732 = 12,267.32. Because (1 + i)^96 equals 1.015^32, the same 1.610324 appears again.
5. Future value: 8,051.62 + 12,267.32 = 20,318.94. You paid in 14,600, so interest is 5,718.94.

Skipping step 3 and using 6% ÷ 12 = 0.5% a month for the contributions gives 12,282.85, which is 15.54 too much, because it switches the contributions to monthly compounding. The calculator converts the rate for every pairing of compounding and contribution frequency, so the two parts always use the same underlying rate.

## Solving for the rate or the time

Rearranging A = P(1 + r/k)^(kt) answers the two other common questions:

- **Rate needed:** r = k × ((A ÷ P)^(1 ÷ kt) − 1). Turning 10,000 into 15,000 in 6 years with monthly compounding takes 12 × (1.5^(1/72) − 1) = 6.78% a year.
- **Time needed:** t = ln(A ÷ P) ÷ (k × ln(1 + r/k)). At 5% compounded monthly, 10,000 reaches 25,000 after ln 2.5 ÷ (12 × ln 1.0041667) = 18.36 years, on the 221st monthly compounding.

Spreadsheets have the conversions built in. `=EFFECT(6%, 4)` returns the 6.1364% effective rate of the worked example, and `=NOMINAL(EFFECT(6%, 4), 12)` returns 5.9702%, which is 12 times the monthly contribution rate from step 3. The [time value of money calculator](/finance/time-value-of-money-calculator) solves for the rate or the time of a lump sum directly.

## Effective annual rate by compounding frequency

The effective annual rate is what the nominal rate actually earns in a year once compounding is included: (1 + r/k)^k − 1 for k periods a year, or e^r − 1 for continuous compounding. Each cell below is that percentage:

| Nominal rate | Yearly | Half-yearly | Quarterly | Monthly | Daily (365) | Continuous |
|---|---|---|---|---|---|---|
| 2% | 2.0000 | 2.0100 | 2.0151 | 2.0184 | 2.0201 | 2.0201 |
| 4% | 4.0000 | 4.0400 | 4.0604 | 4.0742 | 4.0808 | 4.0811 |
| 6% | 6.0000 | 6.0900 | 6.1364 | 6.1678 | 6.1831 | 6.1837 |
| 8% | 8.0000 | 8.1600 | 8.2432 | 8.3000 | 8.3278 | 8.3287 |
| 10% | 10.0000 | 10.2500 | 10.3813 | 10.4713 | 10.5156 | 10.5171 |
| 12% | 12.0000 | 12.3600 | 12.5509 | 12.6825 | 12.7475 | 12.7497 |

Frequency matters more as the rate rises. At 2% the whole spread from yearly to continuous is 0.02 points; at 12% it is 0.75 points. Past daily compounding there is almost nothing left to gain.

## APY and the effective annual rate

US banks must quote savings yields as an annual percentage yield. Regulation DD defines it as APY = 100 × ((1 + interest ÷ principal)^(365 ÷ days in term) − 1) ([12 CFR 1030, Appendix A](https://www.consumerfinance.gov/rules-policy/regulations/1030/a/)). The regulation's own example: 30.37 of interest on 1,000 over a 182-day certificate is an APY of 6.18%. For an account with no fees and a constant rate, the APY equals this calculator's effective annual rate. A 5% rate compounded daily shows 5.1267%, and 1,000 left for a year earns 51.27.

Two offers are only comparable on the effective rate. A 4.95% account compounded monthly (5.0639% effective) beats a 5% account compounded yearly (5.0000%), even though its headline rate is lower. Where a lender quotes only a nominal rate, enter it with the stated compounding and read the effective annual rate off the result.

## Rule of 72: how accurate it is

Dividing 72 by the annual rate estimates how many years money takes to double. The exact time with yearly compounding is ln 2 ÷ ln(1 + r):

| Rate | Rule of 72 (years) | Exact (years) | Error |
|---|---|---|---|
| 1% | 72.00 | 69.66 | +3.4% |
| 2% | 36.00 | 35.00 | +2.8% |
| 3% | 24.00 | 23.45 | +2.3% |
| 4% | 18.00 | 17.67 | +1.9% |
| 6% | 12.00 | 11.90 | +0.9% |
| 8% | 9.00 | 9.01 | −0.1% |
| 10% | 7.20 | 7.27 | −1.0% |
| 12% | 6.00 | 6.12 | −1.9% |
| 15% | 4.80 | 4.96 | −3.2% |
| 20% | 3.60 | 3.80 | −5.3% |

At 4% and below, dividing 70 instead of 72 is closer: 70 ÷ 2 = 35.00 matches the exact figure, and 70 ÷ 4 = 17.50 is 0.17 years off where the rule of 72 is 0.33 off. With continuous compounding the exact doubling time is 69.3 ÷ the rate, because ln 2 = 0.693. The rule also works for inflation: at 3% a year, prices double in about 24 years.

## Contributions and their timing

Paying in at the start of each period instead of the end gives every contribution one more period of growth, which multiplies the contribution total by (1 + i). For 300 a month at 6% compounded monthly:

| Years | Paid in | End of month | Start of month | Difference |
|---|---|---|---|---|
| 10 | 36,000 | 49,163.80 | 49,409.62 | 245.82 |
| 20 | 72,000 | 138,612.27 | 139,305.33 | 693.06 |
| 30 | 108,000 | 301,354.51 | 302,861.29 | 1,506.77 |
| 40 | 144,000 | 597,447.22 | 600,434.46 | 2,987.24 |

The timing choice is worth 0.5% of the balance at any horizon, since it is one month's growth. The horizon matters far more: interest is 26.8% of the end-of-month balance after 10 years and 75.9% after 40. Frequency matters too. Paying the same 3,600 a year as one lump at each year end grows to 47,826.41 after 10 years, 1,337.40 less than 300 a month, because the monthly payments start earning sooner. Set the timing to when money actually reaches the account: a transfer on the first of each month is a start-of-period payment, and a deduction from pay at month end is an end-of-period one.

## Reading the result

- **Nominal versus real.** The future value is in future money. At 7% growth and 3% inflation, the real return is 1.07 ÷ 1.03 − 1 = 3.88% a year, not 4%. The [inflation calculator](/finance/inflation-calculator) turns the result back into today's money.
- **Tax and fees.** Nothing is deducted. A fund charging 1% a year on a 7% return grows at roughly 6%, so enter the return after costs.
- **Whole periods.** With contributions, the term must be a whole number of contribution periods, so 18 months works with quarterly payments and 20 months does not.
- **Rates that change.** The calculator holds one rate for the whole term. For a fixed-term deposit followed by a variable rate, run the fixed part first and use its result as the starting deposit for the second run.

For a deposit with no compounding, use the [simple interest calculator](/finance/simple-interest-calculator). To find the monthly amount that reaches a set target, use the [savings goal calculator](/finance/savings-goal-calculator). For monthly mutual fund investments that step up each year, use the [SIP calculator](/finance/sip-calculator).

## Inputs

- **Initial deposit**
- **Interest rate (per year)**
- **Term**
- **Term in** (options: Years, Months)
- **Compounding** (options: Yearly, Half-yearly, Quarterly, Monthly, Weekly, Daily (365), Continuous)
- **Regular contribution**
- **Contribution every** (options: Week, Month, Quarter, Half-year, Year)
- **Contributions made at** (options: End of period, Start of period)

## Results

- Future value — main result
- Total deposits
- Interest earned
- Effective annual rate
- Number of contributions

## Formula

$$
A = P\left(1+\tfrac{r}{k}\right)^{kt} + C\,\frac{(1+i)^{N}-1}{i}\,(1+i)^{\delta},\quad i = \left(1+\tfrac{r}{k}\right)^{k/m}-1,\ N = mt
$$

## Worked examples

### 10,000 at 7% monthly for 10 years plus 200 a month

- Initial deposit: 10,000
- Interest rate (per year): 7%
- Term: 10
- Term in: Years
- Compounding: Monthly
- Regular contribution: 200
- Contribution every: Month
- Contributions made at: End of period
- **Future value: 54,713.58**
- **Total deposits: 34,000.00**
- **Interest earned: 20,713.58**
- **Effective annual rate: 7.2290%**
- Checked against: Python decimal (prec 50) of P(1+r/12)^120 + C((1+i)^120−1)/i: 54713.5752536…

### Excel FV example: 500 plus 200 at the start of each month, 6%, 10 months

- Initial deposit: 500
- Interest rate (per year): 6%
- Term: 10
- Term in: Months
- Compounding: Monthly
- Regular contribution: 200
- Contribution every: Month
- Contributions made at: Start of period
- **Future value: 2,581.40**
- Checked against: Microsoft FV documentation, =FV(0.06/12, 10, -200, -500, 1) = $2,581.40; Python decimal gives 2581.40337…

### Continuous compounding, no contributions

- Initial deposit: 1000
- Interest rate (per year): 5%
- Term: 10
- Term in: Years
- Compounding: Continuous
- Regular contribution: 0
- Contribution every: Month
- Contributions made at: End of period
- **Future value: 1,648.72**
- **Effective annual rate: 5.1271%**
- Checked against: 1000·e^0.5 = 1648.7212707… and e^0.05 − 1 = 5.12710964 % (Python decimal exp)

### Zero rate

- Initial deposit: 1000
- Interest rate (per year): 0%
- Term: 2
- Term in: Years
- Compounding: Monthly
- Regular contribution: 100
- Contribution every: Month
- Contributions made at: End of period
- **Future value: 3,400.00**
- **Interest earned: 0.00**
- **Number of contributions: 24**
- Checked against: 1000 + 24 × 100 by definition

### Quarterly compounding with monthly contributions

- Initial deposit: 5000
- Interest rate (per year): 8%
- Term: 5
- Term in: Years
- Compounding: Quarterly
- Regular contribution: 100
- Contribution every: Month
- Contributions made at: End of period
- **Future value: 14,767.33**
- **Effective annual rate: 8.2432%**
- Checked against: Python decimal with i = 1.02^(1/3) − 1 per month: 14767.3288154…; EAR 1.02^4 − 1 = 8.243216 %

### Excel FV example: 1,000 a month at 12% for 12 months

- Initial deposit: 0
- Interest rate (per year): 12%
- Term: 12
- Term in: Months
- Compounding: Monthly
- Regular contribution: 1000
- Contribution every: Month
- Contributions made at: End of period
- **Future value: 12,682.50**
- Checked against: Microsoft FV documentation, =FV(0.12/12, 12, -1000) = $12,682.50

## Questions

### What is the formula for compound interest?

A = P(1 + r/n)^(nt), where P is the deposit, r the annual rate, n the number of compounding periods a year and t the years. 10,000 at 7% compounded monthly for 10 years grows to 10,000 × (1 + 0.07/12)^120 = 20,096.61, so the interest is 10,096.61. Excel's =FV(0.07/12, 120, 0, -10000) returns the same amount.

### How much difference does the compounding frequency make?

Less than most people expect. 10,000 at 7% for 10 years grows to 19,671.51 compounded yearly, 20,015.97 quarterly, 20,096.61 monthly, 20,136.18 daily and 20,137.53 continuously. The matching effective annual rates are 7%, 7.1859%, 7.2290%, 7.2501% and 7.2508%, so going from monthly to daily adds about 40 over the decade.

### How long does it take to double money with compound interest?

Divide 72 by the annual rate for an estimate (the rule of 72); the exact time with yearly compounding is ln 2 ÷ ln(1 + r). At 7% the rule gives 72 ÷ 7 = 10.29 years and the exact answer is ln 2 ÷ ln 1.07 = 10.24 years. The rule is most accurate near 8%.

### What is the difference between simple and compound interest?

Simple interest is paid only on the original deposit; compound interest is also paid on interest already earned. On 10,000 at 7% for 10 years, simple interest is 10,000 × 0.07 × 10 = 7,000, while compounding yearly earns 9,671.51 and compounding monthly earns 10,096.61. The gap widens with time because compound growth is exponential.

### How much does starting to save earlier matter?

A great deal, because the earliest deposits compound the longest. Saving 200 a month at 7% compounded monthly builds 104,185.33 in 20 years from 48,000 of deposits, but 243,994.20 in 30 years from 72,000. The extra 10 years add 24,000 of deposits and 139,808.87 to the final balance.

### How accurate is the compound interest calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “10,000 at 7% monthly for 10 years plus 200 a month” is checked against Python decimal (prec 50) of P(1+r/12)^120 + C((1+i)^120−1)/i: 54713.5752536….

### Where does the method come from?

Microsoft Excel FV function; U.S. SEC Investor.gov — Compound interest calculator.

## Sources

- [Microsoft Excel FV function](https://support.microsoft.com/office/fv-function-2eef9f44-a084-4c61-bdd8-4fe4bb1b71b3)
- [U.S. SEC Investor.gov — Compound interest calculator](https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator)

_Note: financial information, not professional advice._