# What is the future value?

Computes the future value of a present amount plus a level payment each period, at a yearly rate compounded 1, 2, 4, or 12 times a year, with payments at the start or end of each period.

- Page: https://www.acalculator.org/finance/future-value-calculator
- JSON spec: https://www.acalculator.org/finance/future-value-calculator.json
- Version: 57aae22dc8c2

## Default answer

Example with the default inputs (Starting amount (present value) $10,000.00, Payment each period $100.00, Yearly interest rate 6%, Periods a year Yearly, Number of years 10, Payments are made at the End of each period): At 6% a year for 10 years, $11,000.00 put in grows to a future value of $19,226.56.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| pv | Starting amount (present value) | The amount you have at the start, before the first period. |
| pmt | Payment each period | The amount added every period (the annuity payment). Leave it empty or 0 for a single sum. |
| rate | Yearly interest rate | The nominal yearly rate. The rate per period is this divided by the periods a year. |
| freq | Periods a year | How many times a year interest is added and a payment is made: yearly (1), twice a year (2), quarterly (4), or monthly (12). |
| years | Number of years | How many years the money grows. The number of periods is years × periods a year. |
| when | Payments are made at the | End of each period (an ordinary annuity) or the start of each period (an annuity due). A payment at the start earns one more period of interest. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| fv | Future value | The balance at the end of the last period. |
| paidIn | Money put in | The starting amount plus every payment. |
| interest | Interest | All the interest earned: the future value minus the money put in. |
| fvPv | Future value of the starting amount | What the starting amount alone grows to: PV × (1 + r)^n. |
| fvPmt | Future value of the payments | What the payments alone grow to: PMT × ((1 + r)^n − 1) ÷ r, times (1 + r) at the start; 0 with no payment. |
| periods | Number of periods | Years × periods a year. |

## Method

FV = PV × (1 + r)^n + PMT × ((1 + r)^n − 1) ÷ r, times (1 + r) on the payments when they are made at the start of each period, with r = yearly rate ÷ periods a year and n = years × periods a year; worked period by period.

## Assumptions

- The rate stays the same for every period, and interest is added once per period.
- All payments are the same size, one per period.
- No fees or taxes are taken out.
- This is an estimate for planning, not financial advice.

## Worked examples

1. pv = $1,000.00, rate = 5%, freq = 1, years = 10, when = end gives fv = $1,628.89, interest = $628.89, fvPv = $1,628.89. Source: FV = PV × (1 + r)^n, hand calculation in content.mdx.
2. pv = $0.00, pmt = $100.00, rate = 6%, freq = 1, years = 10, when = end gives fv = $1,318.08, paidIn = $1,000.00, fvPv = $0.00, fvPmt = $1,318.08. Source: future value of an ordinary annuity, PMT × ((1 + r)^n − 1) ÷ r; hand calculation in content.mdx.
3. pv = $0.00, pmt = $100.00, rate = 6%, freq = 1, years = 10, when = start gives fv = $1,397.16. Source: future value of an annuity due, times (1 + r); hand calculation in content.mdx.
4. pv = $0.00, pmt = $100.00, rate = 6%, freq = 12, years = 10, when = end gives fv = $16,387.93, periods = 120. Source: monthly: r = 0.5%, n = 120, 100 × (1.005^120 − 1) ÷ 0.005; hand calculation in content.mdx.
5. pv = $1,000.00, pmt = $100.00, rate = 0%, freq = 1, years = 12, when = end gives fv = $2,200.00, interest = $0.00. Source: hand calculation in content.mdx: 1,000 + 12 × 100.

## FAQ

### What is future value?

Future value (FV) is what an amount of money today, plus any regular payments, will be worth at a later date when it earns interest. $1,000 at 5% a year is worth $1,628.89 after 10 years.

### What is the future value formula?

For a single amount: FV = PV × (1 + r)^n, where PV is the amount today, r the rate per period, and n the number of periods. For a level payment PMT at the end of each period, add PMT × ((1 + r)^n − 1) ÷ r. When payments are at the start of each period, multiply that payment part by (1 + r).

### What is the difference between an ordinary annuity and an annuity due?

In an ordinary annuity each payment is made at the end of the period. In an annuity due it is made at the start, so every payment earns one more period of interest and the future value is (1 + r) times larger. $100 a year for 10 years at 6% grows to $1,318.08 as an ordinary annuity and $1,397.16 as an annuity due.

### How do I use monthly payments?

Pick "Monthly" under periods a year. The rate per period is then the yearly rate ÷ 12 (6% a year is 0.5% a month), and there are 12 periods each year. $100 a month for 10 years at 6% grows to $16,387.93. If your rate is an effective yearly return instead, the monthly rate would be 1.06^(1/12) − 1 = 0.4868%.

### How is future value different from present value?

They are the same equation read in opposite directions. Future value moves money forward in time by multiplying by (1 + r)^n. Present value moves it back by dividing by (1 + r)^n: $1,628.89 in 10 years at 5% is worth $1,000 today.

## Sources

- U.S. Securities and Exchange Commission, Investor.gov compound interest calculator and formula. https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator
- Future value of an annuity immediate and an annuity due: S. A. Broverman, Mathematics of Investment and Credit, chapter 2.