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Finance maths

Future value of an annuity

What a series of equal regular deposits grows to, with interest.

FV=PMT·(1+i)N−1i

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The live calculator needs JavaScript. The worked example below shows the same steps it would.

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  • After 120 payments: 32,775.87, of which 24,000.00 paid in (blue) and 8,775.87 interest (coral).

What the symbols mean

FV
the future value: the total at the end
PMT
the payment made each period (at the end of each period)
i
the interest rate per period as a decimal (6% a year paid monthly: 6%/12 = 0.005)
N
the number of payments

When to use it

Use it for regular saving: a pension contribution, a monthly savings plan, a sinking fund. Solving for PMT gives the deposit needed to reach a goal; solving for i (numerically) gives the rate a plan implies. It assumes payments at the end of each period (an ordinary annuity).

Worked example

200 is saved at the end of every month for 120 months at 0.5% a month. How much is there at the end?

  1. Start from the formula.

    FV=PMT·((1+i)N−1)i

  2. Substitute and work it out:

    FV=200.00·((1+0.005)120−1)0.005
    =200.00·(1.005120−1)0.005
    ≈32,775.87

  3. Of that, you paid in:

    paid in=PMT⁢N
    =200.00·120
    =24,000.00

  4. and interest added:

    interest=FV−PMT⁢N
    ≈32,775.87−200.00·120
    ≈32,775.87−24,000.00
    ≈8,775.87

Answer FV ≈ 32,775.87, of which 24,000 was paid in and 8,775.87 is interest

Why it works

The first payment earns interest for N − 1 periods, the next for N − 2, and the last for none. Adding PMT(1+i)N−1+…+PMT is a geometric series with ratio 1+i, and its sum is this formula.

Common mistakes

  • Using the annual rate with monthly payments; i must be the rate per payment period.
  • Using this formula for payments at the start of each period (an annuity due); multiply by 1+i for that.
  • Applying it to a single lump sum, which needs the compound interest formula.