letx=

Finance maths

Present value of an annuity

The lump sum today that is worth the same as a series of equal future payments.

PV=PMT·1−(1+i)−Ni

Calculator

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

See it move

012345678910112004006008001000
  • Each blue bar is what one payment of 1,000.00 is worth today; later payments are discounted more.
  • Together: 7,721.73 today, against 10,000.00 paid in total.

What the symbols mean

PV
the present value of all the payments
PMT
the payment each period
i
the interest rate per period as a decimal
N
the number of payments

When to use it

Use it to value a stream of payments (a pension, lottery instalments, a lease), to see how much income a lump sum can provide, and for loans: the amount borrowed is the present value of the repayments. Solving for PMT is the loan repayment formula.

Worked example

What is a stream of 10 yearly payments of 1,000 worth today at 5% a year?

  1. Start from the formula.

    PV=PMT·(1−(1+i)−N)i

  2. Substitute and work it out:

    PV=1,000.00·(1−(1+0.05)−10)0.05
    =1,000.00·(1−1.05−10)0.05
    ≈7,721.73

  3. The payments themselves add up to more:

    total paid=PMT⁢N
    =1,000.00·10
    =10,000.00

Answer PV ≈ 7,721.73, less than the 10,000 paid in total

Why it works

Each payment is discounted back: PMT/(1+i)+PMT/(1+i)2+…+PMT/(1+i)N. That geometric series sums to this formula.

Common mistakes

  • Adding the payments without discounting (10,000 here), which overvalues them.
  • Mixing periods: monthly payments need a monthly rate and N in months.
  • Expecting PV to keep growing with N: it levels off at PMT/i for payments that go on for ever.