Annuity present value calculator

Ten payments of 1000 add to 10000 in nominal money, and they are worth $7,360.09 today at 6 percent. The first payment loses a year of discounting, the tenth loses ten, and the total is the sum of ten different present values.

The closed form does that sum in one step, which is all the annuity factor is. Discounting each payment by hand and adding them gives the same $7,360.09, and doing it once is the fastest way to see why the answer is so far below the nominal total.

Late payments are discounted hardest, so adding years to the end of an annuity adds less and less. Changing the rate, by contrast, moves every payment at once. The second example, $500 for 5 years at 8 percent, is worth $1,996.36, and a change in the rate would move that figure noticeably.

At the limit, an annuity that never ends is a perpetuity, worth payment over rate. The annuity factor approaches that limit as $n$ grows, which is why very long annuities are priced close to perpetuities.

This calculation assumes payments arrive at the end of each period, which is the ordinary annuity and the convention for loans and bonds. Payments at the start of each period, as with rent, form an annuity due and are worth one period of discounting more.

The conversion is a single multiplication by one plus the rate. Knowing which convention a quoted figure used is the practical part; the arithmetic between them is trivial.

The same factor prices a mortgage, values a bond's coupon stream and turns a pension into a lump sum. The annuity present value table lists the factors directly, and present value covers the idea underneath. This is educational material, not financial advice.