Present Value of an Ordinary Annuity

The present value of an annuity is the single lump sum, invested today at the same rate of interest, that would be exactly enough to generate all the future annuity payments. It answers: what is a stream of equal future payments worth right now? This is the figure that matters whenever a future series of payments has to be compared with a price paid today — the cash price of an asset bought on instalments, or the amount of a loan being repaid in equal instalments.

Each future payment is worth less than its face value today, because a rupee received later is worth less than a rupee in hand now (it has lost the interest it could have earned). Discounting each of the $n$ end-of-period payments of $P$ back to today and adding them gives:

where $V$ is the present value, $P$ the periodic payment, $i$ the rate per period and $n$ the number of payments. The bracketed factor $\\dfrac{1-(1+i)^{-n}}{i}$ is the present-value annuity factor — the present worth of a payment of ₹1 per period.

As with future value, this formula is just compound-interest discounting applied payment-by-payment and totalled, which is the independent check used on every present-value numerical here (Worked Example 3 discounts each payment separately and confirms the sum equals the formula's answer).

Capital recovery — the reverse question. Rearranging the present-value formula for $P$ gives the equal instalment that will exactly repay a present sum $V$ (a loan, or the cash price of an asset) over $n$ periods:

$P = \\dfrac{V \\cdot i}{1-(1+i)^{-n}}$ …