Present Value of an Annuity

The present value of an annuity is the single lump sum today that is financially equivalent to the whole future stream of payments — i.e. the amount which, invested now at rate $i$, would exactly fund all the payments. Because a loan is lent as one lump sum and repaid in instalments, the present value is what links a loan amount to its EMI.

Present value — immediate annuity

For $n$ end-of-period payments of $C$ at rate $i$,

$P=C\left[\frac{1-(1+i)^{-n}}{i}\right].$

Present value — annuity due

Each payment is one period earlier, hence discounted one period less:

$P_{\text{due}}=C\left[\frac{1-(1+i)^{-n}}{i}\right](1+i).$

Present value vs accumulated value