Editing Present value (section)

====Present value of an annuity====
{{See also|Annuity#Valuation}}
Many financial arrangements (including bonds, other loans, leases, salaries, membership dues, annuities including annuity-immediate and annuity-due, straight-line depreciation charges) stipulate structured payment schedules; payments of the same amount at regular time intervals. Such an arrangement is called an [[annuity]]. The expressions for the present value of such payments are [[summation]]s of [[geometric series]].

There are two types of annuities: an annuity-immediate and annuity-due. For an annuity immediate, <math>\, n \, </math> payments are received (or paid) at the end of each period, at times 1 through <math>\, n \, </math>, while for an annuity due, <math>\, n \, </math> payments are received (or paid) at the beginning of each period, at times 0 through <math>\, n-1 \, </math>.<ref name="Ross"/> This subtle difference must be accounted for when calculating the present value.

An annuity due is an annuity immediate with one more interest-earning period. Thus, the two present values differ by a factor of <math>(1+i)</math>:

:<math> PV_\text{annuity due} = PV_\text{annuity immediate}(1+i) \,\!</math><ref name="Broverman"/>

The present value of an annuity immediate is the value at time 0 of the stream of cash flows:

:<math>PV = \sum_{k=1}^{n} \frac{C}{(1+i)^{k}} = C\left[\frac{1-(1+i)^{-n}}{i}\right], \qquad (1) </math>

where:

:<math>\, n \, </math> = number of periods,

:<math>\, C \, </math> = amount of cash flows,

:<math>\, i \, </math> = effective periodic interest rate or rate of return.