n = LN ((FV x i + Pmt) / Pmt) / LN(1 + i)
FV = Future Value
Pmt = Periodic payment
i = Discount rate
n = Number of periods
LN = Natural logarithm
This number of periods annuity formula FV calculates the number of periods required for an annuity payment (Pmt) made at the end of each period to produce a future value (FV) when a discount rate (i) is applied.
The number of periods annuity formula FV can be used for example, to determine the number of periods is will take for a savings account balance to reach a given value assuming regular periodic deposits are made into the account at the end of each period.
The Excel NPER function can be used instead of the number of periods annuity formula FV, and has the syntax shown below.
*In this instance, the PV and type arguments are not used when using the Excel number or periods function.
Example Using Number of Periods Annuity Formula FV
An amount of 5,000 (Pmt) is deposited into a savings account at the end of each period. If the discount rate (i) is 4%, calculate the number of periods (n) it will take from the saving account to reach a balance of 120,000 (FV). The number of periods is given as follows:
n = LN ((FV x i + Pmt) / Pmt) / LN(1 + i) n = LN ((120000 x 4% + 5000) / 5000) / LN(1 + 4%) n = 17.16 periods
The same answer can be obtained using the Excel NPER function as follows:
n = NPER(i,pmt,PV,FV,type) n = NPER(4%,-5000,,120000) n = 17.16 periods
The number of periods annuity formula FV is one example of an annuity formula used in time value of money calculations, discover another at the link below.