n = LN (Pmt / (Pmt - PV x i)) / LN(1 + i)
PV = Present Value
Pmt = Periodic payment
i = Discount rate
n = Number of periods
LN = Natural logarithm
This number of periods annuity formula PV calculates the number of periods required for an annuity payment (Pmt) made at the end of each period to produce a present value (PV) when a discount rate (i) is applied.
The number of periods annuity formula PV can be used for example to determine the number of annuity payments required to clear the balance on a loan account.
The Excel NPER function can be used instead of the number of periods annuity formula PV, and has the syntax shown below.
*In this instance, the FV and type arguments are not used when using the Excel number or periods function.
Example Using Number of Periods Annuity Formula PV
A loan account balance of 50,000 (PV) is being paid off with regular periodic payments of 6,000 (Pmt) at the end of each period. If the interest rate on the loan is 6%, the number of periods (n) it would take to clear the loan balance is given as follows:
n = LN (Pmt / (Pmt - PV x i)) / LN(1 + i) n = LN (6000 / (6000 - 50000 x 6%)) / LN(1 + 6%) n = 11.90 periods
The same answer can be obtained using the Excel NPER function as follows:
n = NPER(i,pmt,PV,FV,type) n = NPER(6%,-6000,50000,,) n = 11.90 periods
The number of periods annuity formula PV is one of many annuity formulas used in time value of money calculations, discover another at the link below.