Detailed Solutions for Interest Rate Formulas (123757)

SUMMARY

In Microsoft Excel, the functions FV, NPER, PMT, PV, and RATE are all elements of the following two general interest formulas.

   If rate is not 0
   ----------------

   PV*((1+ rate)^NPER)+ PMT*(1+rate*type)*(((1+ rate)^NPER)-1)/rate+FV = 0

   If rate = 0
   -----------

   (PMT*NPER)+PV+FV = 0

If you look in Microsoft Excel Help to find detailed information about one of the functions listed above, these two formulas are referenced. However, Microsoft Excel Help does not contain solutions for the individual functions. The information in the "More Information" section of this article contains solutions for each of these functions.

MORE INFORMATION

For each function below, a definitions is given, and then general solutions are provided with the special case of rate=0 solutions given last. The solutions are in alphabetical order by derived function name.

NOTE: All of the general terms are shown as positive, but the actual value will be positive (receiving) or negative (paying) cashflow. Input variables must always be entered with the appropriate sign.

Definitions

 FV   - Future value of the investment

 NPER - Number of periods in the investment

 PMT  - Payment amount of the investment

 PV   - Present value of the investment

 rate - Interest rate of the investment

 type - 0 if payment is at the end of a period,
        1 if payment is at the start

Formulas If Rate Is Not 0

 FV   = (PMT*(1+rate*type)*(1-(1+ rate)^NPER)/rate)-PV*(1+rate)^NPER

 NPER = LOG10((PMT*(1+rate*type)-FV*rate)/(PMT*(1+rate*type)+PV*rate))/ 
        LOG10(1+rate)

 PMT  = (rate*(FV+PV*(1+ rate)^NPER))/((1+rate*type)*(1-(1+ rate)^NPER))

 PV   = (PMT*(1+rate*type)*(1-(1+rate)^NPER)-rate*FV)/(rate*(1+rate)^NPER)

 rate = Must be solved using iterative processes since it is an equation
        of variable order depending on the value of NPER.

Formulas If Rate Is 0

 FV   = -1(PV+PMT*NPER)

 NPER = -1(FV+PV)/PMT

 PMT  = -1(FV+PV)/NPER

 PV   = -1(FV+PMT*NPER)

These solutions are a derivation of the model. If you test the solutions against the actual functions in Microsoft Excel, be sure the terms are not forced to be integer at any point because rounding errors become more significant as the order of the equation increases. (If you test the function programmatically, define the terms as type double to reduce computation errors).