Interest Formulas
September 13th, 1997, Last updated 13 July 2007
These formulas are of limited interest (sic) and are included here because someone may wish to do their own thing. Most lenders now use daily interest calculations, and most lenders do not follow the Interest Act in expressing the annual rate used.
Symbols. The following symbols or variables will be used in discussing interest and interest formulas.
- int represents an interest rate per interest period, as a decimal ie. 12% = 0.12 if the interest period is a year If the interest period is a month, it must be calculated.
- nper represents a number of interest periods.
- PV represents a Present Value or present sum of money.
- FV represents a Future Value or a sum of money nper from the present date that is equivalent to PV with interest rate int.
- PMT represents the payment, or more technically, represents a payment in a series of equal payments continuing for the future nper periods at interest rate int.
These formulas apply to any length of interest period, but if the period is other than a year, then the interest must be calculated or converted from the yearly or annual rate to the period rate, or vice versa. An example will follow the formulas and discussion.
The fundamental interest formulas which express the relationship between Present Value, PV, Future Value, FV, and the Payment, PMT, in terms of interest per period, int, and number of periods, nper are as follows:
(1) Given PV, to find FV.
Or to use in a spreadsheet:Appleworks: FV = PV*(1+int)^nper, where ^ = a caret (over the 6)
or Excel: FV = PV*(1+int)^nper, same as Appleworks
or Excel: FV = PV*POWER((1+int),nper, using the Excel POWER function
(2) Given FV, to find PV.
Or to use in a spreadsheet:
PV = FV*(1/(1+int)^nper)
(3) Given FV, to find PMT.
Or to use in a spreadsheet:
PMT = FV*(int/((1+int)^nper-1))
or PMT = FV*(int/(POWER((1+int),nper)-1))
(4) Given PV, to find PMT.
OR
Or to use in a spreadsheet:
PMT = PV*((int*(1+int)^nper)/((1+int)^nper-1))
(5) Given PMT, to find FV.
Or to use in a spreadsheet:
FV = PMT*(((1+int)^nper-1)/int)
(6) Given PMT, to find PV.
Or to use in a spreadsheet:
PV = PMT*((1+int)^nper-1)/int*(1+int)^nper)
NOTES: Explanations of the formulas assume an interest period of one year. For comparison of interest charges made by different lenders interest is always compared by using effective annual rates. The formulas are general and apply to any period or periods, but comparisons must be made using the same rate and equivalent periods.
Where examples are used the variables will be assigned these values:
PV = $12000.
nper = 240 months (20 Years) depending on the formula.
int = 0.12 (12%) per year, or 0.0097587942 per month calculated half-yearly, 0.0094887929 per month calculated yearly. (NOT 0.01) All effective rates. Use the monthly rates as A2 below.
You will get:
FV = $115,755.52, for single payment formulas.
The interest used must be calculated, and an example is found in the spreadsheet
Mortgage calculation using a Spreadsheet
Spreadsheet notation will be used, * for multiply, ^ to the POWER of. A Spreadsheet for this calculations may be found at
formula (2) is easy. Just use formula (1) and express PV in terms of FV.
To test these formulas use a spreadsheet: for formula (1), put PV in cell A1, int in A2, nper in A3, and use the following formula for cell, say C4.
C4 =A1*(1+A2)^A3
If all went well C4 will display the Future Value, and if the above values were used the result should be:$115,755.52.
For formula (2), use a new cell, say E4, for the formula and the following:
E4 =C4*(1/(1+A2)^A3)
If all went well E4 should display the PV you started with, or $12,000.
Present value to Future value, and vice versa.
A Short digression:
Starting with formula (1):
If PV is invested for several years, then at the end of the first year the interest (in decimal) is PV*int and the total amount at the end of the first year is PV+PV*int or PV*(1+int), and the interest at the end of the second year is PV*(1+int)*int, and the total amount at the end of the second year PV*(1+int) + PV*(1+int)*int. Factoring out PV*(1+int) we get PV*(1+int)*(1+int) or PV*(1+int)^2. If you wish to repeat that for year three, and four, or more, go ahead, but in the end the formula (1) is:
FV = PV*(1+int)^nper
The expression (1+int)^nper is called the single payment compound factor.
Its reciprocal 1/(1+int)^nper is called the single payment present worth factor.
The following formula may be used for calculating equivalent interest factors:
(1+int1)^nper1 = (1+int2)^nper2
Interest rate ir is in percent
monthly interest factor, half-yearly, mifhy = ((1+ir/200)^1/6)-1
monthly interest factor, yearly, mify = ((1+ir/100)^1/12)-1
To see a use of this formula
Only formula (1), FV = PV*(1+int)^nper, to use in a spreadsheet has been checked!
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