| FV = | PMT * ((1 + R / N)N * T - 1) | or | FV = | PMT * ((1 + r)N * T - 1) |
| R / N | r |
FV = N * T * PMT + I = A + I
r = R / N
where:
FV = Future value
I = Interest amount
A = Total payment amount
R = Nominal interest rate per year (as a decimal, not in percentage)
r = Interest rate per period (as a decimal, not in percentage)
T = Time period in years
N = Number of compounding periods in one year
PMT = Periodic payment amount, paid at the end of each payment period
Other variations of the savings annuity equation are used to calculate:
- The future value (FV)
FV = PMT * ((1 + R/N)N * T - 1) R / N
- The periodic payment amount (PMT)
PMT = FV * (R/N) and FV = N * T * PMT + I (1 + (R / N))N * T -1
Note:
A sinking fund is when we know the future value of the annuity and wish to calculate the periodic payment amount.
- The nominal annual interest rate (R) (as a decimal, not in percentage)
FV = (1 + (R / N))N * T - 1 PMT R / N
Note:
Keep substituting different values for R until we get successively closer to the desired value of FV / PMT.
- The time period (T) in years
T = log((R / N) * ((FV / PMT) + (N / R))) and FV = N * T * PMT + I N * log(1 + (R / N))
- The compounding periods (N) in one year
N = log(((FV * r) / PMT) + 1) and FV = N * T * PMT + I T * log(1 + r)
Note:
r = interest rate per period
For example, if you borrow $1000 for 2 years at 12% interest compounded quarterly, you must divide the annual interest rate by 4 to obtain the interest rate per period
(r = R / N = 12 / 4 = 3%).
- The effective interest rate (Re) (as a decimal, not in percentage)
Re = (1 + (R / N))N - 1 or Re = (1 + r)N - 1
Note:
The effective interest rate is the equivalent rate of compound interest earned over a period of one year for a nominal interest rate per year which is compounded twice or more over the year.