Annuities

An annuity is a sequence of payments made at regular time intervals.
The formulas in this section make the following simplifying assumptions.

The terms are fixed time intervals.
Periodic payments of equal size.
Payments are made at the end of each payment period.
Payment periods coincide with interest conversion periods.

Future value

The future value of an annuity is the value of the annuity just after the last payment has been made.

       æ (1 + i)ⁿ - 1 ö
F = PS ç             ÷
       è       i     ø

where

F = future value OR accumulated amount
PS = payment size
r = annual interest rate (a.p.r.)
m = number of compounding periods per year
t = time in years
     r
i =     = interest rate per compounding period
     m
n = m t = total number of payments

Present value

The present value of an annuity is the lump sum that must be invested now to accumulate money equal to the future value of the annuity.

       æ  1 - (1 + i)^-n ö
P = PS ç              ÷
       è       i      ø

where

P = present value
remaining terms are the same as above

Example: Compute the future value of an annuity of 60 monthly payments of $100
earning interest at an annual rate of 8% compounded monthly.

i = .08/12 = 0.006666667
n = 60
F = PS((1 + i)ⁿ - 1)/i =  ((1 + 0.006666667)⁶⁰ - 1)/0.006666667 = $7,347.69

Example: Compute the present value of an annuity of 60 monthly payments of $100
earning interest at an annual rate of 8% compounded monthly.

i = .08/12 = 0.006666667
n = 60
P = PS(1 - (1 + i)^-n)/i = (1 - (1 + 0.006666667)^-60)/0.006666667 = $4,931.84

Exercises


(1) Compute the future value of an annuity of $5,000 per quarter
    for 10 years earning an annual interest rate of 6% compounded
    quarterly. 
     $53,513.61
     $271,339.47
     $8,895,451.54


(2) Compute the present value of an annuity of $5,000 per quarter
    for 10 years earning an annual interest rate of 6% compounded
    quarterly. 
     $33,208.89
     $46,110.92
     $149,579.23


(3) Retirement plan.  An employer has contributed $200 per month 
    towards an emmployee's retirement account paying 8% interest
    compounded monthly for 20 years.  How much has the employee
    accumulated towards retirement ? 
     $23,910.86
     $117,804.08


(4) Installment plan.  A home gym is sold under two different
    payment plans.  Cash or $55 per month for 4 years with interest
    charged on the unpaid balance at a rate of 15% per year.  Find 
    the cash price of the home gym. 
     $1,976.23
     $3,587.56

Calculator

Payment size
Annual interest rate as a decimal
Number of payments per year, also Number of interest conversion periods per year
Length of the annuity in years