Home > Economics & Finance > Present Value of Annuities And Annuity Due

Present Value of Annuities And Annuity Due

Present Value of Annuities

An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples.  The payments or receipts occur at the end of each period for an ordinary annuity   while they occur at the beginning of each period.for an annuity due.

Present Value of an Ordinary Annuity

The Present Value of an Ordinary Annuity (PVoa) is the value of a stream of expected or promised future payments that have been discounted to a single equivalent value today.  It is extremely useful for comparing two separate cash flows that differ in some way. 

PV-oa can also be thought of as the amount you must invest today at a specific interest rate so that when you withdraw an equal amount each period, the original principal and all accumulated interest will be completely exhausted at the end of the annuity.

The Present Value of an Ordinary Annuity could be solved by calculating the present value of each payment in the series using the present value formula and then summing the results. A more direct formula is:

PVoa = PMT [(1 – (1 / (1 + i)n)) / i]

Where:
    PVoa = Present Value of an Ordinary Annuity
    PMT = Amount of each payment
    i = Discount Rate Per Period
    n = Number of Periods

Example: What amount must you invest today at 6% compounded annually so that you can withdraw $5,000 at the end of each year for the next 5 years?

PMT = 5,000  i = .06  n = 5

PVoa = 5,000 [(1 – (1/(1 + .06)5)) / .06] = 5,000 (4.212364) = 21,061.82

Year                   1               2             3             4             5
Begin       21,061.82  17,325.53  13,365.06  9,166.96  4,716.98
Interest     1,263.71    1,039.53      801.90     550.02     283.02
Withdraw      -5,000       -5,000       -5,000      -5,000     -5,000
End         17,325.53  13,365.06    9,166.96  4,716.98         .00

Present Value of an Annuity Due (PVad)

The Present Value of an Annuity Due is identical to an ordinary annuity except that  each payment occurs at the beginning of a period rather than at the end. Since each payment occurs one period earlier, we can calculate the present value of an ordinary annuity and then multiply the result by (1 + i).

PVad = PVoa (1+i)

Where:
    PV-ad = Present Value of an Annuity Due
    PV-oa = Present Value of an Ordinary Annuity
    i = Discount Rate Per Period

Example: What amount must you invest today a 6% interest rate compounded annually so that you can withdraw $5,000 at the beginning of each year for the next 5 years?

PMT = 5,000  i = .06  n = 5

PVoa = 21,061.82 (1.06) = 22,325.53

Year                      1              2              3            4             5
Begin         22,325.53  18,365.06  14,166.96  9,716.98  5,000.00
Interest        1,039.53      801.90      550.02     283.02            0
Withdraw     -5,000.00  -5,000.00  -5,000.00  -5,000.00 -5,000.00
End            18,365.06  14,166.96   9,716.98   5,000.00         .00

Categories: Economics & Finance
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