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:
Example 1: 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
PVoa = 5,000 [(1 - (1/(1 + .06)5)) / .06] = 5,000 (4.212364) = 21,061.82
Example 2: In practical problems, you may need to calculate both the present value of an annuity (a stream of future periodic payments) and the present value of a single future amount:
For example, a computer dealer offers to lease a system to you for $50 per month for two years. At the end of two years, you have the option to buy the system for $500. You will pay at the end of each month. He will sell the same system to you for $1,200 cash. If the going interest rate is 12%, which is the better offer?
You can treat this as the sum of two separate calculations:
PMT = 50 per period
PVoa = 50 [ (1 - ( 1/(1.01)24)) / .01] = 50 [(1- ( 1 / 1.26973)) /.01] = 1,062.17
FV = 500 Future value (the lease buy out)
PV = FV [ 1 / (1 + i)n ] = 500 ( 1 / 1.26973 ) = 393.78
The present value (cost) of the lease is $1,455.95 (1,062.17 + 393.78). So if taxes are not considered, you would be $255.95 better off paying cash right now if you have it.
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).
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
PVoa = 21,061.82 (1.06) = 22,325.53
Combined FormulaYou can also combine these formulas and the present value of a single amount formula into one.
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