Time Value of Money (Annuity)

Future Value of an Annuity
An annuity is a series of equal payments made at fixed intervals for a specified number of periods. If the payments occur at the end of each period, as they typically do, the annuity is called an ordinary or deferred annuity.

Equation 3
FVAn = PMT (FVIFAi,n)
Where
FVAn = The future value of an annuity over n periods
PMT = The payments at the end of each period
FVIFAi,n = Future value interest factor for a annuity

If you deposit $100 at the end of each year for three years in a savings account that pay 5 percent interest per year, how much will you have at the end of three years?

Spreadsheet Solution
In Excel, click the function wizard, Financial, FV, and OK to get the FV dialog box. Then, we would enter 0.05 for Rate, 3 for Nper, and -100 for Pmt (the payment is entered as a negative number to show that it is a cash outflow). We would leave PV blank because there is no initial payment, and we would leave Type blank to signify that payments come at the end of the periods. Then, we clicked OK, we would get the FV of the annuity, $315.25.

If payments are made at the beginning of each period, the annuity is an annuity due.

Equation 4
FVAn (Annuity due) = PMT (FVIFAi,n)(1+i)

For the annuity due, proceed just as for the ordinary annuity except enter 1 for Type to indicate that we now have an annuity due. Then, when you click OK, the answer $331.01 will appear.

Present Value of an Annuity
If the payments come at the end of each year, then the annuity is an ordinary annuity.

Equation 5
PVAn = PMT (PVIFAi,n)
Where
PVAn = The present value of an annuity of n periods
PMT = The payments at the end of each period
PVIFAi,n = Present value interest factor for a annuity

Suppose you were offered the following alternatives: (1) a three-year annuity with payments of $100 or (2) a lump sum payment today. You have no need for the money during the next three years, so if you accept the annuity, you would deposit the payments in a bank account that pays 5 percent interest per year. Similarly, the lump sum payment would be deposit into a bank account. How large must the lump sum payment today be to make it equivalent to the annuity?

Spreadsheet Solution
In Excel, click the function wizard, Financial, PV, and OK. Then enter 0.05 for Rate, 3 for Nper, -100 for Pmt, leave blank for FV, and 0 or leave blank for Type. Then, when you click OK, you get the answer, $272,32.

Had the three $100 payments in the preceding example been made at the beginning of each year, the annuity would have been an annuity due

Equation 6
PVAn (Annuity due) = PMT (PVIFAi,n)(1+i)

For an annuity due, proceed exactly as for regular annuity except enter 1 rather than 0 for Type to indicate that we now have an annuity due.

Source: Eugene F. Brigham, Joel F. Houston, Fundamentals of Financial Management, Harcourt, Inc, 2001.