Present value:
(annuity)
Example:PVA of 3 payments of $1,000 when i 4% per period:
PVA 3 $1,000(2.7751) $2,775.10.
An annuity whose payments occur at the endof each period is called an ordinary
annuity.The formulas above are for ordinary annuities.
If each payment occurs at the beginning of the period rather than at the end, then
we have anannuity due.In Figure 2-3, the payments would be shown at Years 0,
1, and 2 rather than at Years 1, 2, and 3. The PV of each payment would be larger,
because each payment would be discounted back one year less, so the PV of the
annuity would also be larger. Similarly, the FV of the annuity due would also be
larger because each payment would be compounded for an extra year. The follow-
ing formulas can be used to convert the PV and FV of an ordinary annuity to an
annuity due:
PVA (annuity due) PVA of an ordinary annuity (1 i).
FVA (annuity due) FVA of an ordinary annuity (1 i).
Example:PVA of 3 beginning-of-year payments of $1,000 when i 4%:
PVA (annuity due) $1,000(2.7751)(1.04) $2,886.10.
Example:FVA of 3 beginning-of-year payments of $1,000 when i 4%:
FVA (annuity due) $1,000(3.1216)(1.04) $3,246.46.
If the time line in Figure 2-3 were extended out forever so that the $1,000 pay-
ments went on forever, we would have a perpetuitywhose value could be found as
follows:
.
If the cash flows in Figure 2-3 were unequal, we could not use the annuity formu-
las. To find the PV or FV of an uneven series, find the PV or FV of each individual
cash flow and then sum them. Note, though, that if some of the cash flows
constitute an annuity, then the annuity formula can be used to calculate the present
value of that part of the cash flow stream.
Financial calculatorshave built-in programs that perform all of the operations
discussed in this chapter. It would be useful for you to buy such a calculator and to
learn how to use it.
Spreadsheet programsare especially useful for problems with many uneven cash
flows. They are also very useful if you want to solve a problem repeatedly with dif-
Value of perpetuity
PMT
i
$1,000
0.04
$25,000
PMT(PVIFAi,n).
PMT
°
1
1
(1i)n
i
¢
PMT a
n
t 1
c
1
1 i
d
t
PVAn
PMT
(1i)^1
PMT
(1i)^2
PMT
(1i)n
92 CHAPTER 2 Time Value of Money
90 Time Value of Money
