Annuities Due
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.Each payment would be shifted
to the left one year, so each payment would be discounted for one less year. Here is the
time line:
Time Line:
05%1 2 3
100 100 100
95.24
90.70
PVA 3 (Annuity due) 285.94
Again, we find the PV of each cash flow and then sum these PVs to find the PV of the
annuity due. This procedure is illustrated in the lower section of the time line dia-
gram. Since the cash flows occur sooner, the PV of the annuity due exceeds that of the
ordinary annuity, $285.94 versus $272.32.
Equation:
(2-5a)
- NUMERICAL SOLUTION:
The lower section of the time line shows the numerical solution, $285.94, calculated
by using the first line of Equation 2-5a, where the present value of each cash flow is
found and then summed to find the PV of the annuity due. If the annuity has many
payments, it is easier to use the third line of Equation 2-5a:
(2-5a)
- FINANCIAL CALCULATOR SOLUTION
BEGIN
Inputs: 3 5 100 0
Output: 285.94
$100(2.7232)(10.05)$285.94.
$100
°
1
1
(10.05)^3
0.05
¢
(10.05)
PVAn(Due)PMT
°
1
1
(1i)n
i
¢
(1i)
PMT(PVIFAi,n)(1i).
PMT
°
1
1
(1i)n
i
¢
(1i)
PMT a
n
t 1
a
1
1 i
b
t 1
PVAn(Due)PMTa
1
1 i
b
0
PMTa
1
1 i
b
1
PMTa
1
1 i
b
n 1
76 CHAPTER 2 Time Value of Money
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74 Time Value of Money
