Constant Growth Annuities
Example 1: Cash Flow Starts at Year 0; Growth = 10%;
Interest Rate = 5%
Year
CF: g = 10%
PV: r = 5%
0
$100.00
$100.00
1
$110.00
$104.76
2
$121.00
$109.75
3
$133.10
$114.98
Present Value
=
$429.49
P
0
= $100*(1.10/1.05)
0
+ $100*(1.10/1.05)
1
+ $100*(1.10/1.05)
2
+ $100*(1.10/1.05)
3
1.10/1.05
=
1.047619048
=
1.0476 (rounded off for this example only)
Therefore, finding present value by growing a cash flow at 10 percent, then discounting it back at
5 percent, is the same as finding the present value by growing the cash flow at 4.7619048
percent.
P
0
= $100*(1.0476)
0
+ $100*(1.04786)
1
+ $100*(1.0476)
2
+ $100*(1.0476)
3
Even though we are finding the present value, MATHEMATICALLY, it looks like we are finding a
future value of an annuity.
What’s more, since we are compounding 1 of the annuity payments
forward zero periods, MATHEMATICALLY, it looks like we are finding the future value as of the
last payment (a regular annuity).
Therefore, set your calculator to END of Period
N = 4; I/YR = 4.7619048; PMT = 100; Solve for FV
=
$429.49
=
Present Value
____________________
Example 2: Cash Flow Starts at Year 0; Growth = 5%;
Interest Rate = 10%
Year
CF: g = 5%
PV: r = 10%
0
$100.00
$100.00
1
$105.00
$95.45
2
$110.25
$91.12
3
$115.76
$86.97
Present Value
=
$373.54
P
0
= $100*(1.05/1.10)
0
+ $100*(1.05/1.10)
1
+ $100*(1.05/1.10)
2
+ $100*(1.05/1.10)
3
Method 1
:
1.05/1.10
=
(1/05/1.05) / (1.10/1.05)
=
1 / 1.047619048
=
1 / 1.0476 (rounded off for this
example only)
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Therefore, finding present value by growing a cash flow at 5 percent, then discounting it back at
10 percent, is the same as finding the present value by discounting the cash flow at 4.7619048
percent.
P
0
= $100 / (1.0476)
0
+ $100 / (1.04786)
1
+ $100 / (1.0476)
2
+ $100 / (1.0476)
3
This looks like we are finding the present value as of the first payment (an annuity due).
Therefore, set your calculator to BEGIN of Period
N = 4; I/YR = 4.7619048; PMT = 100; Solve for PV
=
$373.54
=
Present Value
Method 2
:
1.05/1.10
=
0.95454545
If we allow 0.95454545 to represent (1+i), then
i
=
4.545454546
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 Spring '06
 Tapley
 Finance, Time Value Of Money, Annuity, Interest, Interest Rate, Net Present Value, Mathematical finance

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