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Constant Growth Annuity

# Constant Growth Annuity - Constant Growth Annuities Example...

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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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Constant Growth Annuity - Constant Growth Annuities Example...

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