Section _1 Time Value Solutions

# Section _1 Time Value Solutions - Section#1 Solutions Time...

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Section #1 Solutions Time Value Exercises August 31st, 2010 Copyright 2010 by Rich Curtis

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1. Michelle and “The Time Value of Money” a. After agents’ fees and taxes, Michelle will have \$5.28 M: \$10 M - Agent's Fees - Tax = \$10 M - .04(\$10 M) - .45(Taxable Income) = \$10 M - .04(\$10 M) - .45(\$10 M - .04(\$10 M)) = [\$10 M - .04(\$10 M)] [1-.45] = [\$10 M] [1-.04] [1-.45] = \$5.28 M
If Michelle can invest to earn 5.5% annually after-tax for 25 years, the \$5.28 million will become \$20.13 million: FV = PV 1+r Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ N = \$5.28 M Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ 1.055 Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ 25 = (\$5.28 M) (3.81339) = \$20.1347 M Is this consistent with the Rule of 72? 72 5.5 = 13.09 Note :

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b. If we consider the cash flows at times 0, 1, 2, 3, and 4, her total Time 25 future value could be calculated as follows: \$5.28 M Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ (1.055) 25 + \$5.28 M Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ (1.055) 24 + \$5.28 M Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ (1.055) 23 + \$5.28 M Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ (1.055) 22 + \$5.28 M Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ (1.055) 21 = \$90.710 M
Time 0 1 2 3 4 25 \$5.28 M \$5.28 M \$5.28 M \$5.28 M \$5.28 M

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Is there a more elegant way to get the same answer?
One could also use the formula for the future value of an annuity to get the Time 4 value of the annuity: FV t=4 = \$5.28 M 1.055 Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ 5 - 1 .055 Λি Νয় Μ৏ Μ৏ Μ৏ Μ৏ Μ৏ Μ৏ Ξ৯ Πਏ Ο৿ Ο৿ Ο৿ Ο৿ Ο৿ Ο৿ = \$29.4682 M Remember that the formula for the future value of an annuity gives you the value as of the date of the final cash flow, Time 4. Then use the formula for the future value of a single cash flow to find the Time 25 future value of the annuity: FV t=25 = \$29.4682 M 1.055 Λি Νয় Μ৏ Ξ৯ Πਏ Ο৿ 21 = \$90.710 M

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Time 0 1 2 3 4 25 \$5.28 M \$5.28 M \$5.28 M \$5.28 M \$5.28 M (1.055) 21 29.5 M 90.7 M
One could also a) PV the Time 0 cash flow and b) add the PV of the Time 1-4 annuity, if one was very careful: PV t=0 = \$5.28 M + \$5.28 M 1 - 1 1.055 ( ) 4 .055 Λি Νয় Μ৏ Μ৏ Μ৏ Μ৏ Μ৏ Μ৏ Μ৏ Μ৏ Μ৏ Μ৏ Ξ৯

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