chap005 - Solutions to Chapter 5 The Time Value of Money 1...

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Solutions to Chapter 5 The Time Value of Money 1. a. $100/(1.08) 10 = $46.32 b. $100/(1.08) 20 = $21.45 c. $100/(1.04) 10 = $67.56 d. $100/(1.04) 20 = $45.64 2. a. $100 × (1.08) 10 = $215.89 b. $100 × (1.08) 20 = $466.10 c. $100 × (1.04) 10 = $148.02 d. $100 × (1.04) 20 = $219.11 3. $100 × (1.04) 113 = $8,409.45 $100 × (1.08) 113 = $598,252.29 4. With simple interest, you earn 4% of $1,000 or $40 each year. There is no interest on interest. After 10 years, you earn total interest of $400, and your account accumulates to $1,400. With compound interest, your account grows to: $1,000 × (1.04) 10 = $1480.24 Therefore $80.24 is interest on interest. 5. PV = $700/(1.05) 5 = $548.47 5-1
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6. Present Value Years Future Value Interest Rate a. $400 11 $684 % 00 . 5 1 400 684 ) 11 / 1 ( = - b. $183 4 $249 % 00 . 8 1 183 249 ) 4 / 1 ( = - c. $300 7 $300 % 0 1 300 300 ) 7 / 1 ( = - To find the interest rate, we rearrange the basic future value equation as follows: FV = PV × (1 + r) t r = 1 PV FV ) t / 1 ( - 7. You should compare the present values of the two annuities. a. 73 . 721 , 7 $ (1.05) 0.05 1 0.05 1 $1,000 PV 10 = × - × = 73 . 303 , 8 $ (1.05) 0.05 1 0.05 1 $800 PV 15 = × - × = b. 47 . 192 , 4 $ (1.20) 0.20 1 0.20 1 $1,000 PV 10 = × - × = 38 . 740 , 3 $ (1.20) 0.20 1 0.20 1 $800 PV 15 = × - × = c. When the interest rate is low, as in part (a), the longer (i.e., 15-year) but smaller annuity is more valuable because the impact of discounting on the present value of future payments is less significant. 8. $100 × (1 + r) 3 = $115.76 r = 5.00% $200 × (1 + r) 4 = $262.16 r = 7.00% $100 × (1 + r) 5 = $110.41 r = 2.00% 5-2
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9. PV = ($200/1.06) + ($400/1.06 2 ) + ($300/1.06 3 ) = $188.68 + $356.00 + $251.89 = $796.57 10. In these problems, you can either solve the equation provided directly, or you can use your financial calculator, setting: PV = ( - )400, FV = 1000, PMT = 0, i as specified by the problem. Then compute n on the calculator. a. $400 × (1.04) t = $1,000 t = 23.36 periods b. $400 × (1.08) t = $1,000 t = 11.91 periods c. $400 × (1.16) t = $1,000 t = 6.17 periods 11. APR Compounding period Effective annual rate a. 12% 1 month (m = 12/yr) 1.01 12 - 1 = 0.1268 = 12.68% b. 8% 3 months (m = 4/yr) 1.02 4 - 1 = 0.0824 = 8.24% c. 10% 6 months (m = 2/yr) 1.05 2 - 1 = 0.1025 = 10.25% 12. Effective R at e Compounding period Per period rate APR a. 10.00% 1 month (m = 12/yr) 1.10 (1/ 12) - 1 = 0.0080 0.096 = 9.6% b. 6.09% 6 months (m = 2/yr) 1.0609 (1/ 2) - 1 = 0.0300 0.060 = 6.0% c. 8.24% 3 months (m = 4/yr) 1.0824 (1/ 4) - 1 = 0.0200 0.080 = 8.0% 13. Solve the following for t: 1.08 t = 2 t = 11.9 years 5-3
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On a financial calculator, enter: PV = ( - )1, FV = 2, PMT = 0, i = 6 and then compute n. 5-4
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14. Semiannual compounding means that the 8.6 percent loan really carries interest of 4.3 percent per half year. Similarly, the 8.4 percent loan has a monthly rate of 0.7 percent. APR Compounding period Effective annual rate 8.6% 6 months (m = 2/yr) 1.043 2 - 1 = 0.0878 = 8.78% 8.4% 1 month (m = 12/yr) 1.007 12 - 1 = 0.0873 = 8.73% Choose the 8.4 percent loan for its slightly lower effective rate. 15. APR = 1% × 52 = 52% EAR = (1.01) 52 - 1 = 0.6777 = 67.77% 16. Since we are assuming that it is currently 2007, 107 years have passed since 1900. a. $1,000 × (1.05) 107 = $185,035.50 b. PV × (1.05) 107 = $1,000,000 PV = $5,404.37 17. $1,000 × 1.04 = $1,040.00 interest = $40 $1,040 × 1.04 = $1,081.60 interest = $1,081.60 - $1,040 = $41.60 After 9 years, your account has grown to: $1,000 × (1.04) 9 = $1,423.31 After 10 years, your account has grown to: $1,000 × (1.04) 10 = $1,480.24 Interest earned in tenth year = $1,480.24 - $1,423.31 = $56.93 18. If you earned simple interest (without compounding), then the total growth in your account after 25 years would be: 4% per year × 25 years = 100% Therefore, your money would double. With compound interest, your money would
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