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chapter 6 solution

# chapter 6 solution - Basic 1 To solve this problem we must...

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Basic 1. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we use: PV = FV / (1 + r) t [email protected]% = \$950 / 1.10 + \$1,040 / 1.10 2 + \$1,130 / 1.10 3 + \$1,075 / 1.10 4 = \$3,306.37 [email protected]% = \$950 / 1.18 + \$1,040 / 1.18 2 + \$1,130 / 1.18 3 + \$1,075 / 1.18 4 = \$2,794.22 [email protected]% = \$950 / 1.24 + \$1,040 / 1.24 2 + \$1,130 / 1.24 3 + \$1,075 / 1.24 4 = \$2,489.88 2. To find the PVA, we use the equation: PVA = C ({1 – [1/(1 + r) ] t } / r ) At a 5 percent interest rate: PVA = \$6,000{[1 – (1/1.05) 9 ] / .05 } = \$42,646.93 PVA = \$8,000{[1 – (1/1.05) 6 ] / .05 } = \$40,605.54

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And at a 15 percent interest rate: [email protected]%: PVA = \$6,000{[1 – (1/1.15) 9 ] / .15 } = \$28,629.50 [email protected]%: PVA = \$8,000{[1 – (1/1.15) 6 ] / .15 } = \$30,275.86 Notice that the PV of cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a 15 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more valuable since it has larger cash flows. At the higher interest rate, these bigger cash flows early are more important since the cost of waiting (the interest rate) is so much greater. 3. To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a lump sum, we use: FV = PV(1 + r) t = \$940(1.08) 3 + \$1,090(1.08) 2 + \$1,340(1.08) + \$1,405 = \$5,307.71 [email protected]% = \$940(1.11) 3 + \$1,090(1.11) 2 + \$1,340(1.11) + \$1,405 = \$5,520.96 [email protected]% = \$940(1.24) 3 + \$1,090(1.24) 2 + \$1,340(1.24) + \$1,405 = \$6,534.81 Notice we are finding the value at Year 4, the cash flow at Year 4 is simply added to the FV of the other cash flows. In other words, we do not need to compound this cash flow. 4. To find the PVA, we use the equation: PVA = C ({1 – [1/(1 + r) ] t } / r ) PVA = \$5,300{[1 – (1/1.07) 15 ] / .07} = \$48,271.94 PVA = \$5,300{[1 – (1/1.07) 40 ] / .07} = \$70,658.06 PVA = \$5,300{[1 – (1/1.07) 75 ] / .07} = \$75,240.70 To find the PV of a perpetuity, we use the equation: PV = C / r PV = \$5,300 / .07 = \$75,714.29 Notice that as the length of the annuity payments increases, the present value of the annuity approaches the present value of the perpetuity. The present value of the 75 year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years is only \$473.59.
5. Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the PVA equation: PVA = C ({1 – [1/(1 + r) ] t } / r ) PVA = \$34,000 = \$ C {[1 – (1/1.0765) 15 ] / .0765} We can now solve this equation for the annuity payment. Doing so, we get: C = \$34,000 / 8.74548 = \$3,887.72 6. To find the PVA, we use the equation: PVA = C ({1 – [1/(1 + r) ] t } / r ) PVA = \$73,000{[1 – (1/1.085) 8 ] / .085} = \$411,660.36 7. Here we need to find the FVA. The equation to find the FVA is: FVA = C {[(1 + r) t – 1] / r } FVA for 20 years = \$4,000[(1.112 20 – 1) / .112] = \$262,781.16 FVA for 40 years = \$4,000[(1.112 40 – 1) / .112] = \$2,459,072.63 Notice that because of exponential growth, doubling the number of periods does not merely double the FVA. 8. Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the FVA equation: FVA = C {[(1 + r ) t – 1] / r } \$90,000 = \$ C [(1.068 10 – 1) / .068] We can now solve this equation for the annuity payment. Doing so, we get: C = \$90,000 / 13.68662 = \$6,575.77 9. Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the PVA equation: PVA = C ({1 – [1/(1 + r) ] t } / r ) \$50,000 = C {[1 – (1/1.075) 7 ] / .075} We can now solve this equation for the annuity payment. Doing so, we get: C = \$50,000 / 5.29660 = \$9,440.02 10. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:

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