Homework 6

You have your choice of two investment accounts. Investment A is a 13-year annuity that features end-of-month $1,100 payments and has an interest rate of 6.7 percent compounded monthly. Investment B is a 6.2 percent continuously compounded lump sum investment, also good for 13 years. How much money would you need to invest in B today for it to be worth as much as Investment A 13 years from now? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Here we are trying to find the dollar amount invested today that will equal the FVA with a known interest rate, and payments. First we need to determine how much we would have in the annuity account. Finding the FV of the annuity, we get: FVA = $1,100[{[1 + (.067/12)]^156 - 1}/(.067/12)] FVA = $272,573.66 Now we need to find the PV of a lump sum that will give us the same FV. So, using the FV of a lump sum with continuous compounding, we get: FV = $272,573.66 = PVe^(-.062(13)) PV = $272,573.66e^(-.806) PV = $121,742.59

Investment X offers to pay you $6,000 per year for 9 years, whereas Investment Y offers to pay you $8,200 per year for 5 years. If the discount rate is 8 percent, what is the present value of these cash flows? (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.) If the discount rate is 23 percent, what is the present value of these cash flows? (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.)

To find the PVA, we use the equation: PVA = C({1 - [1/(1 + r)^t]}/r) At an interest rate of 8 percent: X@8%: PVA = $6,000{[1 - (1/1.08)^9]/.08} = $37,481.33 Y@8%: PVA = $8,200{[1 - (1/1.08)^5]/.08} = $32,740.22 And at an interest rate of 23 percent: X@23%: PVA = $6,000{[1 - (1/1.23)^9]/.23 } = $22,038.61 Y@23%: PVA = $8,200{[1 - (1/1.23)^5]/.23 } = $22,988.48 Notice that the PV of Investment X has a greater PV than Investment Y at an interest rate of 8 percent, but a lower PV at an interest rate of 23 percent. 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 annual payments. At a higher interest rate, getting these payments early is more important since the cost of waiting (the interest rate) is so much greater.

An investment offers $5,500 per year, with the first payment occurring one year from now. The required return is 7 percent. a.) What would the value be today if the payments occurred for 20 years? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b.) What would the value be today if the payments occurred for 45 years? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c.) What would the value be today if the payments occurred for 70 years? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) d.) What would the value be today if the payments occurred forever? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

To find the PVA, we use the equation: PVA = C({1 − [1/(1 + r)^t]}/r) PVA@20 yrs: PVA = $5,500{[1 − (1/1.07^20)]/.07} = $58,267.08 PVA@45 yrs: PVA = $5,500{[1 − (1/1.07^45)]/.07} = $74,830.37 PVA@70 yrs: PVA = $5,500{[1 − (1/1.07^70)]/.07} = $77,882.14 To find the PV of a perpetuity, we use the equation: PV = C/r PV = $5,500/.07 = $78,571.43 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 70-year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 70 years is only $689.29.

Fuente, Inc., has identified an investment project with the following cash flows. Year | Cash Flow 1 $920 2 1,150 3 1,370 4 2,110 a.) If the discount rate is 9 percent, what is the future value of these cash flows in Year 4? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b.) If the discount rate is 12 percent, what is the future value of these cash flows in Year 4? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c.) If the discount rate is 23 percent, what is the future value of these cash flows in Year 4? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

To solve this problem, we must find the FV of each cash flow and sum. To find the FV of a lump sum, we use: FV = PV(1 + r)^t FV@9% = $920(1.09)^3 + $1,150(1.09)^2 + $1,370(1.09) + $2,110 = $6,161.04 FV@12% = $920(1.12)^3 + $1,150(1.12)^2 + $1,370(1.12) + $2,110 = $6,379.49 FV@23% = $920(1.23)^3 + $1,150(1.23)^2 + $1,370(1.23) + $2,110 = $7,246.93 Notice, since we are finding the value at Year 4, the cash flow at Year 4 is added to the FV of the other cash flows. In other words, we do not need to compound this cash flow.

McCann Co. has identified an investment project with the following cash flows. Year | Cash Flow 1 $760 2 1,010 3 1,270 4 1,375 a.) If the discount rate is 11 percent, what is the present value of these cash flows? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b.) What is the present value at 18 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c.) What is the present value at 24 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

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 PV@11% = $760/1.11 + $1,010/1.11^2 + $1,270/1.11^3 + $1,375/1.11^4 = $3,338.79 PV@18% = $760/1.18 + $1,010/1.18^2 + $1,270/1.18^3 + $1,375/1.18^4 = $2,851.60 PV@24% = $760/1.24 + $1,010/1.24^2 + $1,270/1.24^3 + $1,375/1.24^4 = $2,517.46