Homework 2 Solutions [PDF]

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Financial Economics Spring 2018 Assignment 2of 5: Household Saving and Investment Decisions, Analysis of Investment Projects This homework will be due in class on April 10, 2018. Late papers will not be accepted. Answer all questions completely in A4 size paper. 1) Suppose you are 25 years old and expect to retire at 65. You earn $35,000 every year (assume you receive it at the end of each year) in real terms net of taxes, and expect to live for 15 more years after retirement. Suppose the real rate of interest on borrowing and lending is 4%. Take each sub-question independently. a) What is the optimal consumption level for this level of real income and real interest rate? (2 points) 4% Solve the equation (35000 − 𝐶) × 𝐹𝑉𝐼𝐹𝐴4% 40 𝑦𝑒𝑎𝑟𝑠 = 𝑃𝑉𝐼𝐹𝐴15 𝑦𝑒𝑎𝑟𝑠 × 𝐶 to find C=31333.81 b) For this example, what is the target replacement rate implied by the consumption level under consumption smoothing? (1 point) 31333.81 Target replacement rate = = 89.53% 35000

c) What is the value of your human capital at 25 years old? (1 point) Human capital = 35000 × 𝑃𝑉𝐼𝐹𝐴4% 40 𝑦𝑒𝑎𝑟𝑠 = 692,748 d) Suppose you want to keep the same level of consumption every year until you die. If you want to die leaving $100,000 to your favorite niece, how much should your annual consumption be? (1 point) 100000 4% Solve it from age 65: the equation (35000 − 𝐶) × 𝐹𝑉𝐼𝐹𝐴4% 40 𝑦𝑒𝑎𝑟𝑠 = 𝑃𝑉𝐼𝐹𝐴15 𝑦𝑒𝑎𝑟𝑠 × 𝐶 + 1.0415 to find C=30810.68 100000 OR solve it from age 25: 𝐻𝑢𝑚𝑎𝑛 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 − 1.0455 = 𝑃𝑉𝐼𝐹𝐴4% 55 𝑦𝑒𝑎𝑟𝑠 × 𝐶 to find C=30810.75 e) Suppose you receive an inheritance of $100,000 now. How would your consumption and savings behavior change under consumption smoothing? (1 point) 40 Solve it from age 65: the equation (35000 − 𝐶) × 𝐹𝑉𝐼𝐹𝐴4% 40 𝑦𝑒𝑎𝑟𝑠 + (100000 × 1.04 ) = 4% 𝑃𝑉𝐼𝐹𝐴15 𝑦𝑒𝑎𝑟𝑠 × 𝐶 to find C=35857 OR solve it from age 25: 𝐻𝑢𝑚𝑎𝑛 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 + 100000 = 𝑃𝑉𝐼𝐹𝐴4% 55 𝑦𝑒𝑎𝑟𝑠 × 𝐶 to find C=35856.93 f) Suppose you have to pay social security taxes of $2,000 per year, and the implied real interest rate paid to you by the social security agency is 5%. Under consumption smoothing, what is your optimal consumption level, and the optimal savings amount if you knew at age 25 that the implied real interest paid to you by the social security agency is 5%? (2 points) 5% 4% Solve the equation (35000 − 2000 − 𝐶) × 𝐹𝑉𝐼𝐹𝐴4% 40 𝑦𝑒𝑎𝑟𝑠 + (2000 × 𝐹𝑉𝐼𝐹𝐴40 𝑦𝑒𝑎𝑟𝑠 ) = 𝑃𝑉𝐼𝐹𝐴15 𝑦𝑒𝑎𝑟𝑠 × 𝐶 to find C=31819.46. Savings would be 35000-31819.46-2000=1180.54. 2) Dickinson Brothers, Inc., is considering investing in a machine to produce computer keyboards. The price of the machine will be $500,000 and its economic life is five years. The machine will be fully depreciated by the straight-line method. The machine will produce 10,000 keyboards each year. The price of each keyboard will be $40 in the first year, and will increase by 5 percent per year. The production cost per keyboard will be $20 in the first year and will increase by 10 percent per year. There are no other costs for the project. The corporate tax rate for the company is 34 percent. If the appropriate discount rate is 15 percent, what is the NPV of the investment? (5 points)

3) Suppose you have the following two mutually exclusive projects: Build a day care center or a health spa. Suppose the day care center has the following cash flows: An immediate cash outlay of $5000 followed by inflows of $2500 in each of the next 3 years and zero thereafter. Suppose the health spa has the following cash flows: An immediate outlay of $5000 followed by inflows of nothing in year one, $1000 in year 2 and $7100 in year 3 and zero thereafter. Answer the following questions: a) Confirm that the IRR of the day care project is 23.375%. (1 point) Plug in k=23.375% into the NPV formula and find that NPV=0 b) Is the IRR for the health spa lower or higher than the IRR for the day care project? If you base your investment decision on which project has the highest IRR, which do you choose? (Hint: you don't have to calculate the IRR for the health spa for this question.) (2 points) If you plug in k=23.375% in the health spa project, you should find that the NPV is -$564.85, which means that the IRR of the project is lower than 23.375%. In particular, the IRR is 18.33%. Using the IRR rule, choose the day care project. c) If you base your investment decision on which investment has the highest NPV, which do you choose when the cost of capital is 15% and which do you choose if the cost of capital is 5%? (2 points) d) Suppose you could triple the size of the health spa project and triple its revenues but you can’t change the size of the day care center. Would your decisions regarding the choice of project change? (2 points) c) From the table here, pick the day care when the cost of capital is 15%. Pick the health spa when the cost of capital is 5%. d) If you could triple the size of the health spa project, you should pick the health spa when the cost of capital is 15%, and also when the cost of capital is 5%.