Last Updated 14 Apr 2020

# Hw Chapter4

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5. 4. You have found three investment choices for a one-year deposit: 10% APR Compounded monthly, 10% APR compounded annually, and 9% APR compounded daily. Compute the EAR for each investment choice. (Assume that there are 365 days in the year. ) Sol: 1+EAR= (1+r/k)k So, for 10% APR compounded monthly, the EAR is 1+EAR= (1+0. 1/12)12 = 1. 10471 => EAR= 10. 47% For 10% compounded annually, the EAR is 1+EAR= (1+0. 1)=1. 1 * EAR= 10% (remains the same). For 9% compounded daily 1+EAR= (1+0. 09/365)365 = 1. 09416 * EAR= 9. 4% 5-8. You can earn $50 in interest on a $1000 deposit for eight months.

If the EAR is the same regardless of the length of the investment, how much interest will you earn on a $1000 deposit for a. 6 months. b. 1 year. c. 1 1/2 years. Sol: Since we can earn $50 interest on a $1000 deposit, Rate of interest is 5% Therefore, EAR = (1. 05)12/8 -1 =7. 593% a) 1000(1. 075936/12 – 1) = 37. 27 b) 1000(1. 07593? 1) = 75. 93 c) 1000(1. 075933/2 ? 1) = 116. 03 5-12. Capital One is advertising a 60-month, 5. 99% APR motorcycle loan. If you need to borrow $8000 to purchase your dream Harley Davidson, what will your monthly payment be? Sol: Discount rate for 12 months is, 5. 99/12 = 0. 499167%

C= 8000/[1/0. 004991(1-1/(1+0. 004991)60)] = $154. 63 5-16. You have just purchased a home and taken out a $500,000 mortgage. The mortgage has a 30-year term with monthly payments and an APR of 6%. a. How much will you pay in interest, and how much will you pay in principal, during the first year? b. How much will you pay in interest, and how much will you pay in principal, during the 20th year (i. e. , between 19 and 20 years from now)? Sol: a. APR of 6%/12 = 0. 5% per month. Payment = 500,000/[(1/. 005)(1- 1/1. 005360)]= $2997. 75 Total annual payments = 2997. 75 ? 12 = $35,973. Loan Balance after 1 year is 2997. 5[1/0. 005(1- 1/1. 005348)] = $493,860. Therefore, 500,000 – 493,860 = $6140 is principal repaid in first year. Interest paid in 1st year is 35,973 – 6140 = $29833. b. Loan balance in 19 years (or 360 – 19? 12 = 132 remaining pmts) is 2997. 75[1/0. 005(1- 1/1. 005192)]= $289,162 Loan Balance in 20 years = 2997. 75[1/0. 005(1- 1/1. 005120)] = $270,018 Therefore, Principal repaid = 289,162 – 270,018 = $19,144, and Interest repaid =$35,973 – 19,144 = $16,829. 5-20. Oppenheimer Bank is offering a 30-year mortgage with an APR of 5. 25%. With this mortgage your monthly payments would be $2000 per month.

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In addition, Oppenheimer Bank offers you the following deal: Instead of making the monthly payment of $2000 every month, you can make half the payment every two weeks (so that you will make 52 ? 2 = 26 payments per year). With this plan, how long will it take to pay off the mortgage of $150,000 if the EAR of the loan is unchanged? Sol: For every 2 weeks payment = 2000/2 = 1000. 1 year = 26 weeks. Therefore, (1. 0525)1/26 = 1. 001970. So, discount rate = 0. 1970%. Here, PV of loan payments is the outstanding balance. 150, 000= (1000/0. 001970)[1- 1/(1. 001970)N] If we solve for N,

We get N= 177. 98. So, it takes 178 months to pay off the mortgage. If we decide to pay for 2 weeks, then 178*2= 356 weeks. 5-24. You have credit card debt of $25,000 that has an APR (monthly compounding) of 15%. Each month you pay the minimum monthly payment only. You are required to pay only the outstanding interest. You have received an offer in the mail for an otherwise identical credit card with an APR of 12%. After considering all your alternatives, you decide to switch cards, roll over the outstanding balance on the old card into the new card, and borrow additional money as well.

How much can you borrow today on the new card without changing the minimum monthly payment you will be required to pay? Sol: Here the discount rate = 15/12 = 1. 25%. Assuming that monthly payment is the interest we get, 25,000*0. 15/12= $312. 50. This is perpetuity. So the amount can be borrowed at the new interest rate is this cash flow discounted at the new discount rate. The new discount rate is 12/12 = 1%. So, PV = 312. 50/0. 01 = $31,250. So by switching credit cards we are able to spend an extra 31, 250 ? 25, 000 = $6, 250. We do not have to pay taxes on this amount of new borrowing, so this is our after-tax benefit of switching cards. -28. Consider a project that requires an initial investment of $100,000 and will produce a single cash flow of $150,000 in five years. a. What is the NPV of this project if the five-year interest rate is 5% (EAR)? b. What is the NPV of this project if the five-year interest rate is 10% (EAR)? c. What is the highest five-year interest rate such that this project is still profitable? Sol: a. NPV = –100,000 + 150,000 / 1. 055 = $17,529. b. NPV = –100,000 + 150,000 / 1. 105 = –$6862. Here we need to calculate the IRR. Therefore, IRR = (150,000 / 100,000)1/5 – 1 = 8. 45%. 5-32. Suppose the current one-year interest rate is 6%.

One year from now, you believe the economy will start to slow and the one-year interest rate will fall to 5%. In two years, you expect the economy to be in the midst of a recession, causing the Federal Reserve to cut interest rates drastically and the one-year interest rate to fall to 2%. The one-year interest rate will then rise to 3% the following year, and continue to rise by 1% per year until it returns to 6%, where it will remain from then on. a. If you were certain regarding these future interest rate changes, what two-year interest rate would be consistent with these expectations? . What current term structure of interest rates, for terms of 1 to 10 years, would be consistent with these expectations? c. Plot the yield curve in this case. How does the one-year interest rate compare to the 10-year interest rate? Sol: a. The one-year interest rate is 6%. If rates fall next year to 5%, then if you reinvest at this rate over two years you would earn (1. 06)(1. 05) = 1. 113 per dollar invested. This amount corresponds to an EAR of (1. 113)1/2 – 1 = 5. 50% per year for two years. Thus, the two-year rate that is consistent with these expectations is 5. 0%. b. Year| Future Interest Rate| FV from re-investing| EAR| 1| 6%| 1. 0600| 6. 00%| 2| 5%| 1. 1130| 5. 50%| 3| 2%| 1. 1353| 4. 32%| 4| 3%| 1. 1693| 3. 99%| 5| 4%| 1. 2161 | 3. 99%| 6| 5%| 1. 2769 | 4. 16%| 7| 6%| 1. 3535 | 4. 42%| 8| 6%| 1. 4347 | 4. 62%| 9| 6%| 1. 5208 | 4. 77%| 10| 6%| 1. 6121 | 4. 89%| c. We can get the yield curve by considering all EARs above. It is an inverted curve. 5-36. You are enrolling in an MBA program. To pay your tuition, you can either take out a standard student loan (so the interest payments are not tax deductible) with an EAR of 5 ? or you can use a tax-deductible home equity loan with an APR (monthly) of 6%. You anticipate being in a very low tax bracket, so your tax rate will be only 15%. Which loan should you use? Sol: APR is given, So we can get EAR by, (1+0. 06/12)12 = 1. 06168. So, EAR = 6. 168%. We have to convert the before tax rate to after tax rate. 6. 168? (1- 0. 15) = 5. 243% Since student loan is higher after tax rate, it is better to use home equity loan. 5-40. You firm is considering the purchase of a new office phone system. You can either pay $32,000 now, or $1000 per month for 36 months. . Suppose your firm currently borrows at a rate of 6% per year (APR with monthly compounding). Which payment plan is more attractive? b. Suppose your firm currently borrows at a rate of 18% per year (APR with monthly compounding). Which payment plan would be more attractive in this case? Sol: a. The payments are as risky as the firm’s other debt. So, opportunity cost = debt rate. PV(36 month annuity of 1000 at 6%/12 per month) = $32,871. So we need to pay cash. b. PV(annuity at 18%/12 per months) = $27,661. So we can pay over time.

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