Business Math

Chapter 5 Interest Rates 5-1. Your bank is polish offering you an grudge that go out give 20% please in total for a two- stratum nonplus. Determine the kindred discount vagabond for a period space of a. Six months. b. One year. c. One month. a. Since 6 months is pic of 2 old age, victimization our rule pic So the equivalent 6 month drift is 4. 66%. b. Since one year is half of 2 geezerhood pic So the equivalent 1 year send is 9. 54%. c. Since one month is pic of 2 years, using our rule pic So the equivalent 1 month mark is 0. 63%. 5-2. Which do you prefer a bank account that pays 5% per year (EAR) for three years or a. An account that pays 2pic every six months for three years? b. An account that pays 7pic every 18 months for three years? c. An account that pays pic per month for three years? If you deposit $1 into a bank account that pays 5% per year for 3 years you will have pic afterward 3 years. a. If the account pays pic per 6 months and so you will have pic afte r 3 years, so you prefer pic every 6 months. b.If the account pays pic per 18 months then you will have pic after 3 years, so you prefer 5% per year. c. If the account pays pic per month then you will have pic after 3 years, so you prefer pic every month. 5-3. Many academic institutions offer a sabbatical policy. Every s regular(a) years a professor is habituated a year free of teaching and other administrative responsibilities at full pay. For a professor piddleing $70,000 per year who works for a total of 42 years, what is the present value of the amount she will earn trance on sabbatical if the provoke tramp is 6% (EAR)? Timeline 0 7 14 42 0 1 2 8 0 1 2 304 48 0. 75 % 20,092. 9 euchre 0 Thus, your remaining equalizer is $20,092. 39. If you prepay an extra $ ampere-second today, your will lower your remaining dimension to $20,092. 39 100 = $19,992. 39. Though your balance is pared, your required monthly allowance does not change. Inste ad, you will pay off the impart faster that is, it will reduce the recompenses you need to cultivate at the very end of the add. How much smaller will the final payment be? With the extra payment, the timeline changes That is, we will pay off by paying $500 per month for 47 months, and some smaller amount, $500 X, in the last month.To solve for X, recall that the PV of the remaining cash flows equals the swell balance when the loan enliven step is employd as the discount rate pic Solving for X gives pic So the final payment will be lower by $143. 14. You can also use the rente spreadsheet to determine this solution. If you prepay $100 today, and make payments of $500 for 48 months, then your final balance at the end will be a credit of $143. 14 N I PV PMT FV 48 0. 75 % 19,992. 9 -500 143. 14 (2) The extra payment effectively lets us exchange $100 today for $143. 14 in four years. We claimed that the return on this investment should be the loan interest rate. allows see if this is the case pic, so it is. Thus, you earn a 9% APR (the rate on the loan). 5-19. contract again the setting of Problem 18. Now that you make water your best investment is to prepay your student loan, you decide to prepay as much as you can each month. Looking at your budget, you can yield to pay an extra $250 per month in addition to your required monthly payments of $500, r $750 in total each month. How persistent will it point you to pay off the loan? The timeline in this case is and we want to determine the number of monthly payments N that we will need to make. That is, we need to determine what length annuity with a monthly payment of $750 has the same present value as the loan balance, using the loan interest rate as the discount rate. As we did in Chapter 4, we set the outstanding balance equal to the present value of the loan payments and solve for N. pic We can also use the annuity spreadsheet to solve for N. N I PV PMT FV 30. 02 0. 75 % 20,092. 39 750 0 So, b y prepaying the loan, we will pay off the loan in about 30 months or 2 ? years, quite an than the four years originally scheduled. Because N of 30. 02 is larger than 30, we could either increase the 30th payment by a small amount or make a very small 31st payment. We can use the annuity spreadsheet to determine the remaining balance after 30 payments. N I PV PMT FV 30 0. 75 % 20,092. 39 750 13. 86 If we make a final payment of $750. 00 + $13. 86 = $763. 86, the loan will be paid off in 30 months. 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. In addition, Oppenheimer Bank offers you the following deal Instead of making he 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? If we make pic ev ery 2 weeks the timeline is as follows. Timeline 0 1 2 3 Using the formula for the loan payment, pic Next we write out the cash flows with the extra payment. Timeline 2 0 1 67 18 To determine the outstanding balance we discount at the original rate, i. e. , pic pic Next we calculate the loan payment on the new mortgage. Timeline 2 0 1 2 360 The discount rate on the new loan is the new loan rate pic Using the formula for the loan payment pic b. pic c. pic (You can use trial and error or the annuity calculator to solve for N. ) d. pic (Note results may differ slightly due to rounding. ) 5-24. You have credit card debt of $25,000 that has an APR (monthly compounding) of 15%. for each one month you pay the minimum monthly payment only. You are required to pay only the outstanding interest. You have received an offer in the broadcast for an otherwise identical credit card with an APR of 12%. After considering all your alternatives, you decide to switc h cards, roll all over the outstanding balance on the sexagenarian 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? The discount rate on the original card is pic Assuming that your current monthly payment is the interest that accrues, it equals pic Timeline 0 1 2 312. 50 312. 50 This is a perpetuity. So the amount you can borrow at the new interest rate is this cash flow discounted at the new discount rate. The new discount rate is pic So,pic So by switching credit cards you are able-bodied to spend an extra pic You do not have to pay taxes on this amount of new borrowing, so this is your after-tax benefit of switching cards. 5-25. In 1975, interest grade were 7. 85% and the rate of inflation was 12. 3% in the United States. What was the real interest rate in 1975? How would the purchasing power of your savings have changed over t he year? pic The purchasing power of your savings declined by 3. 96% over the year. 5-26. If the rate of inflation is 5%, what nominal interest rate is necessary for you to earn a 3% real interest rate on your investment? pic implies pic. Therefore, a nominal rate of 8. 15% is required. 5-27. Can the nominal interest rate available to an investor be significantly negative? (Hint Consider the interest rate earned from saving cash under the mattress. ) Can the real interest rate be negative? Explain. By holding cash, an investor earns a nominal interest rate of 0%.Since an investor can always earn at least 0%, the nominal interest rate cannot be negative. The real interest rate can be negative, however. It is negative whenever the rate of inflation exceeds the nominal interest rate. 5-28. 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? . NPV = 100,000 + 150,000 / 1. 055 = $17,529. b. NPV = 100,000 + 150,000 / 1. 105 = $6862. c. The answer is the IRR of the investment IRR = (150,000 / 100,000)1/5 1 = 8. 45%. 5-29. count the term structure of risk-free interest rates is as shown below pic a. Calculate the present value of an investment that pays $ guanine in two years and $2000 in five years for certain. b. Calculate the present value of receiving $500 per year, with certainty, at the end of the next five years. To keep the rates for the missing years in the table, linearly interpolate between the years for which ou do know the rates. (For example, the rate in year 4 would be the average of the rate in year 3 and year 5. ) c. Calculate the present value of receiving $2300 per year, with certainty, for the next 20 years. Infer rates for the missing years using linear interpolation. (Hint Use a spreadsheet. ) a. Timeline 0 1 2 3 4 5 Since the opportunity cost of capital is dissimilar for investments of distinct maturities, we must use the cost of capital associated with each cash flow as the discount rate for that cash flow. Unfortunately, we do not have a rate for a 4-year cash flow, so we linearly interpolate. pic pic c. Timeline 0 1 2 3 20 2,300 2,300 2,300 2,300 Since the opportunity cot of capital is different for investments of different maturities, we must use the cost of capital associated with each cash flow as the discount rate for that cash flow.Unfortunately, we do not have a rate for a number of years, so we linearly interpolate. pic pic pic pic 5-30. Using the term structure in Problem 29, what is the present value of an investment that pays $100 at the end of each of years 1, 2, and 3? If you wanted to value this investment correctly using the an nuity formula, which discount rate should you use? PV = 100 / 1. 0199 + 100 / 1. 02412 + 100 / 1. 02743 =$285. 61. To determine the single discount rate that would compute the value correctly, we solve the following for r PV = 285. 1 = 100/(1 + r) + 100 / (1 + r)2 + 100/(1 + r)3 = $285. 61. This is clean an IRR calculation. Using trial and error or the annuity calculator, r = 2. 50%. Note that this rate is between the 1, 2, and 3-yr rates given. 5-31. What is the shape of the yield abridge given the term structure in Problem 29? What expectations are investors likely to have about future interest rates? The yield curve is increasing. This is much a sign that investors expect interest rates to rise in the future. 5-32. Suppose the current one-year interest rate is 6%.One year from now, you study 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 i nterest 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? 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. We can do the same logic for future years c . We can plot the yield curve using the EARs in (b) note that the 10-year rate is below the 1-year rate (yield curve is inverted). 5-33. Figure 5. 4 shows that Wal-Marts five-year borrowing rate is 3. 1% and GE Capitals is 10%. Which would you prefer? $500 from Wal-Mart paid today or a promise that the firm will pay you $700 in five years? Which would you choose if GE Capital offered you the same alternatives? We can use the interest rates each company must pay on a 5-year loan as the discount rate.PV for GE Capital = 700 / 1. 105 = $434. 64 $500 today, so take the money now. PV for Wal-Mart = 700 / 1. 0315 = $600. 90 $500 today, so take the promise. 5-34. Your best taxable investment opportunity has an EAR of 4%. You best nontaxable investment opportunity has an EAR of 3%. If your tax rate is 30%, which opportunity provides the high after-tax interest rate? After-tax rate = 4%(1 . 30) = 2. 8%, which is slight than your tax-free investment with pays 3%. 5-35. Your uncle Fred j ust purchased a new boat.He brags to you about the low 7% interest rate (APR, monthly compounding) he obtained from the dealer. The rate is even lower than the rate he could have obtained on his syndicate equity loan (8% APR, monthly compounding). If his tax rate is 25% and the interest on the sign of the zodiac equity loan is tax deductible, which loan is truly cheaper? After-tax cost of home equity loan is 8%(1 . 25) = 6%, which is cheaper than the dealers loan (for which interest is not tax-deductible). Thus, the home equity loan is cheaper. (Note that this could also be done in terms of EARs. ) 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 5pic 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? Using the formula t o convert an APR to an EAR pic So the home equity loan has an EAR of 6. 168%. Now since the rate on a tax deductible loan is a before-tax rate, we must convert this to an after-tax rate to compare it. picSince the student loan has a larger after tax rate, you are better off using the home equity loan. 5-37. Your best wizard consults you for investment advice. You learn that his tax rate is 35%, and he has the following current investments and debts A car loan with an outstanding balance of $5000 and a 4. 8% APR (monthly compounding) Credit cards with an outstanding balance of $10,000 and a 14. 9% APR (monthly compounding) A regular savings account with a $30,000 balance, paying a 5. 50% EAR A money market savings account with a $100,000 balance, paying a 5. 25% APR (daily compounding) A tax-deductible home equity loan with an outstanding balance of $25,000 and a 5. 0% APR (monthly compounding) a. Which savings account pays a higher after-tax interest rate? b. Should your friend use his savings to pay off any of his outstanding debts? Explain. a. The regular savings account pays 5. 5% EAR, or 5. 5%(1 . 35) = 3. 575% after tax. The money-market account pays (1 + 5. 25%/365)365 1 = 5. 39% or 5. 39%(1 . 35) = 3. 50% after tax. Therefore, the regular savings account pays a higher rate. b. Your friend should pay off the credit card loans and the car loan, since they have after-tax costs of 14. % APR and 4. 8% APR respectively, which exceed the rate earned on savings. The home equity loan should not be repaid, as its EAR = (1 + 5%/12)12 1 = 5. 12%, for an after-tax rate of only 5. 125(1 . 35) = 3. 33%, which is below the rate earned on savings. 5-38. Suppose you have outstanding debt with an 8% interest rate that can be repaid anytime, and the interest rate on U. S. Treasuries is only 5%. You plan to repay your debt using any cash that you dont invest elsewhere. Until your debt is repaid, what cost of capital should you use when evaluating a new risk-free i nvestment opportunity? Why?The discriminate cost of capital for a new risk-free investment is 8%, since you could earn 8% without risk by paying off your existing loan and avoiding interest charges. 5-39. In the summer of 2008, at Heathrow Airport in London, Bestofthebest (BB), a private company, offered a lottery to win a Ferrari or 90,000 British pounds, equivalent at the time to about $180,000. Both the Ferrari and the money, in 100 pound notes, were on display. If the U. K. interest rate was 5% per year, and the dollar interest rate was 2% per year (EARs), how much did it cost the company in dollars each month to keep the cash on display?That is, what was the opportunity cost of keeping it on display rather than in a bank account? (Ignore taxes. ) Because the prize is in pounds, we should use the pound interest rate (comparable risk). (1. 05)(1/12) 1 = . 4074%. 0. 4074% x 90k = 366. 7 pounds per month, or $733 per month at the current exchange rate. 5-40. You firm is consideri ng the purchase of a new office phone system. You can either pay $32,000 now, or $ curtilage per month for 36 months. a. Suppose your firm currently borrows at a rate of 6% per year (APR with monthly compounding).Which payment plan is more sweet? 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? a. The payments are as risky as the firms other debt. So opportunity cost = debt rate. PV(36 month annuity of 1000 at 6%/12 per month) = $32,871. So pay cash. b. PV(annuity at 18%/12 per mo) = $27,661. So pay over time. 47 500 0 2 500 1 500 48 500 47 500 0 19,992. 39 2 500 1 500 48 (500 X) N -750 0 20,092. 39 2 -750 1 -750 pic