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Mortgage with Points

35. Mortgage with Points. Home loans typically involve ¬points, ­ which are fees charged by the lender. Each point charged means that the borrower must pay 1 percent of the loan amount as a fee. For example, if the loan is for $100,000, and two points are charged, the loan repayment schedule is calculated on a $100,000 loan, but the net amount the borrower receives is only $98,000. What is the effective annual interest rate charged on such a loan assuming loan repayment occurs over 360 months? Assume the interest rate is 1 percent per month.

36. Amortizing Loan. You take out a 30-year $100,000 mortgage loan with an APR of 8 percent and monthly payments. In 12 years you decide to sell your house and pay off the mortgage. What is the principal balance on the loan?

37. Amortizing Loan. Consider a 4-year amortizing loan. You borrow $1,000 initially, and repay it in four equal annual year-end payments.

a. If the interest rate is 10 percent, show that the annual payment is $315.47.

b. Fill in the following table, which shows how much of each payment is comprised of interest versus principal repayment (that is, amortization), and the outstanding balance onthe loan at each date.

38. Annuity Value. You`ve borrowed $4,248.68 and agreed to pay back the loan with monthly payments of $200. If the interest rate is 12 percent stated as an APR, how long will it take you to pay back the loan? What is the effective annual rate on the loan?

39. Annuity Value. The $40 million lottery payment that you just won actually pays $2 million per year for 20 years. If the discount rate is 10 percent, and the first payment comes in 1 year,what is the present value of the winnings? What if the first payment comes immediately?

40. Real Annuities. A retiree wants level consumption in real terms over a 30-year retirement. If the inflation rate equals the interest rate she earns on her $450,000 of savings, how much can she spend in real terms each year over the rest of her life?

41. EAR versus APR. You invest $1,000 at a 6 percent annual interest rate, stated as an APR. Interest is compounded monthly. How much will you have in 1 year? In 1.5 years?

42. Annuity Value. You just borrowed $100,000 to buy a condo. You will repay the loan in equal monthly payments of $804.62 over the next 30 years. What monthly interest rate are you paying on the loan? What is the effective annual rate on that loan? What rate is the lender more likely to quote on the loan?

43. EAR. If a bank pays 10 percent interest with continuous compounding, what is the effective annual rate?

44. Annuity Values. You can buy a car that is advertised for $12,000 on the following terms: (a) pay $12,000 and receive a $1,000 rebate from the manufacturer; (b) pay $250 a month for 4 years for total payments of $12,000, implying zero percent financing. Which is the better deal if the interest rate is 1 percent per month?

45. Continuous Compounding. How much will $100 grow to if invested at a continuously compounded interest rate of 10 percent for 6 years? What if it is invested for 10 years at 6 percent?

46. Future Values. I now have $20,000 in the bank earning interest of .5 percent per month. I need $30,000 to make a down payment on a house. I can save an additional $100 per month. How long will it take me to accumulate the $30,000?

47. Perpetuities. A local bank advertises the following deal: ¬Pay us $100 a year for 10 years and then we will pay you (or your beneficiaries) $100 a year forever. ­ Is this a good deal if the interest rate available on other deposits is 8 percent?

48. Perpetuities. A local bank will pay you $100 a year for your lifetime if you deposit $2,500 in the bank today. If you plan to live forever, what interest rate is the bank paying?

49. Perpetuities. A property will provide $10,000 a year forever. If its value is $125,000, what must be the discount rate?

50. Applying Time Value. You can buy property today for $3 million and sell it in 5 years for $4 million. (You earn no rental income on the property.)

a. If the interest rate is 8 percent, what is the present value of the sales price?

b. Is the property investment attractive to you? Why or why not?

c. Would your answer to (b) change if you also could earn $200,000 per year rent on the property?

51. Applying Time Value. A factory costs $400,000. You forecast that it will produce cash inflows of $120,000 in Year 1, $180,000 in Year 2, and $300,000 in Year 3. The discount rate is 12 percent. Is the factory a good investment? Explain.

52. Applying Time Value. You invest $1,000 today and expect to sell your investment for $2,000 in 10 years.

a. Is this a good deal if the discount rate is 5 percent?

b. What if the discount rate is 10 percent?

53. Calculating Interest Rate. A store will give you a 3 percent discount on the cost of your purchase if you pay cash today. Otherwise, you will be billed the full price with payment duein 1 month. What is the implicit borrowing rate being paid by customers who choose to defer payment for the month?