22.1 Simple Interest

1. Suppose that you take out a federal direct loan on September 1 before your senior year for $7500 (the maximum allowed for a dependent student) and plan to begin paying it back on December 1 after graduation (so you will have had the loan for 15 months, including the six-month grace period after leaving school). The interest rate is 4.29% and you pay the interest every quarter until that December 1. How much will you owe on that December 1, and how much of that will be interest?

2. Suppose that you borrow $7500 on September 1 before your junior year. You will graduate 21 months later, on June 1, and you will have 6 months of grace period. So you plan to begin paying the loan back on December 1, 27 months after you took it out. But you do not pay any of the interest as it accumulates. How much will you owe on that December 1, and how much of that will be interest?

3. Suppose that you borrow $5500 for your first year and $6500 for your second year (the maximum amounts for a dependent student), as federal direct student loans at a 4.29% interest rate. Suppose that each loan begins on September 1 of its year, that you finish college in four years, that you do not pay the accruing interest in the meantime, and that you begin repayment on December 1 after graduation. What is your total debt on that December 1, and how much of that is interest?

4. Assume the same situation as in Exercise 3, but you also borrow $75 00 for each of your third and fourth years (again, the maximum amounts), again on September 1, all at a 4.29% interest rate. You finish college in four years, and you begin repayment on December 1 after graduation. What is your total debt on that December 1, and how much of that is interest?

22.2 Compound Interest

5. If you borrowed $15,000 to buy a new car at 4.9% interest per year, compounded annually, and paid back all the principal and interest at the end of 5 years, how much would you pay back?

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6. Assume the same situation as in Exercise 5, but the interest is compounded monthly. How much would you pay back?

7. If you borrowed $200,000 to buy a house at 6% interest per year, compounded annually, and paid back the principal and interest at the end of 30 years, how much would you pay back?

8. Assume the same situation as in Exercise 7, but the interest is compounded monthly (this is the usual case). How much would you pay back?

9. A credit card bill showed an APR of 17.24%.

10. You receive an offer for a credit card with 0% fixed APR for the first 12 months, after which the card would have one of several rates depending on credit history. The highest rate was a 22.74% APR (and the company reserves the right to change the APR “at any time for any reason”).

22.3 Conventional Loans

11. Suppose that your federal direct student loans plus accumulated interest total $20,000 at the time that you start repayment, and the interest rate is 4.29%.

12. Suppose that your federal direct student loans plus accumulated interest total $31,811 at the time that you start repayment, and the interest rate is 4.29%.

However, because your accumulated outstanding federal loans total more than $30,000, you can elect to repay over 25 years instead. If you do that:

13. Suppose that your parents take out a PLUS loan on your behalf on September 1 before your senior year for $10,000, at the rates mentioned above, and begin paying it back six months after you graduate on June 1. How much is their monthly payment?

14. If your parents instead take out a PLUS loan for $10,000 on September 1 before each of your four years of college, how much is their monthly payment if they begin paying it back six months after your graduation?

15. In late November 2014, a car dealership in southern Wisconsin was offering a new 2014 Toyota Corolla LE sedan for $18,299 (not including sales tax, registration, license plates, title, and $125 “dealer documentation fee”) at 0% annual interest over 36 months. What would be the monthly payment?

16. Repeat Exercise 15, but opt instead for 1.9% over 60 months. What would be the monthly payment?

17. Repeat Exercise 15, but opt instead for a further $500 manufacturer rebate and pay 2.9% interest over 60 months. What would be the monthly payment?

18. In December 2010, Kevin Lauterbach, 29, of Coral Springs, Florida, who had “mildly damaged credit,” bought a 2008 Jeep Liberty with no money down and a 72-month loan for $19,000 with a 4.75% rate. (New York Times, February 28, 2011, p. A3).

19. Suppose that you have good credit and can get a 30-year mortgage for $100,000 at 5%. What is your monthly payment?

20. Assume the same situation as in Exercise 19, except that your credit is not as good and the rate that you are offered is 7.125%. What is your monthly payment?

21. Assume the same situation as in Exercise 19, but you inquire about a 15-year loan instead. You are offered 3.75%. What is your monthly payment?

22. Assume the same situation as in Exercise 21, but your credit is not as good, and you are offered 6.75%. What is your monthly payment?

23. For the mortgage in Exercise 19, how much equity would you have after 5 years?

24. For the mortgage in Exercise 20, how much equity would you have after 5 years?

25. For the mortgage in Exercise 21, how much equity would you have after 5 years?

26. For the mortgage in Exercise 22, how much equity would you have after 5 years?

27. The first offer is from a local bank for a 30-year, fixed-rate mortgage at 4.25% with closing costs of $2639.07. (You must pay the closing costs right away— you cannot include them in the mortgage.)

28. The second offer is on the Internet (from a company you have never heard of) for a 30-year, fixed-rate mortgage at 3.99% with closing costs of $5000.

29. In a 2/28 “hybrid” adjustable-rate mortgage (ARM), the initial interest rate is fixed for 2 years and then is adjusted every 6 months. (You usually pay “points” up front at closing in exchange for the “rate lock” for the first 2 years.) Suppose you buy a house with a $200,000 mortgage, with a 2/28 ARM with initial rate of 3%; and suppose that 2 years later, the interest rate goes up to 5%.

30. In a 5/1 “hybrid” adjustable-rate mortgage (ARM), the initial interest rate is fixed for 5 years and then is adjusted annually. (You usually pay “points” up front at closing in exchange for the “rate lock” for the first 5 years.) Suppose that you buy a house with a $200,000 mortgage with a 5/1 ARM with initial rate of 4%; suppose that 5 years later, the interest rate goes up to 6%.

31. (Requires a spreadsheet) Suppose that your credit card has an APR of 18% interest rate, corresponding to approximately 1.5% per month. (The actual interest applied is daily interest, at a daily rate of 18%/365; but for simplicity we use a uniform approximate monthly rate. Also, the amount of interest owed depends on exactly when in the month your payment is received.)

32. (Requires a spreadsheet) Repeat Exercise 31, but this time you miss the first payment, incurring a $35 late fee and an increase to a penalty APR of 30%, corresponding to approximately 2.5% per month.

33. The purpose of such a complicated formula for the minimum payment on a credit card is to avoid the situation of a customer who makes just the minimum payment but nevertheless falls farther and farther behind. For example, formerly some banks set the minimum payment at balance due or else the larger of $10 or 2% of the total new balance (including interest). However, for a high enough interest rate, paying 2% of the balance due will not cover the interest, so the balance actually would increase (this is called negative amortization). How high would the APR have to be to make this happen? (Hint: It’s not just )

34. (Requires a spreadsheet) A well-known national credit card calculates minimum payment due as the new balance (if less than $35), or else the greatest of the following:

Then any late fees are added and the total is rounded to the nearest whole dollar.

35. (Requires a spreadsheet) Repeat Exercise 34, but for a monthly interest rate of 2.5%.

36. (Requires a spreadsheet) A bank or credit union may offer to let you agree in advance to skip a payment (e.g., on a car loan but usually not on a mortgage or a credit card)—in exchange for a processing fee (such as $35) to be added to the principal. If you skip the payment, interest continues to accrue for that month on the remaining principal plus the added processing fee. You continue regular payments as usual in the same amount as before, except that the last payment is a larger “balloon” payment to pay off the loan. Suppose that you borrowed $11,158.05 from your credit union for a 60-month home improvement loan at 9%. Verify that your monthly payment is $231.62 and that after 12 months of payments you still owe $9307.74. You receive an offer to skip the 13th payment, for $35 added to the principal, and you do so. How much will the balloon payment be?

37. (Requires a spreadsheet) Ads for purchasing cars often cite the monthly payment per $1000 borrowed. For example, a recent ad quoted $17.48 per $1000 borrowed for a 60-month loan.

38. (Requires a spreadsheet) An ad I saw on TV quoted 0.9% interest and $17.05 monthly payment per $1000 for an auto loan—but the ad went by too fast for me to see the term of the loan. For how many months would it be? (Hint: Use the NPER function in your spreadsheet.)

39. As we noted, the Consumer Financial Protection Bureau feels that almost all of the up-front costs that the consumer must pay to get a loan should be included as part of the “finance charge” and consequently factor into the quoted APR. Citicorp (a mortgage lender) argues that “the APR calculation includes interest that you will never pay and spreads the closing costs over too many years.” Consider the closing costs noted in Figure 22.2 (page 928). Which of these costs do you feel should be included as part of the fee for getting the loan and hence should be entered into the APR?

40. Should “truth in lending” require disclosure of the EAR rather than the APR? Why or why not?

22.4 Other Loans

41. You need to buy a car and finance $5000 of the cost. The dealer offers you a 5.9% add-on loan to be repaid in monthly installments over four years. How much is your monthly payment?

42. You have to make some home improvements—well, really maintenance that you can’t defer any longer!—and need to borrow $3000 to pay for them. You can get an 8.5% add-on loan from a savings and loan association to be repaid in installments over 2 years. How much is your monthly payment?

43. Repeat Exercise 42, except that you can get a 9% discounted loan from a loan company to be repaid in monthly installments over 4 years. What is the monthly payment on this loan?

44. Repeat Exercise 42, except that you can get an 8.5% discounted loan from a loan company to be repaid in monthly installments over 5 years. What is the monthly payment on this loan?

45. Suppose you can get either an add-on loan or a discounted loan, both for the same proceeds (principal), at the same interest rate, and for the same period. Show in general that the add-on loan has a lower monthly payment.

46. For one payday lender, the fee for a $100 loan for up to two weeks is $15. What is the APR if the loan is for the full two weeks?

47. Another payday lender charges $26.10 for a $100 loan for 7 to 14 days. What is the APR if the loan is for 7 days?

48. The leading British payday lender Wonga made profits of $1.9 billion on loans in 2012. Wonga charges a daily rate of 1%, compounded daily.

49. All Credit Lenders (with storefronts in Illinois, Wisconsin, and South Carolina) offers “line of credit” loans. With such a loan, you receive a cash advance, much as you might from using a credit card. The interest, calculated on a daily basis, comes to the currently advertised APR of 24% (below the Illinois usury cap of 36%). But if you have not paid back the loan by the end of the month, you are charged a “required account protection fee,” usually $15 per $50 borrowed, whose alleged purpose is to protect the borrower in case the borrower becomes unemployed and is unable to make payments. In March 2012, Loralty Harden (who is retired and disabled) borrowed $100 under such an arrangement at the Machesney Park, Illinois, office, with an interest rate of 18%. During the subsequent year, she paid $360 in protection fees and $18 in interest. She still owed $100. Using her case, the attorney general of Illinois sued parent company CMK Investments for “unfair and deceptive business practices.”

50. Should your state cap the interest rate on short-term loans, such as payday loans? According to one source, the average payday loan is “flipped” eight times, so the loan system is trapping borrowers in a “cycle of debt.” Lenders counter that they are doing the borrowers a favor because some borrowers have no other alternative (except theft or robbery!), and that small loans and high rates of nonrepayment make high rates of interest essential. One representative said that capping interest rates at a proposed 36% in Wisconsin would “eliminate the industry.” Another objected to figuring the interest rate on an annual basis, claiming that doing so is like calculating the cost of staying in a hotel for a year even though you stay only a couple of nights.

51. For the 12-month loan:

52. For a 24-month loan, the Rule of 78s uses 300 instead of . (The general formula for the sum of the first integers is .) Suppose that you take out an add-on loan for a car that costs $10,000 at a 3.0% interest rate, with monthly payments over a term of 24 months and a contract that specifies use of the Rule of 78s for early repayment. How much will you have to pay if you pay off the balance on the loan after 12 months?

53. Put off by the high monthly car payments of Exercises 15–17, you might be attracted to leasing a car instead of buying one. With the “college grad discount,” in May 2014 in Rhode Island you could lease a 2014 Toyota Corolla LE A4 for $65 per month for 24 months (plus tax, tag, title, registration, and “dealer documentation fee” of $200). You would also get no-cost maintenance and 24-hour roadside assistance during the 24 months. Because this was a lease, not a loan, the dealer does not have to disclose anything about interest rates. However, the “capitalized cost” was the buy-it price of $15,490 minus the required down payment of $1900, so $13,590. You would have the option at the end of the lease to purchase the car for $13,542. So in effect, your monthly payments would have paid for the difference in value of the car of . However, those payments would have totaled only .

The purchase price, if you had bought instead of leased, would have been $15,490, with (after the down payment) financing needed for $13,590.

54. The manufacturer’s suggested retail price (MSRP) on the car in Exercise 53 was $19,335. How do the manufacturer and the dealer make money on leasing? Why are the payments for leasing a car so much lower than for purchasing?

22.5 Annuities

55. (Requires a spreadsheet) Jack Whittaker of West Virginia, on Christmas Day in 2002, won the jackpot worth $314.9 million in a lottery with “annuity value.” (His subsequent life has been far from a fairy tale, as a Google search will reveal.) Instead of receiving $314.9 million in 30 equal annual payments, including one immediately, he chose a lump sum, which came to $170 million. What was the corresponding interest rate of the annuity?

56. (Requires a spreadsheet) Winners of the Powerball lottery can elect either an immediate lump sum (almost all do) or an annuity. In the latter case, the advertised jackpot amount is paid in 30 annual payments, including one immediate payment. To keep up with inflation, each payment is 4% more than the previous year’s; such an annuity is called a graduated annuity. On October 10, 2007, Eugene and Stanislawa Markiewicz took their prize of $20 million in the form of a graduated annuity.

57. Suppose a man retires at age 65, and in addition to Social Security, he needs $2000 per month in income. Based on an expected lifetime of 204 more months, how much would he have to invest in a life income annuity earning 3% APR to pay that much per year?

58. Repeat Exercise 57, but for a woman at age 65, whose expected lifetime is 237 more months.

59. (Requires a spreadsheet) Long ago, Darryl Strawberry played for the New York Mets baseball team. Part of his compensation was an annuity of $8,891.82 to be paid out over the 30 years after the end of his career. But to pay off his tax debts, in January 2015 the government auctioned off the right to receive the 12 years of remaining payments. The IRS set a minimum bid of $550,000. What interest rate would that correspond to?

60. Some annuity companies offer “merged gender” rate schedules, so that for the same purchase price, a man or a woman at the same age receives the same monthly annuity payment. (This is in effect what happens with the “life annuity” of Social Security payments.) Since a woman can expect to live longer, is that fair?

Chapter Review

61. You get a 30-year mortgage for $150,000 at 4.5%. What is your monthly payment?

62. The simple interest rate on a federal PLUS loan was 6.84% for 2015-2016. There was also a loan origination fee of 4.292%, so you receive as proceeds only of the loan amount. Assume that your parents make interest payments until repayment starts and then repay with fixed monthly payments over 10 years. For the equivalent no-fee loan:

63. A payday lender charges $25 for a $100 loan for 7 to 14 days. What is the APR if the loan is for 14 days?

64. Suppose that you bought your house with a 30-year adjustable-rate mortgage (ARM) at 4.5% for $150,000. After 5 years, the rate was raised to 6.5%. What was your new payment?

65. In July 2015, you could buy a 2015 Toyota Camry LE for $22,790 at 0% interest for 60 months, or you could get a $1000 Toyota rebate but then pay 2.9% over 60 months. Which deal offers a lower monthly payment?