CHAPTER 5

Use the following to answer question 1:

Given the recent drop in mortgage interest rates, you have decided to refinance your home. Exactly five years ago, you obtained a $100,000 30-year mortgage with a fixed rate of 10%. Today you can get a 30-year loan for the currently outstanding balance at 8%. This loan, however, requires you to pay a $250 appraisal fee and 3 points at the time of the refinancing. (Hints: (a) 1 point equals 1% of the amount borrowed, and (b) ignore tax considerations.)

        1.     How much will your monthly payments be after you refinance?

        A)        $443.96

        B)        $505.67

        C)        $652.90

        D)        $708.63

        E)        $733.76

        2.     You notice a local consumer finance company is offering 20% APR loans, but compounds interest daily. What is the EAR?

        A)        12.21%

        B)        22.13%

        C)        23.61%

        D)        24.97%

        E)        25.83%

        3.        There are three factors that affect the future value of an annuity. Explain what these three factors are and discuss how an increase in each will impact the future value of the annuity.

        4.        Deryl wishes to save money to provide for his retirement. Beginning one month from now, he will begin depositing a fixed amount into a retirement savings account that will earn 12% compounded monthly. He will make 360 such deposits. Then, one year after making his final deposit, he will withdraw $100,000 annually for 25 years. The fund will continue to earn 12% compounded monthly. How much should his monthly deposits be?

        A)        $205.28

        B)        $209.58

        C)        $214.21

        D)        $234.89

        E)        $249.38

        5.     Moe purchases a $100 perpetuity on which payments begin in one year. Larry purchases a $100 perpetuity on which payments begin immediately. Both make annual payments and a 10% interest rate is appropriate for both cash flow streams. Which of the following statements is true?

        A)        Moe's perpetuity is worth $100 more than Larry's

        B)        Larry's perpetuity is worth $100 more than Moe's

        C)    The perpetuities are of equal value today

        D)        Larry's perpetuity is worth $90.91 more than Moe's

        E)        Moe's perpetuity is worth $90.91 more than Larry's

        6.     Mr. Dubofsky just won a "Name That Tune" contest with a grand prize of $250,000. However, the contest stipulates that the winner will receive $100,000 immediately, and $15,000 at the end of each of the next 10 years. Assuming that he can earn 5% on his money, how much has he actually won?

        A)        $92,156.46

        B)        $98,225.11

        C)        $115,826.02

        D)        $215,826.02

        E)        $250,000.00

        7.     You have $10,000 to invest. The First National Bank offers one-year certificates of deposit with a stated rate of 5.50% compounded quarterly. What rate compounded semiannually would provide you with the same amount of money at the end of one year?

        A)        5.487%

        B)        5.500%

        C)        5.507%

        D)        5.512%

        E)        5.538%

        8.     You are considering two annuities, both of which make total annuity payments of $10,000 over their life. Which would be worth more today, annuity A which pays $1,000 at the end of each year for the next 10 years, or annuity B which pays $775 at the end of the first year, but the annuity payment grows by $50 each year, reaching $1,225 at the end of the 10th year? Are there any circumstances in which the two would be equal? Explain your reasoning.

        9.        Should lending laws be changed to require lenders to report the EAR rather than the APR? Explain.

Use the following to answer questions 10-11:

Rob and Laura wish to buy a new home. The price is $187,500 and they plan to put 20% down. New Rochelle Savings and Loan will lend them the remainder at a 10% fixed rate for 30 years, with monthly payments to begin in one month. (Ignore taxes.)

        10.        Assume that, in order to receive the 30-year loan from Brady Financing, Rob and Laura must pay 3 "points" at the time the loan is originated. (One point equals 1% of the amount to be borrowed.) What is the effective interest rate (EAR) on this loan, after taking the points into account? (Hint: find the discount rate that equates the loan amount with the present value of the loan payments plus the points paid.)

        A)        9.11%

        B)        10.00%

        C)        10.37%

        D)        10.47%

        E)        10.87%

        11.        Suppose Rob wants to pay off the loan in 15 years. How much extra must he pay each month to do so?

        A)        $11.25

        B)        $201.99

        C)        $295.55

        D)        $311.55

        E)        $314.47

        12.        What is the present value of $1,000 payments received at the beginning of each year for the next 10 years? Assume an interest rate of 5.625%.

        A)        $7,069.13

        B)        $7,093.62

        C)        $7,492.64

        D)        $7,914.10

        E)        $8,165.12

Use the following to answer questions 13-14:

You and your spouse have found your dream home in Rapid City, South Dakota. The selling price is $120,000; you will put $20,000 down and obtain a 30-year fixed-rate mortgage at 8.25% for the rest.

        13.        Although you will get a 30-year mortgage, you plan to prepay the loan by making an additional payment each month along with your regular payment. How much extra must you pay each month if you wish to pay off the loan in 20 years?

        A)        $24.56

        B)        $54.88

        C)        $100.80

        D)        $103.28

        E)        $106.86

        14.   How much interest will you pay (in dollars) over the life of the loan? (Assume you make each of the required 360 payments on time.)

        A)        $135,101

        B)        $145,583

        C)        $170,457

        D)        $190,457

        E)        $270,457

        15.   If you deposit $2,500 at the end of each six months into an account which earns 5.5% interest compounded quarterly, how much will be in the account in 5 years?

        A)        $13,953

        B)        $16,931

        C)        $26,605

        D)        $28,357

        E)        $32,188

        16.        Annuity A makes annual payments of $813.73 for each of the next 10 years, while annuity B makes annual payments of $500 per year forever. At what interest rate would you be indifferent between the two? At interest rates above this break-even rate, which annuity would you choose? How about below?

        17.   Your brother-in-law borrowed $2,000 from you 4 years ago and then disappeared. Yesterday he returned and expressed a desire to pay back the loan, including the interest accrued. Assuming that you had agreed to charge him 10% compounded annually, and assuming that he wishes to make five equal annual payments beginning in one year, how much would your brother-in-law have to pay you annually in order to extinguish the debt? (Assume that the loan continues to accrue interest at 10% per year.)

        A)        $697.43

        B)        $738.63

        C)        $751.46

        D)        $772.45

        E)        $798.24

Use the following to answer question 18:

With auto loans extending 5,6,7 or more years these days, it is common for buyers who wish to trade their cars in after a few years to find themselves to be "upside down". In other words, the outstanding principal on the auto loan exceeds the value of the car being traded. Suppose you buy a new Toyota for $20,000, paying nothing down. You agree to a repayment schedule of 6 equal annual payments beginning one year from today. The banker's required return is 9%, compounded annually. Assume the car will lose 20% of its value the first year, and 10% ($2,000) each year thereafter.

        18.        Given the depreciation schedule above, how much will the car be worth after 3 years?

        A)        $10,000

        B)        $12,000

        C)        $14,000

        D)        $16,000

        E)        $20,000

        19.        Which of the following describes the equation for finding the annuity present value factor?

        A)    (1 minus present value factor) times the interest rate

        B)    (1 plus present value factor) divided by the interest rate

        C)    (1 plus present value factor) times the interest rate

        D)    (1 minus present value factor) divided by the interest rate

        E)        (present value factor minus 1) divided by the interest rate

        20.   You are going to withdraw $1,000 at the end of each year for the next three years from an account that pays interest at a rate of 8% compounded annually. The account balance will reduce to zero when the last withdrawal is made. How much money will be in the account immediately after the second withdrawal is made?

        A)        $925.93

        B)        $977.10

        C)        $982.29

        D)        $1,000.00

        E)        $2,000.00

Answer Key

        1.     D    

        2.     B     

        3.     The factors are the interest rate, payment amount, and number of payments. An increase in any of these three will increase the future value of the annuity.

        4.     C    

        5.     B     

        6.     D    

        7.     E     

        8.     The second annuity weights its payments more toward the back of the period, rather than the front, making it less valuable unless the discount rate is zero. Some students may get tripped up by the fact that the two annuities have the same total payments. This would clearly demonstrate a lack of understanding of the time value of money.

        9.     It would be more meaningful for consumers to know the EAR rather than the APR. The EAR is slightly more difficult to compute and also more difficult to explain, and may add confusion to the loan process. However, regardless of the costs, it would appear that consumers would benefit from learning what the EAR is as opposed to the APR.

        10.   E     

        11.   C    

        12.   D    

        13.   C    

        14.   C    

        15.   D    

        16.   This requires the students to actually use the present value formulas, setting the present value annuity equal to the present value of a perpetuity and solving for the interest rate that makes the two equivalent. The major first step is recognizing that the indifference point occurs when the two present values are equal. The break-even rate is 10%, below that rate, the perpetuity is better, while above that rate, the 10-year annuity is preferred.

        17.   D    

        18.   B     

        19.   D