Chapter 4, annuitites and loans PDF

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Chapter questions, solutions and the formulas used if necessary....

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1. You are 62 and planning your retirement. If you retire immediately, then your Social Security benefit is $1,963 per month. If you retire at age 67 then the benefit is $2,157 per month. If you expect to live to age 88, when should you retire and start claiming retirement benefits? The interest rate is 2.8% per annum. Assume that benefits are collected at the end of each month. Express your answer as the difference in the future value of the benefits."(Early minus Late.)# $160,726.84

2. Calculate the future value of a 8-year annuity due. The first annual payment of $12,000 occurs immediately. Use an interest rate of 7%. a. b. c. d.

$131,736 $103,848# $123,736# $111,118#

3. You want to borrow $60,000. Your bank offers an amortized loan with annual, end-of year payments over a 25-year term. The bank charges an interest rate of 11%. What are the annual loan payments?# $7,124.41

4. Sabrina deposits $400 in an account at the end of each year for 2 years. If the account pays 2% interest annually, how much money will be in Sabrina's account at the end of 2 years?# $808.00

5. You have just purchased a $250,000 home, and the bank has quoted you an interest rate of 6.5% on a 25-year mortgage. You chose to make 26 payments per year. What periodic rate should you use to calculate the mortgage payments?# 0.2463%

6. South Penn Tracking is financing a new truck with a loan of $10,000 to be repaid in 5 annual end-of-year installments of $2,504.56. What annual interest rate is the company paying?# a. b. c. d. e.

10%# 8% 7%# 9%# 11%#

7. The AMC Gremlin has a sticker price of $95,000. You are trying to choose between a lease or a loan and you wonder if there is a tax advantage to the lease. The lease rate is 5.9%. The term of the lease is four years. The buyout is $52,250 at the end of the lease. The lease payments are monthly. With leases, the lessee pays sales tax on all payments including the buyout."The loan charges the same rate as the lease and payments are monthly over a 4-year term. With loans, the sales tax is paid on the price of the car and forms part of the loan principal. What is the difference between the tax on the lease and the loan? (Discount all taxes to time zero, the date of the lease/loan, and assume a sales tax rate of 13%.)# $0

8. Yesterday, you invested $40 in an investment. Assume that investment repays you a lumpsum of $45 in one year."Use this information to answer the"following questions. If you receive an additional cash payment from the investment of $8 at the end of the year, then what interest rate have you earned over the year?# 32.5%

9. Angela owes $9,951.56. She has made arrangements to repay the debt in installments of $248.73 per month and will be charged monthly interest of 1.62 % on the overdue balance. How many months will it take her to repay the debt and interest? Assume end-of-month payments. Round your answer to months.# 65 months

10. In three years, you will begin receiving an annual payment of $600 that will be made for two years. If the annual interest rate is 12%, what will be the balance in your account at the end of the fourth year? a. b. c. d. e.

$1,260# $1,344# $2,024 $1,272 $1,200#

11. You have just borrowed $120,000 to buy a new home. The term of the amortized loan is 30 years. You have decided to make monthly payments. If the stated (nominal) annual interest rate is 14.75%, what is the amount of each of the monthly payments? Round to the nearest whole dollar. a. b. c. d. e.

$1,472# $1,515# $1,493 $1,440# $1,522#

12. If a 5-year regular annuity has a present value of $1,000, and if the interest rate is 10%, what is the amount of each annuity payment? a. b. c. d. e.

$315.38# $263.80 $346.87# $300.20# $240.42#

13. You would like to lease a car. The car costs $65,816.28. You have $5,000 for a down payment. The dealer has offered you a 48-month lease with monthly payments of $750. At the end of the term, the buyout for the lease is 49% of the purchase price. What is the APR on this lease?# 4%

14. Tony Hawk plans to retire in 35 years. Upon retirement, Tony wants to buy a home in the prestigious Westbrook Manor Estates. He has already selected the home of his dreams and it sells, today, for $60,000. House price inflation is expected to cause this price to increase at a rate of 2% per year until his retirement. How much does Tony have to save at the end of each year (starting in one year) to have enough money to buy his dream home? Assume Tony can earn 8% on his savings. a. b. c. d.

$696 $1,200# $348# $538#

15. You are going to borrow $97,000 to fund the start of your new restaurant. The bank wants to be repaid with a balloon payment in 4 years. They charge 9% interest. How much will your payment be?# $136,923.42 16. You are going to lease a new Hudson Hornet. The sticker price for the car is $31,035.95. The term of the lease is 36 months, the rate on the lease is 3% and the buyout is 45%. If you decide not to make a down payment, then what are the monthly lease payments (before tax)?# $530

17. You bought a ski chalet at Mount Tremblant for $357,367.07. You borrow the full amount over a 25-year term. The bank quotes you a rate of 3 %. You choose to make monthly payments. Over the life of the mortgage, what is the total amount of interest that you pay? a. b. c. d.

$150,000 $175,000# $200,000# $125,000#

18. You are going to lease a new Hudson Hornet. The sticker price for the car is $34,595.86. The term of the lease is 36 months, the rate on the lease is 2% and the buyout is 45%. If you decide not to make a down payment, then what are the monthly lease payments (before tax)?# $570#

19. You borrowed $350,000 12 years ago. The loan is structured as an amortized loan. The interest rate is 10% and you make quarterly (end-of-quarter) payments of $9,559.16. The loan is amortized over 25 years. How much principal have you paid over the first 12 years?# $73,519.72

20. Gina is interested in buying a house beside the university which she will rent to students. She expects to receive $900 a month in rental income. Gina's real estate agent estimates that she will be able to sell the property for $225,000 at the end of 2.25 years. Gina wants a return of 0.5% per month. What is the most that she should pay for the property? Assume that she will purchase the house today and receive the first (beginning of month) rental payment today.# $219443.83

21. Jed wants to borrow $1,000 from you. He is proposing to repay you with three end-of-year payments of $330.69. In addition, he will make a final lump-sum of $150. What rate of return are you earning on the loan? Express your answer in percentage form rounded to two decimal places.# 6.5%

22. Kelvinator Appliances sells its 4.0 cubic foot, stainless steel front-load washer and steam dryer set for $2,400. It offers a 48-month purchase financing arrangement with end-of-month payments of $50. Customers have to make a final lump-sum payment called the “buy-out.” After the buy-out customers own their"washer-dryer. Whirlpool advertises its financing rate of 5% APR. What buyout makes this a fair deal? a. b. c. d.

$194.60# $250.02# $279.40 $182.83#

23. You tried to sneak a cappuccino into the movie theatre. It spilled on your leg and scalded you. Your lawyer says that you can expect a payout from the coffee shop’s insurance company of $10,000 per year for 10 years. The only catch is that the first payment won’t happen until the court case is over in 2 years. What is the present value of your settlement as of today? Assume an interest rate of 3%.# $82,817.50

24. Homer expects to receive $250 next year. He also expects to receive $250 in two years and again in three years. What is the present value of those payments one year from today? Assume an interest rate of 5.6%.# $710.93# 25. You are 30 years old today. You want to retire at the age of 55. You expect to live until age 85. You would like to have a monthly income of $12,000 per month in retirement. How much do you have to save per month during your working years in order to achieve your retirement goal? Assume end of period payments. Assume an annual interest rate of 3 % in retirement and 5 % during your working life.# $4,779.56

26. You bought a house 3 years ago. To finance the purchase, you took out a mortgage for $9962,305.82. The interest rate on the mortgage is 3 % and the amortization period is 30 years. You chose to make 26 payments per year and each payment is $1,866.82. Your last payment was yesterday. How much principal remains owing today?# $899,999.33

27. Starting one month from now, you need to withdraw $240 per month from your bank account to help cover the costs of your university education. You will continue the monthly withdrawals for the next four years. If the account pays 0.2 % interest per month, how much money must you have in your bank account today to support your future needs?# $10,973.87

28. Two years ago you signed a 4-year lease on a DeLorean DMC-12. The sticker price on the car was $39,952.61, the lease rate was 3%, and the monthly payments (starting at the time of signing) were $530. The buyout was $17,978.68 due at the end of the lease term. Now (two years after signing the lease but just before the 25th lease payment) DeLorean has just released the new DMC-13 with better styling and 20% more horsepower. You want to get out of your lease on the old car and lease one of the new ones. DeLorean is happy to take your car from you and cancel the lease if you have positive equity in the car. That is, if the resale value of the car exceeds the principal sill owing (the difference is positive equity). However, if the resale value is less than the principal outstanding, DeLorean will require you to pay the difference in order to exit the lease."The resale value of the old DeLorean is $29,794.75. What is the value of your equity in the car?# $500

29. You are going to lease a brand new Edsel Corsair. The price of the Corsair is $61,998.55. You intend to make a $3,000 down payment. The term of the lease is 48 months and the lease rate is 2% APR. The buyout for the lease is 45% of its purchase price. What are the monthly lease payments (before tax)? a. b. c. d.

680# 700# 720 660#

30. Calculate the present value of a perpetuity with annual payments of $22,000. Use an interest rate of 8% and assume that the first payment occurs one year from today."# $275,000#

31. You would like to lease a car. The car costs $59,337.53. You have $5,000 for a down payment. The dealer has offered you a 48-month lease with monthly payments of $700. At the end of the term, the buyout for the lease is 49% of the purchase price. What is the APR on this lease?# 4%# 7%# 6%# 5%

32. Happy birthday! You are 30 years old today. You want to retire at age 60. You want to have $2,200,000 at retirement. Realistically, you know that the most that you can save from your 31st birthday until your 50th is $6,750 per year (you only save on your birthdays!). How much do you have to save each year from your 51st to your 60th birthday in order to achieve your retirement goal if you can earn 5.5% on your savings?# $139,644.25#

33. You borrow $220,000. The loan is structured as an amortized loan to be repaid over 3 years with 52 (end-of-period) payments per year. The lender is charging you a rate of 15.5 % APR. (Assume that the lender can continue to earn 15.5 % by reinvesting the payments.) How much interest does the lender earn over the life of the loan?"# $130,000.96

34. Sally wants to buy a Ford Mustang. The MSRP is $26,185. Ford offers a purchase financing plan with no money down and 48 end-of-month payments of $550. Should she buy the car for cash or take Ford's purchase financing? Assume that she has the cash and could invest it to earn 4%. Calculate the present value of each alternative and enter your answer as the PV of the cash purchase minus the PV of the financing.# $1,826.14

35. What is the future value of a perpetuity with annual payments of $1,000 compounding at 10%? a. b. c. d.

it depends on the length of the perpetuity# infinity $10,000# $100,000#

36. Homer deposited $250 in the bank today and plans to make the same deposit next year and again the year after that. How much will Homer have in the bank in two years? Homer's bank pays an interest rate of 5.6%. a. b. c. d.

$1,087# $837# $793 $543#

37. You've graduated from college and landed a good job. You want to replace your car, but don't want to take out a car loan. Instead, you decide to invest $ 600 per month in the stock market and hope to earn 12%. If the market performs as you're hoping, how many years will it take to accumulate $ 50,000? Ignore taxes.# 5.07

38. Some banks allow you to skip a loan payment and roll it into your principal. This is especially attractive in January when the Christmas VISA bill arrives."Consider the following simplified example."You renovated your house last year and borrowed $80,000."The term of the loan is three years, the rate is 8% (APR), and the annual (end-of-year) payments are $31,042.68."The end of the first year arrives and you can’t afford to make the first payment."You want to roll it into your principal."Your bank has offered to increase the size of the second and third payments to allow you to do this."Answer the following two questions.# (A) What will your new (second and third) loan payments be? (Hold everything else in your loan constant, e.g., remaining amortization period and interest rate.)# $48,450.46

(B) Compare the amount of interest in the two sets of loan payments."How much more interest do you pay over the life of the loan (if you skip the first payment) compared to what you would have paid if you hadn't skipped the first payment?# $3,772.88

39. The BMW M5 retails for $90,000 (all taxes included). BMW Financial Services offers a 3year car loan at an interest rate of 5.5% with end-of-month payments. The payments are $2,717.63. How much interest will you pay over the life of the loan (in the blended payments)?# $7,834.72 40. One year ago you signed a 4-year lease on a DeSoto Fireflite Sportsman. The sticker price on the car was $46,657.97, the lease rate was 2%, and the monthly payments (starting at the time of signing) were $544.10. The buyout was $23,328.98 due at the end of the lease term. Now (one year after signing the lease but just before the 13th lease payment) DeSoto has just released the new Fireflite Sportsman with better styling and 20% more horsepower. You want to get out of your lease on the old car and lease one of the new ones. How much remains owing on your lease?# $40,999.36 41. The Shelby Cobra retails for $47,684 (all taxes included). What are the monthly loan payments for the car if you make a down payment of $4,001? The term is 5 years and the APR is 6.5%. (Car loan payments are made at the end of each month.)# $854.71

42. You are 30 years old today. You want to retire at the age of 65. You expect to live until age 90. You would like to have annual income of $160,000 in retirement. How much do you have to save (per year) during your working years in order to achieve your retirement goal? Assume end of period payments, so you start saving at age 31, end savings on your 65th birthday, begin retirement withdrawals at age 66, and make your last retirement withdrawal at age 90. Assume an interest rate of 5%.# $24,967.04#

43. Uriah Heep celebrated his 18th birthday by opening a savings account at the Thames River Bank and depositing $1,900. He continued to deposit the same amount on every subsequent birthday until he was 27 years old. After depositing $1,900 on his 27th birthday, Uriah decided to abandon his savings plan. He never saved again, but he left the accumulated savings in the bank account. The bank paid an interest rate of 15%. When Uriah turned 65, he withdrew the money from the bank. What was the amount of his withdrawal?"# $7813526.91#

44. You are trying to calculate how much money you should have at retirement. On your 55th birthday you will retire and immediately make your first withdrawal of $4,615.38. You plan to make 26 such withdrawals each year. You plan to continue withdrawing at that level and frequency until you are 90 years old. (Assume beginning of period withdrawals with the first withdrawal on your retirement date and no withdrawal on your final birthday.) During retirement, your savings will earn only 3.5 % per annum. How much do you have to have saved at retirement to fund these planned withdrawals?# $2,423,828.18#

45. Calculate the present value of a 50-year ordinary annuity. The first payment of $22,000 occurs in one year. Then calculate the present value of a perpetuity with the same payment. (Assume that the first payment occurs in one year.) Use an interest rate of 8%. What is the difference between the PV of the perpetuity and the PV of the annuity? Express the difference as a percentage of the PV of the perpetuity.# 2%

46. Homer plans to deposit $150 in the bank in one year. He plans to make the same deposit two years from today and three years from today. How much will Homer have in the bank in four years? Homer's bank pays an interest rate of 5.6%.# a. b. c. d.

$652# $502 $689# $476#

47. You are trying to choose between investing in an Index mutual fund (that mimics the S&P500) or buying units of an exchange-traded-fund (ETF) that invests directly in the S&P500. Both investments give you exposure to the S&P500, but the difference is in the way fees are assessed. The mutual fund subtracts an annual fee of 1% from the gross returns on the fund. The S&P500 is expected to grow at a rate of 9% (0.75% per month) in the future, so the return on the Index mutual fund will be 8% (0.6667% per month). You are 30 years old today. You plan to save $300 per month at the end of each month until your 60th birthday. So, you expect to accumulate $447,107.83 by investing in the mutual fund. Alternatively, you could invest in the ETF. With the ETF, there are no management expenses which reduce the return--you will earn the same return as the S&P500. However, you pay a brokerage commission ($fee) each time that you trade (so your actual deposit each month is $300 - $fee). How big does the trading commission ($fee) have to be on the ETF in order for the future value of the two strategies to be equal? (Assume monthly periodicity and monthly purchases of the ETF. Ignore any commission on selling at age 60.)"# $55.78#

48. In the following case, the mixed end-of-period cash flow stream has an annuity embedded within it. Calculate the present value of the cash flow stream, assuming a 6% discount rate.#

$19,525.96 49. You are trying to decide how much money you will need at retirement. You expect to retire at age 65. You hope to travel extensively while you are healthy enough. To finance your travels (and cover your basic living expenses), you think you will need income of $127,000 per year (at the end of each year). You will make your first withdrawal on your 66th birthday. You expect to stay healthy enough for travel for the first 13 years after retirement and thus make a withdrawal of $127,000 at the end of each of those 13 years. Once your active travel years are over, you will settle down into a retirement home. During your retirement home years, you will only need $104,000 in income per year. You expect to live in the retirement home for 6 years. You will make the first retirement home withdrawal 14 years after retirement and all subsequent withdrawals will continue to be at year-end. If retirement savings will earn a return of 5%, then how much will you need at retirement to fund these planned withdrawals?# $1,472,925.55

50. You are 40 years old and want to retire at age 60. You have already saved $64,000 for retirement. Each year, starting one year from no...