chapter 04 future value, present value and interest...

Chapter 04 - Future Value, Present Value and Interest Rates

Chapter 04Future Value, Present Value and Interest Rates

Multiple Choice Questions

1. (p. 66) A promise of a $100 payment to be received one year from today is: a. More valuable than receiving the payment todayb. Less valuable than receiving the payment six months from nowC. Equally valuable as a payment received today if the interest rate is zerod. Not enough information is provided to answer the question

AACSB: Reflective ThinkingBLOOMS: ApplicationLOD: 1

2. (p. 66) The future value of $100 at a 5% per year interest rate at the end of one year is: a. $95.00B. $105.00c. $97.50d. 107.50

AACSB: AnalyticBLOOMS: ApplicationLOD: 3

3. (p. 65) Credit: a. Probably came into being at the same time as coinageB. Predates coinage by 2,000 yearsc. Did not exist until the Middle Agesd. First became popular due to the writings of Aristotle

AACSB: AnalyticBLOOMS: KnowledgeLOD: 1

4-1


4. (p. 66) Which of the following expresses 5.65%? a. 0.565b. 0.00565c. 5.65D. 0.0565

AACSB: AnalyticBLOOMS: ComprehensionLOD: 1

5. (p. 66) Which of the following expresses 4.85%? A. 0.0485b. 4.850c. 0.00485d. 0.485


6. (p. 66) Which of the following expresses 5.5%? a. 0.0055b. 5.50c. 0.550D. 0.0550


7. (p. 66) If the interest rate is zero, a promise to receive a $100 payment one year from now is: a. More valuable than receiving $100 todayb. Less valuable than receiving $100 todayC. Equal in value to receiving $100 todayd. Equal in value to receiving $101 today


4-2


8. (p. 66) If a saver is willing to wait a year to receive a $100 payment rather than accept a lesser amount today: a. The $100 is less than the present valueb. The present value must not be calculableC. The present value must be less than $100d. The saver must not know the present value


9. (p. 66) Which of the following best expresses the proceeds a lender receives from a one-year simple loan when the annual interest rate equals i? a. PV + ib. FV/iC. PV(1 + i)d. PV/i


10. (p. 66) Which of the following best expresses the proceeds a lender received from a one-year simple loan when the annual interest rate equals i? a. PV/ib. PV + ic. FV/iD. FV-PV/i


4-3


11. (p. 66) Suppose Tom receives one-year loan from ABC Bank for $5000.00. At the end of the year, Tom repays $5400.00 to ABC Bank. Assuming the simple calculation of interest, the interest rate on Tom's loan was: a. $400B. 8.00%c. 7.41%d. 20%


12. (p. 66) Suppose Mary receives an $8,000 loan from First National Bank. Mary repays $8,480 to First National Bank at the end of one year. Assuming the simple calculation of interest, the interest rate on Mary's loan was: a. 8.00%b. $480C. 6.00%d. 5.66%


13. (p. 66) An investor deposits $400 into a bank account that earns an annual interest rate of 8%. Based on this information, how much interest will he earn during the second year alone? a. $25.60b. $32C. $34.56d. $64


4-4


14. (p. 67) Compound interest means that: a. You get an interest deduction for paying your loan off earlyB. You get interest on interestc. You get an interest deduction if you take out a loan for longer than one yeard. Interest rates will rise on larger loans

AACSB: Reflective ThinkingBLOOMS: ComprehensionLOD: 1

15. (p. 68) Which of the following best expresses the payment a saver receives for investing their money for two years? a. PV + PVb. PV + PV (1 + i)C. PV( 1 + i)2

d. 2PV( 1 +i)


16. (p. 68) Suppose a family wants to save $60,000 for a child's tuition. The child will be attending college in 18 years. For simplicity, assume the family is saving for a one-time college tuition payment. If the interest rate is 6%, then about how much does this family need to deposit in the bank today? A. $10,000b. $21,000c. $42,000d. $57,000


4-5


17. (p. 68) Which of the following best expresses the payment a lender receives for lending money for three years? a. 3PVB. PV(1+i)3

c. PV/(1 + i)3

d. FV/ (1 + i)3


18. (p. 68) Suppose Paul borrows $4000 for one year from his grandfather who charges Paul 7% interest. At the end of the year Paul will have to repay his grandfather: A. $4,280b. $4,290c. $4,350d. $4,820


19. (p. 70) Suppose that Ray Allen, a basketball player for the Seattle Supersonics, will become a free agent at the end of this NBA season. Suppose that Allen is considering two possible contracts from different teams. Note that the salaries are paid at the end of EACH year.

Contract #1 (Seattle) Contract #2 (Portland)

Signing bonus (paid today) $1 million $1 millionFirst-year salary $2 million $4 millionSecond-year salary $4 million $4 millionThird-year salary $5 million $3 million

The interest rate is 10%. Based on this information, which of the following is true? A. Allen should take the Seattle contract because it has a higher present valueb. Allen should take the Portland contract because it has a higher present valuec. Allen is indifferent between the two contracts because they are both worth $12 milliond. Allen is indifferent between the two contracts because they are both worth $10.9 million


4-6


20. (p. 69) Farou invests $2,000 at 8% interest. About how long will it take for Farou to double his investment (e.g., to have $4,000)? a. 4 yearsb. 5 yearsc. 8 yearsD. 9 years


21. (p. 70) A lender is promised a $100 payment (including interest) one year from today. If the lender has a 6% opportunity cost of money, he/she should be willing to accept what amount today? a. $100.00b. $106.20c. $96.40D. $94.34


22. (p. 70) A saver knows that if she put $95 in the bank today she will receive $100 from the bank one year from now, including the interest she will earn. What is the interest rate she is earning? a. 5.10%b. 6.00%c. 5.52%D. 5.26%


4-7


23. (p. 70) Tom deposits funds in his savings account at the bank which is paying 3.5% interest. If he keeps his funds in the bank for one year he will have $155.25. What amount is Tom depositing? a. $151.75B. $150.00c. $148.75d. $147.50


24. (p. 71) Mary deposits funds into a CD at her bank. The CD has an annual interest of 4.0%. If Mary leaves the funds in the CD for two years she will have $540.80. What amount is Mary depositing? a. $520.00b. $514.50C. $500.00d. $512.40


25. (p. 71) Mary deposits funds into a CD at her bank. The CD has an annual interest of 4.0%. If Mary leaves the funds in the CD for two years she will have $540.80. Assuming no penalties for withdrawing the funds early, what amount would Mary have at the end of one year? a. $521.60b. $490.00c. $500.00D. $520.00


4-8


26. (p. 67) Sharon deposits $150.00 in her savings account at the bank. At the end of one year she has $156.38. What was the interest rate that Sharon earned? A. 4.25%b. 6.38%c. 4.52%d. 5.63%


27. (p. 68) The value of $100 left in a savings account earning 5% a year, will be worth what amount after ten years? a. $150.00b. $160.50c. $159.84D. $162.89


28. (p. 68) The value of $100 left in a certificate of deposit for four years that earns 4.5% annually will be: a. $120.00B. $119.25c. $117.00d. $145.00


4-9


29. (p. 68) The future value of $100 that earns 10% annually for n years is best expressed by which of the following? a. $100(0.1)n

b. $100 x n x (1.1)C. $100(1.1)n

d. $100/(1.1)n


30. (p. 68) The future value of $200 that is left in account earning 6.5% interest for three years is best expressed by which of the following? a. $200(1.065) x 3b. $200(1.065)/3c. $200(1.065)n

D. $200(1.065)3


31. (p. 68) Which of the following best expresses the future value of $100 left in a savings account earning 3.5% for three and a half years? A. $100(1.035)3.5

b. $100(0.35)3.5

c. $100 x 3.5 x (1.035)d. $100(1.035)3/2


4-10


32. (p. 68) Which of the following best expresses the present value of $500 that you have to wait four years and three months to receive? a. ($500/4.25) x (1+i)b. $500 x 4.25 x (1 +i)C. $500/(1+i)4.25

d. ($500/4) x (1+i)3


33. (p. 68) If 10% is the annual rate, considering compounding, the monthly rate is: a. 0.0833%b. 0.833%C. 0.00797%d. 1.0833%


34. (p. 68) What is the future value of $1000 after six months earning 12% annually? a. $1050.00b. $1060.00c. $1120.00D. $1058.30


35. (p. 69) In reading the national business news, you hear that mortgage rates increased by 50 basis points. If mortgage rates were initially at 6.5%, what are they after this increase? a. 6.55%B. 7.0%c. 11.5%d. 56.5%


4-11


36. (p. 69) One hundred basis points could be expressed as: a. 0.01%B. 1.00%c. 100.0%d. 0.10%


37. (p. 69) The decimal equivalent of a basis point is: A. 0.0001b. 1.00c. 0.001d. 0.01


38. (p. 69) According to the rule of 72: a. Any amount should double in value in 72 months if invested at 10%B. 72/interest rate is the number of years approximately it will take for an amount to doublec. 72 x interest rate is the number of years it will take for an amount to doubled. The interest rate divided by the number of years invested will always equal 72%


39. (p. 69) The rule of 72 says that at 6% interest $100 should become $200 in about: a. 72 monthsb. 100 monthsC. 12 yearsd. 7.2 years


4-12


40. (p. 71) What is the present value of $200 promised two years from now at 5% annual interest? a. $190.00b. $220.00c. $180.00D. $181.41


41. (p. 70) What is the present value of $100 promised one year from now at 10% annual interest? a. $89.50b. $90.00C. $90.91d. $91.25


42. (p. 71) What is the present value of $500 promised four years from now at 5% annual interest? A. $411.35b. $400.00c. $607.75d. $520.00


43. (p. 72) The higher the future value of the payment: a. The lower the present valueB. The higher the present valuec. The future value doesn't impact the present value, only the interest rate really mattersd. The lower the present value because the interest rate must fall


4-13


44. (p. 72) The shorter the time until a payment: A. The higher the present valueb. The lower the present value because time is valuablec. The lower must be the interest rated. The higher must be the interest rate


45. (p. 73) The lower the interest rate, i: a. The lower is the present valueb. The greater must be nC. The higher is the present valued. The higher is the future value

AACSB: Reflective ThinkingBLOOMS: AnalysisBLOOMS: ComprehensionLOD: 2

46. (p. 72) Doubling the future value will cause: a. The present value to fall by halfb. The interest rate i, to doublec. No change to present value, only the interest rateD. The present value to double

AACSB: AnalyticBLOOMS: AnalysisLOD: 3

47. (p. 72) Doubling the future value will cause: A. The present value to doubleb. The present value to decreasec. The present value to increase by less than 100%d. The interest rate, i, to decrease

AACSB: AnalyticAACSB: Reflective ThinkingBLOOMS: AnalysisBLOOMS: ComprehensionLOD: 2

4-14


48. (p. 73) The present value and the interest rate have: a. A direct relationship; they both move togetherB. An inverse relationship; as i increases, PV decreasesc. An unclear relationship; whether it is direct or inverse depends on the interest rated. No relationship


49. (p. 73) At any fixed interest rate, an increase in time, n, until a payment is made: a. Increases the present valueb. Has no impact on the present value since the interest rate is fixedC. Reduces the present valued. Affects only the future value


50. (p. 73) A change in the interest rate: a. Has a smaller impact on the present value of a payment to be made far into the future than on one to be made soonerb. Will not make a difference in the present values of two equal payments to be made at different timesC. Has a larger impact on the present value of a payment to be made far into the future than on one to be made soonerd. Has a larger impact on the present value of a bigger payment to be made far into the future than on one of lesser value


4-15


51. (p. 74) A monthly growth rate of 0.5% is an annual growth rate of: a. 6.00%b. 5.00%C. 6.17%d. 6.50%


52. (p. 74) A monthly growth rate of 0.6% is an annual growth rate of: a. 7.20%b. 6.00%c. 7.60%D. 7.44%


53. (p. 74) A monthly interest rate of 1% is a compounded annual rate of: a. 12.00%b. 10.00%c. 14.11%D. 6.00%


54. (p. 74) A monthly interest rate of 1% is a compounded annual rate of: A. 12.68%b. 12.00%c. 6.00%d. 24%


4-16


55. (p. 74) An investment has grown from $100.00 to $130.00 or by 30% over four years. What annual increase gives a 30% increase over four years? a. 7.50%b. 6.30%C. 6.78%d. 7.24%


56. (p. 74) An investment grows from $100.00 to $150.00 or 50% over five years. What annual increase gives a 50% increase over five years? a. 12.00%b. 10.00%c. 9.25%D. 8.45%


57. (p. 79) The "coupon rate" is: A. The annual amount of interest payments made on a bond as a percentage of the amount borrowedb. The change in the value of a bond expressed as a percentage of the amount borrowedc. Another name for the yield on a bond, assuming the bond is sold before it maturesd. The total amount of interest payments made on a bond as a percentage of the amount borrowed


4-17


58. (p. 75) Higher savings usually requires higher interest rates because: a. Everyone prefers to save more instead of consumingB. Saving requires sacrifice and people must be compensated for this sacrificec. Higher savings means we expect interest rates to decreased. Of the rule of 72


59. (p. 76) The internal rate of return of an investment is: a. The same as return on investmentb. Zero when the present value of an investment equals its costC. The interest rate that equates the present value of an investment with its costd. Equal to the market rate of interest when an investment is made

AACSB: AnalyticBLOOMS: ComprehensionBLOOMS: KnowledgeLOD: 1

60. (p. 76) The equation above is for: a. A three-year coupon bond with a price equal to $400 and a coupon rate equal to 30%b. A four-year coupon bond with a price equal to $400 and a coupon rate equal to 12%C. A four-year investment with an initial investment of $400 that returns $120 each yeard. A four-year investment with an initial investment of $520 that returns $120 each year


61. (p. 77) If the internal rate of return from an investment is more than the opportunity cost of funds: A. The firm should make the investmentb. The firm should not make the investmentc. The firm should only make the investment using retained earningsd. The firm should only make part of the investment and wait to see if interest rates decrease


4-18


62. (p. 76) A mortgage, where the monthly payments are the same for the duration of the loan, is an example of: a. A variable payment loanb. An installment loanC. A fixed payment loand. An equity security


63. (p. 76) An investment carrying a current cost of $120,000 is going to generate $50,000 of revenue for each of the next three years. To calculate the internal rate of return we need to: a. Calculate the present value of each of the $50,000 payments and multiply these and set this equal to $120,000b. Find the interest rate at which the present value of $150,000 for three years from now equals $120,000C. Find the interest rate at which the sum of the present values of $50,000 for each of the next three years equals $120,000d. Subtract $120,000 from $150,000 and set this difference equal to the interest rate

AACSB: Reflective ThinkingBLOOMS: AnalysisLOD: 2

64. (p. 77) Usually an investment will be profitable if: a. The internal rate of return is less than the cost of borrowingb. The cost of borrowing is equal to the internal rate of returnc. It is financed with retained earningsD. The cost of borrowing is less than the internal rate of return


4-19


65. (p. 79) A coupon bond is a bond that: a. Always sells at a price that is less than the face valueB. Provides the owner with regular paymentsc. Pays the owner the sum of the coupons at the bond's maturityd. Pays a variable coupon rate depending on the bond's price


66. (p. 79) The coupon rate for a coupon bond is equal to: A. The annual coupon payment divided by the face value of the bondb. The annual coupon payment divided by the purchase price of the bondc. The purchase price of the bond divided by the coupon paymentd. The annual coupon payment divided by the selling price of the bond


67. (p. 79) If a bond has a face value of $1000 and a coupon rate of 4.25%, the bond owner will receive annual coupon payments of: a. $425.00b. $4.25C. $42.50d. A value that cannot be determined from the information provided


68. (p. 79) If a bond has a face value of $1,000 and the bondholder receives coupon payments of $27.50 semi-annually, the bond's coupon rate is: a. 2.75%B. 5.50%c. 27.5%d. A value that cannot be determined from the information provided


4-20


69. (p. 81) Consider a bond that costs $1000 today and promises a one-time future payment of $1080 in four years. What is the approximate interest rate on this bond? A. 2%b. 4%c. 8%d. 10.8%


70. (p. 82) Which of the following is necessarily true of coupon bonds? a. The price exceeds the face valueb. The coupon rate exceeds the interest ratec. The price is equal to the coupon paymentsD. The price is the sum of the present value of coupon payments and the face value


71. (p. 82) The price of a coupon bond will increase as: a. The face value decreasesb. The yield increasesc. The coupon payments increaseD. The term to maturity is shorter


72. (p. 84) Suppose the nominal interest rate on a one-year car loan is 8% and the inflation rate is expected to be 3% over the next year. Based on this information, we know: A. The ex ante real interest rate is 5%b. The lender benefits more than the borrower because of the difference in the nominal versus real interest ratesc. At the end of the year, the borrower pays only 5% in nominal interestd. The ex post real interest rate 11%


4-21


73. (p. 84) Interest rates that are adjusted for expected inflation are known as: a. Coupon ratesB. Ex ante real interest ratesc. Ex post real interest ratesd. Nominal interest rates


74. (p. 82) The price of a coupon bond is determined by: a. Taking the present value of the bond's final payment and subtracting the coupon paymentsb. Taking the present value of the coupon payments and adding this to the face valuec. Taking the present value of the bond's final paymentD. Taking the present value of all of the bond's payments

AACSB: AnalyticBLOOMS: ComprehensionBLOOMS: KnowledgeLOD: 2

75. (p. 82) The price of a coupon bond is determined by: a. Taking the present value of the bond's final payment and subtracting the coupon paymentsb. Taking the present value of the coupon payments and adding this to the face valueC. Taking the present value of all of the bond's paymentsd. Estimating its future value


76. (p. 74) Compounding refers to: a. The calculation of after tax interest returnsb. The internal rate of return a firm earns on an investmentc. The real interest return after taxesD. The process of earning interest on both the principal and the interest of an investment


4-22


77. (p. 82) The interest rate used to discount the promised payment from a bond: a. Will vary directly with the value of the bondb. Should be the one that makes the value equal to the par value of the bondC. Will vary inversely with the value of the bondd. Should always be greater than the coupon rate


78. (p. 83) A credit card that charges a monthly interest rate of 1.5% has an effective annual interest rate of: a. 18.0%B. 19.6%c. 15.0%d. 17.50%


79. (p. 84) Which formula below best expresses the real interest rate, (r)? a. i = r - πeb. r = i + πeC. r = i - πed. πe = i + r


80. (p. 84) A borrower who makes a $1000 loan for one year and earns interest in the amount of $75, earns what nominal interest rate and what real interest rate if inflation is two percent? a. A nominal rate of 5.5% and a real rate of 2.0%b. A nominal rate of 7.5% and a real rate of 5.0%c. A nominal rate of 7.5% and a real rate of 9.5%D. A nominal rate of 7.5% and a real rate of 5.5%


4-23


81. (p. 84) As inflation increases, for any fixed nominal interest rate, the real interest rate: a. Also increasesb. Remains the same, that's why it is realC. Decreasesd. Decreases by less than the increase in inflation


82. (p. 84) Considering the data on real and nominal interest rates for the U.S. from 1979 to 2006, which of the following statements is most accurate? a. The real interest rate remains unchanged over timeB. There have been times when the real interest rate has been negativec. Nominal interest rates higher in 2000 than they had been at any other point in timed. The inflation rate is always greater than the real interest rate


83. (p. 86) Which of the following statements is most correct? A. We can always compute the ex post real interest rate but not the ex ante real rateb. We cannot compute either the ex post or ex ante real interest rates accuratelyc. We can accurately compute the ex ante real interest rate but not the ex post real rated. None of the above statements is correct


84. (p. 84) From the Fisher equation we see that the nominal interest rate and expected inflation have: a. An inverse relationshipb. A relationship which is direct but less than one-to-oneC. A relationship which is direct and one-to-oned. No relationship


4-24


85. (p. 84) High rates of inflation are usually associated with: a. Low nominal interest rates but high real interest ratesB. High nominal interest rates and positive real interest ratesc. Low nominal interest rates and low real interest ratesd. High nominal interest rates and negative real rates


86. (p. 84) If a lender wants to earn a real interest rate of 3% and expects inflation to be 3%, he/she should charge a nominal interest rate that: a. Is at least 7%b. Is anything above 0%C. Equals the real rate desired plus expected inflationd. Equals the real rate desired less expected inflation


87. (p. 85) A borrower is offered a choice between a fixed rate mortgage and a variable rate mortgage. The fixed rate mortgage may be more attractive if the borrower expects: a. Inflation to decreaseB. Inflation to increasec. The home price to increased. The home price to decrease

AACSB: Reflective ThinkingBLOOMS: EvaluationLOD: 2

4-25


88. (p. 85) A borrower is offered a choice between a fixed rate mortgage and a variable rate mortgage. The variable rate mortgage may be more attractive to the lender if the lender expects: a. Inflation to decreaseb. The home price to decreasec. The home price to increaseD. Inflation to increase


89. (p. 85) We should expect a country that experiences volatile inflation to also have: A. Volatile nominal interest ratesb. Volatile real interest rates but stable nominal ratesc. Stable nominal interest ratesd. Volatile real interest rates


Short Answer Question

90. (p. 84) A lender expects to earn a real interest rate of 4.5% over the next 12 months. She charges a 9.25% (annual) nominal rate for a 12-month loan. What inflation rate is she expecting? If the lender is in a 30% marginal tax bracket and the borrower is in a 25% marginal tax bracket, what are the real after-tax rates each expects?

For the first part she expected an inflation rate of 4.75%. We obtain this answer using the Fisher equation where i = r + πe. For the second part we need to use a variation of the Fisher equation. The lender receives an after-tax nominal rate of 6.475% from which we subtract the inflation rate of 4.75% and the lender expects a real after-tax rate of 1.725%. The borrower expects to pay an after-tax real rate of 2.188%.


4-26


91. (p. 81) Compute the interest rate for a $1,000 face value a bond that sells for $280 and matures in 20 years. The bond has no coupon payments, only the face value payment.

Using a financial calculator and inserting $280 for the present value, $1,000 for the future values, 20 for n, and solving for i, we can compute this to be 6.57%.


92. (p. 68) Compute the future value of $1,000 at a 6 percent interest rate after three different lengths of time. Use 6, 10 and 20 years into the future.

We can use a calculator and the formula FV = PV(1+i)nto solve this problem. To calculate the future value for six years the formula will be: FV = $1000(1.06)6which equals $1418.52. Using a similar approach for 10 years: FV = $1000(1.06)10which equals $1790.85. And finally for 20 years: FV = $1000(1.06)20which equals $3207.14.


93. (p. 67) Considering the concept of compounding, explain why in determining the future value of a $100 investment at 5 percent annual interest, you can't simply multiply $100 by (1.10) and get the correct answer.

To simply multiply $100 by 1.10 ignores the effect of compounding which is interest paid on the principal and on the interest earned. That is why the correct formula would be FV = $100(1.05)2.


4-27


94. (p. 76) Calculate which has a higher present value: an annual payment of $100 received over 3 years or an annual payment of $50 received over 7 years. In both cases the interest rate is 7% (or 0.07).

We can use the present value formula to answer this question. In the case of the $100 payment, the present value = $262.43. In the case of the $50 payments received over 7 years, the present value is $269.46. So the 7 payments of $50 each have a higher present value.


95. (p. 68) What is the monthly interest rate if you are asked to convert a 12 percent annual rate to a monthly rate (calculate to 4 decimal places)?

It is not as simple as dividing 12 percent by 12 and thus obtaining an answer of 1.000 percent. The monthly rate, imcan be determined by using the following formula: (1 +im)12= (1.12) which we can manipulate to (1 + im) = (1.12)1/12which equals 1.0095. Therefore the monthly interest rate is 0.95%.


96. (p. 69) Convert each of the following basis points amounts to percents: a) 412.5b) 10c) 125.7d) 1075e) 1

Since 1 basis point equals .01 percent, we can determine that:a) 412.5 basis points is 4.125%b) 10 basis points is 0.1%c) 125.7 basis points is 1.257%d) 1075 basis points is 10.75%e) 1 basis point is 0.01%


4-28


97. (p. 69) Using the rule of 72, determine the approximate time it will take $1000 to double given the following interest rates. a) 5.5%b) 10.0%c) 30.0%d) 2.0%e) 4.5%

Since the rule of 72 says if we take 72/i we get the approximate number of years it takes for an amount to double, we can determine the answer for each interest rate.a) 72/5.5 = 13.1 yearsb) 72/10 = 7.2 yearsc) 72/30 = 2.4 yearsd) 72/2 = 36 yearse) 72/4.5 = 16 years


98. (p. 83) What will be the amount owed at the end of one year if a borrower charges $100 on his/her credit card and doesn't make any payments during the year (assume the interest rate is 1.5% per month)?

$119.56. While it is tempting to multiply 1.5 times 12, obtaining 18% and the multiplying this by $100 to determine the interest charge, it would be incorrect since we would be ignoring compounding. The correct answer can be determined by using the following: FV = PV(1 + im)12. This will be FV = $100(1.015)12or $119.56.


4-29


99. (p. 68) Which investment plan will provide the highest future value: $500 invested at 5 percent annually for four years and then that balance invested at 7 percent annually for an additional three years, or $500 invested at 6 percent annually for seven years?

$500 invested for four years at 5 percent interest and then that balance invested at 7% for three additional years will produce a balance of $744.52 at the end of seven years. The future value of $500 invested for seven years at 6 percent interest is $751.82.


4-30


100. (p. 70) Suppose that you have a winning lottery ticket for $100,000. The State of California doesn't pay this amount up front - this is the amount you will receive over time. The State offers you two options. The first pays you $80,000 up front and that will be the entire amount. The second pays you winnings over a three year period. The last option pays you a large payment today with small payments in the future. The payment options are detailed in the table below:

Compute the present value of each payment option, assuming the interest rate is 12%. Now, compute the present values based on an interest rate of 5%. Compare your answers, explaining why they are different when the interest rate changes.

4-31


When the interest rate is 5%, the present values are as follows:

When the interest rate is 12%, the present values are as follows:

From the computations above, when the interest rate is 5%, Option #3 has the highest present value. When the interest rate is 12%, Option #1 has the highest present value. When the interest rate increases from 5% to 12%, the opportunity cost of foregoing future payments is higher. That is, while the winner is waiting to receive his/her future payments, he/she is forgoing interest that could be earned on a bank deposit or other investment. When the interest rate is low, this opportunity cost is relatively low, making Option #3 (with larger fixed payments similar to coupon payments on a bond) more attractive. When the interest rate is relatively high, these future fixed payments have less value, making Option #1 more attractive.


4-32


101. (p. 72) Briefly discuss the relationship between present value and each of the following: a) future valueb) timec) interest rate

Holding time and interest rate constant, any percentage change in the future value will cause the same percentage change in the present value. Holding the future value and the interest rate constant, and increase in the time until payment reduces the present value and any decrease in time increases the present value. Holding future value and time constant, an increase in the interest rate reduces the present value and a decrease in the interest rate increases the present value.


102. (p. 74) An investment grows from $2,000 to $2,750 over the period of 10 years. What average annual growth rate will produce this result?

First we determine the overall percentage change in the investment is 37.5%, [(2750-20000/2000] x 100 = 37.5. Next, we ask what annual growth rate over 10 years produces this result? We can determine this by using the following: (1+i)10= (1.375); which with a little manipulation turns into: i = (1.375)1/10-1; which says i = .03236, or an annual growth rate of 3.24% produces this result. Notice this is different than the answer you would obtain by simply dividing 37.5% by 10.


103. (p. 76) Calculate the internal rate of return for a machine that costs $500,000 and provides annual revenue of $115,000 per year for 5 years. You can assume all revenue is received once a year at the end of the year.

To solve this we equate the cost of the machine to the sum of the present value for each annual payment and solve for the interest rate. Using a financial calculator or a spreadsheet we obtain an internal rate of return of 4.85%.


4-33


104. (p. 76) You win your state lottery. The lottery officials offer you the following options: you can accept annual payments of $50,000 for 20 years or receive an upfront payment of $700,000. Ignoring issues like mortality tables, taxes, etc., what market interest rate would make it more attractive to take the upfront payment?

Using a financial calculator or a spreadsheet we can equate the $700,000 to the sum of the present value flow of receiving $50,000 a year for the next 20 years, and the internal rate of return is 3.67%. If you are confident that you can earn an average annual return greater than 3.67% a year over the next 20 years, the upfront payment may be the option to select.


105. (p. 76) You are considering purchasing a home. You find one that you like but you realize that you will need to obtain a mortgage for $100,000. The mortgage company presents you with two options: a 15-year mortgage at a 6.0% annual rate and a 30-year mortgage at a 6.5% annual rate. What will be the fixed annual payment for each mortgage?

Using a financial calculator or a spreadsheet we can determine the 15-year mortgage will require annual payments of $10,296.28; the 30-year mortgage will require annual payments in the amount of $7,657.74.


106. (p. 81) A bond offers a $50 coupon, has a face value of $1,000, and has 10 years to maturity. If the interest rate is 4.0% what is the value of this bond?

Realizing that the price of the bond is the sum of the present value of all payments we simply calculate the present value of each payment and sum these. With the help of a financial calculator or a spreadsheet if necessary, we see the value of the bond is $1,081.10.


4-34


107. (p. 81) A bond offers a $40 coupon, has a face value of $1000, and 10 years to maturity. If the interest rate is 5.0%, what is the value of this bond?

Realizing that the price of the bond is the sum of the present value of all payments we simply calculate the present value of each payment and sum these. With the help of a financial calculator or spreadsheet if necessary, we see the value of the bond is $922.78.


108. (p. 82) Describe the effects on the value of a bond from the following: length of time to maturity and interest rates (you can ignore the relationship of the coupon rate to market interest rates to simplify the analysis).

Holding everything else constant, like coupon payments, as the length of time until maturity increases, so will the value of the bond and vice versa. Also, holding everything else constant, such as maturity, as interest rates increase the value of the bond decreases and vice versa.


4-35


109. (p. 82) Suppose a two-year coupon bond has payments of $40 and a face value of $800. The interest rate is 8%. Compute the present value of the coupon payments and the principal payment of the bond. What is the price of this bond?

The present values are:

The price of the bond is equal to the present value of future payments on the bond. The future payments include the $40 coupon payments paid over two years (with a present value of $71.33) and the $800 face value payment (with a present value of $685.87). The price of the bond is, therefore, the sum of these two amounts, $757.20.


110. (p. 84) Suppose you negotiate a one-year loan with a principal of $1000 and the nominal interest rate is currently 7%. You expect the inflation rate to be 3% over the next year. When you repay the principal plus interest at the end of the year, the actual inflation rate is 2.5%. Compute the ex ante and ex post real interest rate. Who benefits from this unexpected decrease in inflation? Who loses?

The ex ante real interest rate is 4% (=7% - 3%). The ex post real interest rate is 4.5% (=7% - 2.5%). The unexpected decrease in inflation benefits the lender because he/she receives a higher real interest rate than what was expected. The borrower loses because his/her real interest rate is higher than expected.


4-36


111. (p. 85) In the data, we observe that countries with high inflation rates tend to have high nominal interest rates. What does this imply, if anything, about real interest rates in countries with very high inflation rates?

The higher nominal interest rates are simply a reflection of high inflation rates. The real interest rates in these countries could be equal to (or even less than) those in low-inflation countries.


112. (p. 84) Explain why an increase in expected inflation will result in an increase in nominal interest rates, holding other factors constant.

This follows from the Fisher equation that says the nominal interest rate equals the sum of the real interest rate and the expected rate of inflation. So, for any given real interest rate, an increase in the expected rate of inflation will cause the nominal interest rate to increase.


113. (p. 85) Explain why, if real interest rates are so important, we see most interest rates quoted in nominal terms.

It is almost impossible to quote real interest rates ex ante. For any given nominal interest rate, the real interest rate is the nominal interest rate less the rate of inflation. The problem is no one knows what the rate of inflation will be exactly. As a result it is easier to quote nominal interest rates.


4-37


114. (p. 85) If a borrower and a lender agree on a long-term loan at a nominal interest rate that is fixed over the duration of the loan, how will a higher-than-expected rate of inflation impact the parties if at all?

A higher-than-expected rate of inflation will benefit the borrower who will end up paying a lower real interest rate than planned, and so will be better off. The lender, on the other hand, will end up receiving a real interest rate that is less than what was planned so the lender will be harmed.


115. (p. 85) Explain why countries with high and volatile inflation rates are likely to have volatile nominal interest rates.

Using the Fisher equation (that says the nominal interest rate equals the sum of the real interest rate and the expected rate of inflation), a country where inflation is volatile will have lenders adding a high expected inflation component, thus raising the nominal interest rate. The higher volatility of nominal interest rates is directly the result of the volatility in the inflation rate.


4-38


116. (p. 84) Explain why the Fisher equation is not highly accurate at high rates of inflation. Use an example.

Consider a lender who loans $100 for a year, in an environment of 10% inflation. If the lender wants to earn a real interest rate of 2%, the Fisher equation says he/she should charge a nominal interest rate of 12.0%. The reality is, however, that the lender wanted to have 2% more purchasing power at the end of the loan. Since inflation also impacts the interest earned, we can calculate the actual interest rate he/she needs to charge by realizing that if the lender wanted $2.00 more purchasing power per hundred dollars loaned, we can take $102 and multiply this by 1 + the rate of inflation or 1.1. $102x 1.1 = $112.20. Thus he/she really needs to charge a nominal interest rate of 12.2% or slightly more than the 12.0% of the Fisher equation.


4-39


117. (p. 78) An individual is currently 30 years old, wants to work until the age of 65 and plans on dying at the age of 85. How much will the individual need to have saved by the time he or she is 65 if he or she plans on spending $40,000 per year while retired? You can assume the individual can earn an interest rate of 5.0% and the $40,000 is in addition to any Social Security that may be received.

We can use a financial calculator to determine that in order to determine that the individual will need to amass a fund of $498,488 at the time he/she plans on retiring to obtain $40,000 a year for 20 years. Now since the individual has 35 years to amass this fund, this will require him/her to set aside $5,519 each year for 35 years.

AACSB: Reflective ThinkingBLOOMS: SynthesisLOD: 3

Essay Questions

118. (p. 65) Explain why an investor cannot simply compare the size of promised payments from different investments, even if the interest rates and other risk factors are the same.

The key here is time. Payments that are promised at different times are not equal in value; we could say they are really different units of value. We employ the concept of present value to allow us to make comparisons of promised payments that are due at different time periods. We know that payments that are promised sooner are worth more, other factors held constant (for example interest rates), than payments we have to wait for longer. This is seen from the present value formula. So a saver who is going to make a thorough comparison of different investments must consider the timing of the payments and convert all future payments to present value amounts so they can be compared in the same units.


4-40


119. (p. 71) Historically, many cultural groups have outlawed usury, or the practice of levying interest on loans. Some groups oppose usury because it exacerbates problems of income inequality (as wealthier individuals can afford to lend to poorer individuals), while others claim investment and loans should be made charitably. Evaluate these arguments against usury based on your knowledge of present value. Do such prohibitions make sense?

Prohibitions on interest payments (or usury) are problematic when we apply the concept of opportunity cost. For every dollar that is lent, the lender gives up the use of these funds that could go elsewhere. For example, the funds could be used for consumption, or for earning return on some other investment (such as a bank deposit or bond). If usury is outlawed, then there is no incentive to lend, outside of the perceived benefit the lender receives from charitable contributions to his/her colleagues.

AACSB: Reflective ThinkingBLOOMS: AnalysisBLOOMS: EvaluationLOD: 3

120. (p. 71) How has Islamic banking redefined lending to deal with Islam's prohibition of usury?

Islamic banking has found alternative mechanisms for encouraging the flow of funds from savers to borrowers through banks that pay no interest on deposits, or loans. This provides savers with access to their liquid assets, while capturing the lower transactions costs and risk sharing associated with depository institutions and other financial intermediaries. On the lender side, the bank is entitled to a share of the gains the borrower generates from the loan (e.g., from investing in capital or some other physical asset), OR purchases goods on behalf of the borrower. Since the bank benefits from economies of scale, it is able to generate profits by negotiating lower prices (or lower per-unit cost) than an individual could.


4-41


121. (p. 78) Discussions in recent years about the vulnerability of the Social Security System cause some people to feel the payments promised will not materialize. Discuss the possible changes we might observe now.

If people working now begin to question the viability of Social Security and yet if they want to retire at the planned age and keep their lifestyle during retirement, they will have to increase saving now. The idea is that people will need to build a larger fund at the time of retirement and to do this will require they decrease their current consumption. If people do not alter their saving, they either believe that Social Security will honor their payments and/or they plan on reducing their consumption during retirement.


122. (p. 84) During the early 1980s, the U.S. economy experienced an increase in interest rates quoted on U.S. Treasury debt, business loans, and mortgages. At the same time the inflation rate gradually declined more than expected. What happened to ex ante versus ex post real interest rates during this period? Use the Fisher equation to support your answer.

The Fisher equation is: i = r + πe

The Fisher equation can be used to compute the ex ante real interest rate. The ex post real interest rate is computed using actual inflation in place of expected inflation. If nominal interest rates increase and the inflation rate decreased, this implies the ex post real interest rate must have decreased. If inflation declined more than expected, this would imply that the ex post real interest rate exceeded the ex ante real interest rate.


4-42


123. (p. 85) Explain why countries that have volatile inflation rates are likely to have high nominal interest rates.

You could argue that volatile inflation means the inflation rate changes, but it doesn't always mean it increases. The rate could also decrease, and then the average rate may not be that bad. So why is the nominal interest rate higher? The answer can be found in the positions of the party and counterparty to any agreement. For example, in a country where inflation is low, a change of 1 percentage point, say from 1% to 2% can benefit one party and harm the other party, but the harm/benefit is somewhat minimal. In a country where inflation may average 4% (for example), but is highly volatile, the volatility can cause the rate to change by a larger amount (more percentage points), meaning the potential harm can be much larger. To compensate for this risk, the nominal interest rates will have to be higher.


124. (p. 78) Explain the suggestion that people may have their own "personal discount rate" and how that may affect decisions about borrowing and other financial matters.

A good illustration of this comes from the story of the downsizing by the Defense Department in the 1990s. Military personnel were offered a choice between an annual payment and a lump sum, and were given information about how to calculate the present value of the annual payment using a 7 percent discount rate. The evidence suggests that most people put excessive weight on a "bird in the hand," meaning the "sure thing" of the lump-sum payment, suggesting that for most people their "personal discount rate" is higher. For the military personnel, it seemed to be much higher than 7 percent. This explains why people are more likely to choose lump-sum payments and to borrow at high rates of interest (for example, the rates on credit card balances). Most people seem to be extremely impatient, even to their own financial detriment.


4-43


125. (p. 83) What matters more: having a credit card with a low rate or paying off your balance as quickly as possible? Explain.

Whatever the rate of interest on your credit card, the faster you pay off the balance the better. For example, if you had a balance of $2,000 on a credit card with a 10% interest rate, paying $50 a month will mean it will take you more than 4 years to pay off the principal (and the interest accumulated on it along the way). If you made payments of $75 instead, it would take you only about 30 months to pay off the balance. If the credit card interest rate were twice as high (20%), paying off $2,000 in $50 a month increments will mean it takes over 5 years to achieve a zero balance. But even at that much higher rate, paying $75 a month means it takes about 34 months to pay it off. Therefore, whatever the credit card rate, paying it off sooner is better than paying it off more slowly.


4-44

chapter 04 future value, present value and interest...

Documents