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Challenge Problems

64. Real versus Nominal Dollars. An engineer in 1950 was earning $6,000 a year. Today she earns $60,000 a year. However, on average, goods today cost 6 times what they did in 1950. What is her real income today in terms of constant 1950 dollars?

65. Real versus Nominal Rates. If investors are to earn a 4 percent real interest rate, what nominal interest rate must they earn if the inflation rate is:

a. zero

b. 4 percent

c. 6 percent

66. Real Rates. If investors receive an 8 percent interest rate on their bank deposits, what real interest rate will they earn if the inflation rate over the year is:

a. zero

b. 3 percent

c. 6 percent

67. Real versus Nominal Rates. You will receive $100 from a savings bond in 3 years. The nominal interest rate is 8 percent.

a. What is the present value of the proceeds from the bond?

b. If the inflation rate over the next few years is expected to be 3 percent, what will the real value of the $100 payoff be in terms of today`s dollars?

c. What is the real interest rate?

d. Show that the real payoff from the bond (from part b) discounted at the real interest rate (from part c) gives the same present value for the bond as you found in part a.

68. Real versus Nominal Dollars. Your consulting firm will produce cash flows of $100,000 this year, and you expect cash flow to keep pace with any increase in the general level of prices. The interest rate currently is 8 percent, and you anticipate inflation of about 2 percent.

a. What is the present value of your firm`s cash flows for Years 1 through 5?

b. How would your answer to (a) change if you anticipated no growth in cash flow?

69. Real versus Nominal Annuities. Good news: you will almost certainly be a millionaire by

the time you retire in 50 years. Bad news: the inflation rate over your lifetime will average about 3 percent.

a. What will be the real value of $1 million by the time you retire in terms of today`s dollars?

b. What real annuity (in today`s dollars) will $1 million support if the real interest rate at retirement is 2 percent and the annuity must last for 20 years?

70. Rule of 72. Using the Rule of 72, if the interest rate is 8 percent per year, how long will it take for your money to quadruple in value?

71. Inflation. Inflation in Brazil in 1992 averaged about 23 percent per month. What was the annual inflation rate?

72. Perpetuities. British government 4 percent perpetuities pay Вг4 interest each year forever. Another bond, 21вБД2 percent perpetuities, pays Вг2.50 a year forever. What is the value of 4 percent perpetuities, if the long-term interest rate is 6 percent? What is the value of 21вБД2 percent perpetuities?

73. Real versus Nominal Annuities.

a. You plan to retire in 30 years and want to accumulate enough by then to provide yourself with $30,000 a year for 15 years. If the interest rate is 10 percent, how much must you accumulate by the time you retire?

b. How much must you save each year until retirement in order to finance your retirement consumption?

c. Now you remember that the annual inflation rate is 4 percent. If a loaf of bread costs $1.00 today, what will it cost by the time you retire?

d. You really want to consume $30,000 a year in real dollars during retirement and wish to save an equal real amount each year until then. What is the real amount of savings that you need to accumulate by the time you retire?

e. Calculate the required preretirement real annual savings necessary to meet your consumption goals. Compare to your answer to (b). Why is there a difference?

f. What is the nominal value of the amount you need to save during the first year? (Assume the savings are put aside at the end of each year.) The thirtieth year?

74. Retirement and Inflation. Redo part (a) of problem 63, but now assume that the inflation rate over the next 50 years will average 4 percent.

a. What is the real annual savings the couple must set aside?

b. How much do they need to save in nominal terms in the first year?

c. How much do they need to save in nominal terms in the last year?

d. What will be their nominal expenditures in the first year of retirement? The last?

75. Annuity Value. What is the value of a perpetuity that pays $100 every 3 months forever? The discount rate quoted on an APR basis is 12 percent.

76. Changing Interest Rates. If the interest rate this year is 8 percent and the interest rate nex tyear will be 10 percent, what is the future value of $1 after 2 years? What is the present value of a payment of $1 to be received in 2 years?

77. Changing Interest Rates. Your wealthy uncle established a $1,000 bank account for you when you were born. For the first 8 years of your life, the interest rate earned on the account was 8 percent. Since then, rates have been only 6 percent. Now you are 21 years old and ready to cash in. How much is in your account?