Chapter 5 Introduction to Valuation: The Time Value of Money
1.
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You are investing $100 today in a savings account at your local bank. Which one of the following terms refers to the value of this investment one year from now?
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2.
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Tracy invested $1,000 five years ago and earns 4 percent interest on her investment. By leaving her interest earnings in her account, she increases the amount of interest she earns each year. The way she is handling her interest income is referred to as which one of the following?
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3.
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Steve invested $100 two years ago at 10 percent interest. The first year, he earned $10 interest on his $100 investment. He reinvested the $10. The second year, he earned $11 interest on his $110 investment. The extra $1 he earned in interest the second year is referred to as:
E.
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present value interest.
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4.
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Interest earned on both the initial principal and the interest reinvested from prior periods is called:
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5.
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Sara invested $500 six years ago at 5 percent interest. She spends her earnings as soon as she earns any interest so she only receives interest on her initial $500 investment. Which type of interest is Sara earning?
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6.
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Shelley won a lottery and will receive $1,000 a year for the next ten years. The value of her winnings today discounted at her discount rate is called which one of the following?
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7.
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Terry is calculating the present value of a bonus he will receive next year. The process he is using is called:
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8.
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Steve just computed the present value of a $10,000 bonus he will receive in the future. The interest rate he used in this process is referred to as which one of the following?
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9.
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The process of determining the present value of future cash flows in order to know their worth today is called which one of the following?
A.
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compound interest valuation
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B.
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interest on interest computation
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C.
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discounted cash flow valuation
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D.
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present value interest factoring
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10.
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Andy deposited $3,000 this morning into an account that pays 5 percent interest, compounded annually. Barb also deposited $3,000 this morning into an account that pays 5 percent interest, compounded annually. Andy will withdraw his interest earnings and spend it as soon as possible. Barb will reinvest her interest earnings into her account. Given this, which one of the following statements is true?
A.
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Barb will earn more interest the first year than Andy will.
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B.
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Andy will earn more interest in year three than Barb will.
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C.
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Barb will earn interest on interest.
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D.
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After five years, Andy and Barb will both have earned the same amount of interest.
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E.
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Andy will earn compound interest.
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11.
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Sue and Neal are twins. Sue invests $5,000 at 7 percent when she is 25 years old. Neal invests $5,000 at 7 percent when he is 30 years old. Both investments compound interest annually. Both Sue and Neal retire at age 60. Which one of the following statements is correct assuming that neither Sue nor Neal has withdrawn any money from their accounts?
A.
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Sue will have less money when she retires than Neal.
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B.
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Neal will earn more interest on interest than Sue.
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C.
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Neal will earn more compound interest than Sue.
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D.
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If both Sue and Neal wait to age 70 to retire, then they will have equal amounts of savings.
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E.
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Sue will have more money than Neal as long as they retire at the same time.
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12.
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Samantha opened a savings account this morning. Her money will earn 5 percent interest, compounded annually. After five years, her savings account will be worth $5,600. Assume she will not make any withdrawals. Given this, which one of the following statements is true?
A.
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Samantha deposited more than $5,600 this morning.
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B.
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The present value of Samantha's account is $5,600.
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C.
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Samantha could have deposited less money and still had $5,600 in five years if she could have earned 5.5 percent interest.
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D.
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Samantha would have had to deposit more money to have $5,600 in five years if she could have earned 6 percent interest.
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E.
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Samantha will earn an equal amount of interest every year for the next five years.
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13.
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This afternoon, you deposited $1,000 into a retirement savings account. The account will compound interest at 6 percent annually. You will not withdraw any principal or interest until you retire in forty years. Which one of the following statements is correct?
A.
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The interest you earn six years from now will equal the interest you earn ten years from now.
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B.
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The interest amount you earn will double in value every year.
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C.
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The total amount of interest you will earn will equal $1,000 × .06 × 40.
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D.
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The present value of this investment is equal to $1,000.
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E.
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The future value of this amount is equal to $1,000 × (1 + 40).06.
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14.
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Your grandmother has promised to give you $5,000 when you graduate from college. She is expecting you to graduate two years from now. What happens to the present value of this gift if you delay your graduation by one year and graduate three years from now?
E.
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cannot be determined from the information provided
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15.
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Luis is going to receive $20,000 six years from now. Soo Lee is going to receive $20,000 nine years from now. Which one of the following statements is correct if both Luis and Soo Lee apply a 7 percent discount rate to these amounts?
A.
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The present values of Luis and Soo Lee's monies are equal.
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B.
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In future dollars, Soo Lee's money is worth more than Luis' money.
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C.
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In today's dollars, Luis' money is worth more than Soo Lee's.
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D.
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Twenty years from now, the value of Luis' money will be equal to the value of Soo Lee's money.
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E.
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Soo Lee's money is worth more than Luis' money given the 7 percent discount rate.
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16.
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Which one of the following variables is the exponent in the present value formula?
E.
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There is no exponent in the present value formula.
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17.
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You want to have $1 million in your savings account when you retire. You plan on investing a single lump sum today to fund this goal. You are planning on investing in an account which will pay 7.5 percent annual interest. Which of the following will reduce the amount that you must deposit today if you are to have your desired $1 million on the day you retire?
I. Invest in a different account paying a higher rate of interest.
II. Invest in a different account paying a lower rate of interest.
III. Retire later.
IV. Retire sooner.
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18.
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Which one of the following will produce the highest present value interest factor?
A.
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6 percent interest for five years
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B.
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6 percent interest for eight years
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C.
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6 percent interest for ten years
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D.
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8 percent interest for five years
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E.
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8 percent interest for ten years
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19.
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What is the relationship between present value and future value interest factors?
A.
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The present value and future value factors are equal to each other.
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B.
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The present value factor is the exponent of the future value factor.
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C.
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The future value factor is the exponent of the present value factor.
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D.
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The factors are reciprocals of each other.
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E.
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There is no relationship between these two factors.
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20.
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Martin invested $1,000 six years ago and expected to have $1,500 today. He has not added or withdrawn any money from this account since his initial investment. All interest was reinvested in the account. As it turns out, Martin only has $1,420 in his account today. Which one of the following must be true?
A.
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Martin earned simple interest rather than compound interest.
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B.
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Martin earned a lower interest rate than he expected.
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C.
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Martin did not earn any interest on interest as he expected.
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D.
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Martin ignored the Rule of 72 which caused his account to decrease in value.
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E.
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The future value interest factor turned out to be higher than Martin expected.
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21.
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Gerold invested $5,600 in an account that pays 5 percent simple interest. How much money will he have at the end of ten years?
Ending value = $5,600 + ($5,600 × .05 × 10) = $8,400
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22.
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Alex invested $10,500 in an account that pays 6 percent simple interest. How much money will he have at the end of four years?
Ending value = $10,500 + ($10,500 × .06 × 4) = $13,020
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23.
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You invested $1,400 in an account that pays 5 percent simple interest. How much more could you have earned over a 20-year period if the interest had compounded annually?
Simple interest = $1,400 + ($1,400 × .05 × 20) = $2,800
Annual compounding = $1,400 × (1.05)20 = $3,714.62
Difference = $3,714.62 - $2,800 = $914.62
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24.
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Travis invested $9,250 in an account that pays 6 percent simple interest. How much more could he have earned over a 7-year period if the interest had compounded annually?
Simple interest = $9,250 + ($9,250 × .06 × 7) = $13,135
Compound interest = $9,250 × (1 + .06)7 = $13,908.58
Difference = $13,908.58 - $13,135 = $773.58
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25.
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What is the future value of $6,200 invested for 23 years at 9.25 percent compounded annually?
Future value = $6,200 × (1 + .0925)23 = $47,433.47
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26.
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Today, you earn a salary of $36,000. What will be your annual salary twelve years from now if you earn annual raises of 3.6 percent?
Future value = $36,000 × (1 + .036)12 = $55,032.54
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27.
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You own a classic automobile that is currently valued at $150,000. If the value increases by 6.5 percent annually, how much will the automobile be worth ten years from now?
Future value = $150,000 × (1 + .065)10 = $281,570.62
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28.
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You hope to buy your dream car four years from now. Today, that car costs $82,500. You expect the price to increase by an average of 4.8 percent per year over the next four years. How much will your dream car cost by the time you are ready to buy it?
Future value = $82,500 × (1 + .048)4 = $99,517.41
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29.
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This morning, TL Trucking invested $75,000 to help fund a company expansion project planned for 4 years from now. How much additional money will the firm have 4 years from now if it can earn 5 percent rather than 4 percent on its savings?
Future value = $75,000 × (1 + .05)4 = $91,162.97
Future value = $75,000 × (1 + .04)4 = $87,739.39
Difference = $91,162.97 - $87,739.39 = $3,423.58
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30.
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You just received $225,000 from an insurance settlement. You have decided to set this money aside and invest it for your retirement. Currently, your goal is to retire 25 years from today. How much more will you have in your account on the day you retire if you can earn an average return of 10.5 percent rather than just 8 percent?
Future value = $225,000 × (1 + .105)25 = $2,730,483
Future value = $225,000 × (1 + .08)25 = $1,540,907
Difference = $2,730,483 - $1,540,907 = $1,189,576
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31.
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You just received a $3,000 gift from your grandmother. You have decided to save this money so that you can gift it to your grandchildren 50 years from now. How much additional money will you have to gift to your grandchildren if you can earn an average of 8.5 percent instead of just 8 percent on your savings?
Future value = $3,000 × (1 + .085)50 = $177,258.95
Future value = $3,000 × (1 + .08)50 = $140,704.84
Difference = $177,258.95 - $140,704.84 = $36,554.11
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32.
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You are depositing $1,500 in a retirement account today and expect to earn an average return of 7.5 percent on this money. How much additional income will you earn if you leave the money invested for 45 years instead of just 40 years?
Future value = $1,500 × (1 + .075)45 = $38,857.26
Future value = $1,500 × (1 + .075)40 = $27,066.36
Difference = $38,857.26 - $27,066.36 = $11,790.90
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33.
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You collect old coins. Today, you have two coins each of which is valued at $300. One coin is expected to increase in value by 6 percent annually while the other coin is expected to increase in value by 4.5 percent annually. What will be the difference in the value of the two coins 15 years from now?
Future value = $300 × (1 + .06)15 = $718.97
Future value = $300 × (1 + .045)15 = $580.58
Difference = $718.97 - $580.58 = $138.38
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34.
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Your father invested a lump sum 26 years ago at 4.25 percent interest. Today, he gave you the proceeds of that investment which totaled $51,480.79. How much did your father originally invest?
Present value = $51,480.79 × [1/(1 + .0425)26] = $17,444.86
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35.
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What is the present value of $150,000 to be received 10 years from today if the discount rate is 11 percent?
Present value = $150,000 × [1/1 + .11)10] = $52,827.67
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36.
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You would like to give your daughter $75,000 towards her college education 17 years from now. How much money must you set aside today for this purpose if you can earn 8 percent on your investments?
Present value = $75,000 × [1/(1 + .08)17] = $20,270.17
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37.
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You want to have $25,000 saved 6 years from now to buy a house. How much less do you have to deposit today to reach this goal if you can earn 5.5 percent rather than 5 percent on your savings? Today's deposit is the only deposit you will make to this savings account.
Present value = $25,000 × [1/(1 + .05)6] = $18,655.38
Present value = $25,000 × [1/(1 + .055)6] = $18,131.15
Difference = $18,655.38 - $18,131.15 = $524.24
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38.
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Your older sister deposited $5,000 today at 8.5 percent interest for 5 years. You would like to have just as much money at the end of the next 5 years as your sister will have. However, you can only earn 7 percent interest. How much more money must you deposit today than your sister did if you are to have the same amount at the end of the 5 years?
Future value = $5,000 × (1 + .085)5 = $7,518.28
Present value = $7,518.28 × [1/(1 + .07)5] = $5,360.43
Difference = $5,360.43 - $5,000 = $360.43
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39.
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A year ago, you deposited $40,000 into a retirement savings account at a fixed rate of 5.5 percent. Today, you could earn a fixed rate of 6.5 percent on a similar type account. However, your rate is fixed and cannot be adjusted. How much less could you have deposited last year if you could have earned a fixed rate of 6.5 percent and still have the same amount as you currently will when you retire 38 years from today?
Future value = $40,000 × (1 + .055)38+1 = $322,779.48
Present value = $322,779.48 × [1/(1 + .065)38+1] = $27,686.70
Difference = $40,000 - $27,686.70 = $12,313.30
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40.
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When you retire 40 years from now, you want to have $1.2 million. You think you can earn an average of 12 percent on your investments. To meet your goal, you are trying to decide whether to deposit a lump sum today, or to wait and deposit a lump sum 2 years from today. How much more will you have to deposit as a lump sum if you wait for 2 years before making the deposit?
Present value = $1,200,000 × [1/(1 + .12)40] = $12,896.16
Present value = $1,200,000 × [1/(1 + .12)38] = $16,176.94
Difference = $16,176.94 - $12,896.16 = $3,280.78
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41.
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Theo needs $40,000 as a down payment for a house 6 years from now. He earns 2.5 percent on his savings. Theo can either deposit one lump sum today for this purpose or he can wait a year and deposit a lump sum. How much additional money must he deposit if he waits for one year rather than making the deposit today?
Present value = $40,000 × [1/(1 + .025)6] = $34,491.87
Present value = $26,000 × [1/(1 + .025)5] = $35,354.17
Difference = $35,354.17 - $34,491.87 = $862.30
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42.
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One year ago, you invested $1,800. Today it is worth $1,924.62. What rate of interest did you earn?
$1,924.62 = $1,800 × (1 + r)1; r = 6.92 percent
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43.
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According to the Rule of 72, you can do which one of the following?
A.
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double your money in five years at 7.2 percent interest
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B.
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double your money in 7.2 years at 8 percent interest
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C.
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double your money in 5 years at 14.4 percent interest
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D.
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triple your money in 7.2 years at 5 percent interest
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E.
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triple your money at 10 percent interest in 7.2 years
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Rule of 72 = 72/5 years = 14.4 percent interest
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44.
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Forty years ago, your mother invested $5,000. Today, that investment is worth $430,065.11. What is the average annual rate of return she earned on this investment?
$430,065.11 = $5,000 × (1 + r)40; r = 11.78 percent
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45.
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Sixteen years ago, Alicia invested $500. Eight years ago, Travis invested $900. Today, both Alicia's and Travis' investments are each worth $2,400. Assume that both Alicia and Travis continue to earn their respective rates of return. Which one of the following statements is correct concerning these investments?
A.
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Three years from today, Travis' investment will be worth more than Alicia's.
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B.
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One year ago, Alicia's investment was worth more than Travis' investment.
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C.
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Travis earns a higher rate of return than Alicia.
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D.
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Travis has earned an average annual interest rate of 3.37 percent.
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E.
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Alicia has earned an average annual interest rate of 6.01 percent.
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Alicia: $2,400 = $500 × (1 + r)16; r = 10.30 percent
Travis: $2,400 = $900 × (1 + r)8; r = 13.04 percent
Since both Alicia and Travis have equal account values today and since Travis earns the higher rate of return, his account had to be worth less than Alicia's account one year ago.
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46.
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Penn Station is saving money to build a new loading platform. Two years ago, they set aside $24,000 for this purpose. Today, that account is worth $28,399. What rate of interest is Penn Station earning on this investment?
$28,399 = $24,000 × (1 + r)2; r = 8.78 percent
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47.
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Ten years ago, Jackson Supply set aside $130,000 in case of a financial emergency. Today, that account has increased in value to $330,592. What rate of interest is the firm earning on this money?
$330,592 = $130,000 × (1 + r)10; r = 9.78 percent
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48.
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Fourteen years ago, your parents set aside $7,500 to help fund your college education. Today, that fund is valued at $26,180. What rate of interest is being earned on this account?
$26,180 = $7,500 × (1 + r)14; r = 9.34 percent
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49.
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Some time ago, Julie purchased eleven acres of land costing $36,900. Today, that land is valued at $214,800. How long has she owned this land if the price of the land has been increasing at 6 percent per year?
$214,800 = $36,900 × (1 + .06)t; t = 30.23 years
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50.
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On your ninth birthday, you received $300 which you invested at 4.5 percent interest, compounded annually. Your investment is now worth $756. How old are you today?
$756 = $300 × (1 + .045)t; t = 21 years; Age today = 9 + 21 = 30
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51.
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Assume the total cost of a college education will be $300,000 when your child enters college in 16 years. You presently have $75,561 to invest. What rate of interest must you earn on your investment to cover the cost of your child's college education?
$300,000 = $75,561 (1 + r)16; r = 9 percent
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52.
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At 8 percent interest, how long would it take to quadruple your money?
$4 = $1 × (1 + .08)t; t = 18.01 years
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53.
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Assume the average vehicle selling price in the United States last year was $41,996. The average price 9 years earlier was $29,000. What was the annual increase in the selling price over this time period?
$41,996 = $29,000 × (1 + r)9; r = 4.20 percent
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54.
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You're trying to save to buy a new $160,000 Ferrari. You have $58,000 today that can be invested at your bank. The bank pays 6 percent annual interest on its accounts. How many years will it be before you have enough to buy the car? Assume the price of the car remains constant.
$160,000 = $58,000 × (1 + .06)t; t = 17.41 years
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55.
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Imprudential, Inc. has an unfunded pension liability of $850 million that must be paid in 25 years. To assess the value of the firm's stock, financial analysts want to discount this liability back to the present. The relevant discount rate is 6.5 percent. What is the present value of this liability?
PV = $850,000,000 × [1/(1.065)25] = $176,067,311
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56.
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You have just received notification that you have won the $1.4 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday, 78 years from now. The appropriate discount rate is 8 percent. What is the present value of your winnings?
PV = $1,400,000 × [1/(1.08)78] = $3,459.99
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57.
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Your coin collection contains fifty-four 1941 silver dollars. Your grandparents purchased them for their face value when they were new. These coins have appreciated at a 10 percent annual rate. How much will your collection be worth when you retire in 2060?
FV = $54 × (1.10)119 = $4,551,172
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58.
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In 1895, the winner of a competition was paid $150. In 2006, the winner's prize was $70,000. What will the winner's prize be in 2040 if the prize continues increasing at the same rate?
$70,000 = $150 × (1 = r)111; r = 5.6927277 percent
FV = $70,000 × (1 + .056927277)34 = $459,866
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59.
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Suppose that the first comic book of a classic series was sold in 1954. In 2000, the estimated price for this comic book in good condition was about $340,000. This represented a return of 27 percent per year. For this to be true, what was the original price of the comic book in 1954?
PV = $340,000 × [1/(1 + .27)46; PV = $5.71
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60.
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Suppose you are committed to owning a $140,000 Ferrari. You believe your mutual fund can achieve an annual rate of return of 8 percent and you want to buy the car in 7 years. How much must you invest today to fund this purchase assuming the price of the car remains constant?
PV = $140,000 × [1/(1 + .08)7; PV = $81,688.66
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61.
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You have just made a $1,500 contribution to your individual retirement account. Assume you earn a 12 percent rate of return and make no additional contributions. How much more will your account be worth when you retire in 25 years than it would be if you waited another 10 years before making this contribution?
FV = $1,500 × (1 + .12)25 = $25,500.10
FV = $1,500 × (1 + .12)15 = $8,210.35
Difference = $17,289.75
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62.
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You are scheduled to receive $30,000 in two years. When you receive it, you will invest it for 5 more years, at 6 percent per year. How much money will you have 7 years from now?
FV = $30,000 × (1 + .06)(7-2) = $40,146.77
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63.
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You expect to receive $9,000 at graduation in 2 years. You plan on investing this money at 10 percent until you have $60,000. How many years will it be until this occurs?
$60,000 = $9,000 × (1 + .10)t; t = 19.90 years
Total time = 2 + 19.90 = 21.90 years
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