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Chapter 005 Introduction to Valuation: The Time Value of Money Multiple Choice Questions 1. The amount an investment will be worth after one or more periods of time is the _ value. A. future b. present c. principal d. discounted e. simple SECTION: 5.1 TOPIC: FUTURE VALUE TYPE: DEFINITIONS 2. The process of accumulating interest on an investment over time to earn more interest is called: a. growth B. compounding c. aggregation d. accumulation e. discounting SECTION: 5.1 TOPIC: COMPOUNDING TYPE: DEFINITIONS 3. Interest earned on the reinvestment of previous interest payments is called _ interest. a. free b. annual c. simple D. interest on e. compound SECTION: 5.1 TOPIC: INTEREST ON INTEREST TYPE: DEFINITIONS 5-1 Chapter 005 Introduction to Valuation: The Time Value of Money 4. Interest earned on both the initial principal and the interest reinvested from prior periods is called _ interest. a. free b. annual c. simple d. interest on E. compound SECTION: 5.1 TOPIC: COMPOUND INTEREST TYPE: DEFINITIONS 5. Interest earned only on the original principal amount invested is called _ interest. a. free b. annual C. simple d. interest on e. compound SECTION: 5.1 TOPIC: SIMPLE INTEREST TYPE: DEFINITIONS 6. The future value interest factor is computed as: A. (1 + r)t b. (1 + rt) c. (1 + r) t d. 1 + r t e. (1+ r) rt SECTION: 5.1 TOPIC: FUTURE VALUE INTEREST FACTOR TYPE: DEFINITIONS 5-2 Chapter 005 Introduction to Valuation: The Time Value of Money 7. The value today of future cash flows discounted at the appropriate discount rate is called the _ value. a. principal b. future C. present d. simple e. compound SECTION: 5.2 TOPIC: PRESENT VALUE TYPE: DEFINITIONS 8. The process of finding the present value of some future amount is commonly called: a. growth B. discounting c. accumulation d. compounding e. reduction SECTION: 5.2 TOPIC: DISCOUNTING TYPE: DEFINITIONS 9. The present value interest factor is computed as: a. 1 / (1 + r t) b. 1 / (1 + rt) c. 1 / (1 + r) t D. 1 / (1 + r)t e. 1 + r + t SECTION: 5.2 TOPIC: PRESENT VALUE INTEREST FACTOR TYPE: DEFINITIONS 5-3 Chapter 005 Introduction to Valuation: The Time Value of Money 10. The interest rate used to calculate the present value of future cash flows is called the _ rate. a. free b. annual c. compound d. simple E. discount SECTION: 5.2 TOPIC: DISCOUNT RATE TYPE: DEFINITIONS 11. As the discount rate increases, the present value of $2,000 to be received four years from now: a. remains constant b. also increases C. decreases d. becomes negative e. will vary but the direction of the change is unknown SECTION: 5.2 TOPIC: PRESENT VALUE AND DISCOUNT RATE TYPE: CONCEPTS 12. Jeff is going to receive $10,000 five years from now. Tammy is going to receive $10,000 eight years from now. Which one of the following statements is correct if both Tammy and Jeff apply a 6 percent discount rate to these amounts? a. The present value of Tammy's and Jeff's money will be equal b. The value of Jeff's money will be less than the value of Tammy's money 15 years from now C. In today's dollars, Jeff's money is worth more than Tammy's d. Ten years from now, the value of Jeff's money will be equal to the value of Tammy's money e. Tammy's money is worth more than Jeff's money given the 6 percent discount rate SECTION: 5.2 TOPIC: PRESENT VALUE AND TIME TYPE: CONCEPTS 5-4 Chapter 005 Introduction to Valuation: The Time Value of Money 13. Tracie deposits $5,000 into an account that pays 3 percent interest compounded annually. Chris deposits $5,000 into an account that pays 3 percent simple interest. Both deposits were made this morning. Which of the following statements are true concerning these two accounts? I. At the end of one year, both Tracie and Chris will have the same amount in their accounts II. At the end of five years, Tracie will have more money in her account than Chris has in his III. Chris will never earn any interest on interest IV. All else equal, Chris made the better investment. a. I and II only b. III and IV only C. I, II, and III only d. I, III, and IV only e. II, III, and IV only SECTION: 5.1 TOPIC: SIMPLE VERSUS COMPOUND INTEREST TYPE: CONCEPTS 14. Kate invests $3,000 at 9 percent when she is 20 years old. Kurt invests $3,000 at 9 percent when he is 35 years old. Both investments compound interest annually. Both Kate and Kurt retire at age 60. Which one of the following statements is correct? a. Kate will have less money when she retires than Kurt b. Kurt will earn more interest on interest than Kate c. Kurt will earn more compound interest than Kate d. If Kurt waits to age 70 to retire, then he will have just as much money as Kate E. Kate will have more money when she retires than Kurt SECTION: 5.1 TOPIC: FUTURE VALUE AND TIME TYPE: CONCEPTS 5-5 Chapter 005 Introduction to Valuation: The Time Value of Money 15. Raoul has $800 today. Which one of the following statements is correct if he invests this money at a positive rate of interest for ten years? Assume the interest is compounded annually. a. The higher the interest rate he earns, the less money he will have in the future b. The higher the interest rate, the longer he has to wait for his money to grow to $2,000 in value c. If Raoul can earn 7 percent, he will have to wait about six years to have $1,600 total d. At the end of the ten years, Raoul will have less money if he invests at 9 percent rather than at 8 percent E. At 7.2 percent interest, Raoul should expect to have about $1,600 in his account at the end of the ten years SECTION: 5.1 TOPIC: RULE OF 72 TYPE: CONCEPTS 16. Frank invests $2,500 in an account that pays 6 percent simple interest. How much money will he have at the end of four years? a. $2,650 B. $3,100 c. $3,156 d. $3,163 e. $10,600 Ending value = $2,500 + ($2,500 .06 4) = $3,100.00 AACSB TOPIC: ANALYTIC SECTION: 5.1 TOPIC: SIMPLE INTEREST TYPE: PROBLEMS 5-6 Chapter 005 Introduction to Valuation: The Time Value of Money 17. Faith invests $4,500 in an account that pays 4 percent simple interest. How much money will she have at the end of eight years? a. $4,680 b. $5,367 C. $5,940 d. $6,122 e. $6,159 Ending value = $4,500 + ($4,500 .04 8) = $5,940.00 AACSB TOPIC: ANALYTIC SECTION: 5.1 TOPIC: SIMPLE INTEREST TYPE: PROBLEMS 18. Jessica invests $3,000 in an account that pays 5 percent simple interest. How much more could she have earned over a 7year period if the interest had compounded annually? a. $122.20 b. $129.20 c. $147.80 D. $171.30 e. $221.30 Ending value at 5 percent simple interest = $3,000 + ($3,000 .05 7) = $4,050.00; Ending value at 5 percent compounded annually = $3,000 (1 + .05)7 = $4,221.30; Difference = $4,221.30 $4,050.00 = $171.30 AACSB TOPIC: ANALYTIC SECTION: 5.1 TOPIC: SIMPLE VERSUS COMPOUND INTEREST TYPE: PROBLEMS 5-7 Chapter 005 Introduction to Valuation: The Time Value of Money 19. Jeff invests $3,000 in an account that pays 7 percent simple interest. How much more could he have earned over a 20year period if the interest had compounded annually? a. $2,840.00 b. $3,212.12 c. $3,778.54 d. $4,087.18 E. $4,409.05 Ending value at 7 percent simple interest = $3,000 + ($3,000 .07 20) = $7,200.00; Ending value at 7 percent compounded annually = $3,000 (1 + .07)20 = $11,609.05; Difference = $11,609.05 $7,200.00 = $4,409.05 AACSB TOPIC: ANALYTIC SECTION: 5.1 TOPIC: SIMPLE VERSUS COMPOUND INTEREST TYPE: PROBLEMS 20. What is the future value of $3,497 invested for 15 years at 7.5 percent compounded annually? a. $7,431.13 B. $10,347.19 c. $14,289.16 d. $14,911.08 e. $15,267.21 Future value = $3,497 (1 + .075)15 = $10,347.19 AACSB TOPIC: ANALYTIC SECTION: 5.1 TOPIC: FUTURE VALUE TYPE: PROBLEMS 5-8 Chapter 005 Introduction to Valuation: The Time Value of Money 21. Today, you earn a salary of $42,500. What will be your annual salary 10 years from now if you earn annual raises of 3.2 percent? a. $56,100.00 b. $57,414.06 C. $58,235.24 d. $59,122.08 e. $59,360.45 Future value = $42,500 (1 + .032)10 = $58,235.24 AACSB TOPIC: ANALYTIC SECTION: 5.1 TOPIC: FUTURE VALUE TYPE: PROBLEMS 22. You own a classic automobile that is currently valued at $67,900. If the value increases by 8 percent annually, how much will the automobile be worth 15 years from now? a. $199,801.33 b. $212,524.67 c. $214,740.01 D. $215,390.28 e. $218,887.79 Future value = $67,900 (1 + .08)15 = $215,390.28 AACSB TOPIC: ANALYTIC SECTION: 5.1 TOPIC: FUTURE VALUE TYPE: PROBLEMS 5-9 Chapter 005 Introduction to Valuation: The Time Value of Money 23. You hope to buy your dream house 3 years from now. Today, your dream house costs $247,900. You expect housing prices to rise by an average of 7.5 percent per year over the next 3 years. How much will your dream house cost by the time you are ready to buy it? a. $292,063.48 b. $294,882.01 c. $298,600.00 D. $307,965.40 e. $309,425.45 Future value = $247,900 (1 + .075)3 = $307,965.40 AACSB TOPIC: ANALYTIC SECTION: 5.1 TOPIC: FUTURE VALUE TYPE: PROBLEMS 24. Your grandmother invested one lump sum 42 years ago at 3.5 percent interest. Today, she gave you the proceeds of that investment which totaled $28,204.37. How much did your grandmother originally invest? a. $4,500 B. $6,650 c. $7,200 d. $7,500 e. $9,000 Present value = $28,204.37 [1 / (1 + .035)42] = $6,650.00 AACSB TOPIC: ANALYTIC SECTION: 5.2 TOPIC: PRESENT VALUE TYPE: PROBLEMS 5-10 Chapter 005 Introduction to Valuation: The Time Value of Money 30. Tropical Tans is saving money to build a new salon. Three years ago, they set aside $12,000 for this purpose. Today, that account is worth $16,418. What rate of interest is Tropical Tans earning on this money? a. 10.88 percent b. 10.97 percent C. 11.01 percent d. 11.14 percent e. 11.23 percent $16,418 = $12,000 (1 + r)3; r = 11.01 percent AACSB TOPIC: ANALYTIC SECTION: 5.3 TOPIC: INTEREST RATE FOR MULTIPLE PERIODS TYPE: PROBLEMS 31. Five years ago, Precision Tool set aside $50,000 in case of a financial emergency. Today, that account has increased in value to $64,397. What rate of interest is the firm earning on this money? A. 5.19 percent b. 5.47 percent c. 6.18 percent d. 6.32 percent e. 6.45 percent $64,397 = $50,000 (1 + r)5; r = 5.19 percent AACSB TOPIC: ANALYTIC SECTION: 5.3 TOPIC: INTEREST RATE FOR MULTIPLE PERIODS TYPE: PROBLEMS 5-14 Chapter 005 Introduction to Valuation: The Time Value of Money 32. Six years ago, Home Health Industries (HHI) adopted a plan to expand its services next year. At the time the plan was adopted, HHI set aside $125,000 in excess funds to be held for this purpose. As of today, that money has increased in value to $186,408. What rate of interest is the firm earning on these funds? A. 6.89 percent b. 7.10 percent c. 7.18 percent d. 7.27 percent e. 7.43 percent $186,408 = $125,000 (1 + r)6; r = 6.89 percent AACSB TOPIC: ANALYTIC SECTION: 5.3 TOPIC: INTEREST RATE FOR MULTIPLE PERIODS TYPE: PROBLEMS 33. Some time ago, Richard purchased five acres of land costing $123,400. Today, that land is valued at $189,700. How long has he owned this land if the price of land has been increasing at 5.5 percent per year? a. 6.01 years b. 6.98 years c. 7.42 years D. 8.03 years e. 8.67 years $189,700 = $123,400 (1 + .055)t; t = 8.03 years AACSB TOPIC: ANALYTIC SECTION: 5.3 TOPIC: NUMBER OF TIME PERIODS TYPE: PROBLEMS 5-15 Chapter 005 Introduction to Valuation: The Time Value of Money 34. On your thirteenth birthday, you received $1,000 which you invested at 6.5 percent interest, compounded annually. Your investment is now worth $5,476. How old are you today? a. age 29 b. age 32 c. age 35 d. age 37 E. age 40 $5,476 = $1,000 (1 + .065)t; t = 27 years; Age today = 13 + 27 = 40 Note: You received the money when you were 13 years old. Thus, you will be 40 (13 + 27) years old when the value reaches $5,476 AACSB TOPIC: ANALYTIC SECTION: 5.3 TOPIC: NUMBER OF TIME PERIODS TYPE: PROBLEMS 5-16 Chapter 005 Introduction to Valuation: The Time Value of Money 35. You want to have $260,000 saved 15 years from now. How much less do you have to deposit today to reach this goal if you can earn 8 percent rather than 7 percent on your savings? a. $8,728.44 B. $12,273.13 c. $16,602.12 d. $17,414.41 e. $20,019.27 Present value = $260,000 [1 / (1 + .08)15] = $81,962.84; Present value = $260,000 [1 / (1 + .07)15] = $94,235.97; Difference = $94,235.97 $81,962.84 = $12,273.13 AACSB TOPIC: ANALYTIC SECTION: 5.2 AND 5.3 TOPIC: PRESENT VALUE AND RATE CHANGES TYPE: PROBLEMS 5-17 Chapter 005 Introduction to Valuation: The Time Value of Money 36. Your big brother deposited $10,000 today at 9 percent interest for 6 years. You would like to have just as much money at the end of the next 6 years as your brother. However, you can only earn 7.5 percent interest. How much more money must you deposit today than your brother did if you are to have the same amount at the end of the 6 years? a. $398.68 b. $487.63 c. $575.00 d. $648.21 E. $866.96 Future value = $10,000 (1 + .09)6 = $16,771.00; Present value = $16,771.00 [1 / (1 + 075)6] = $10,866.96; Difference = $10,866.96 $10,000.00 = $866.96 AACSB TOPIC: ANALYTIC SECTION: 5.2 AND 5.3 TOPIC: PRESENT VALUE AND RATE CHANGES TYPE: PROBLEMS 5-18 Chapter 005 Introduction to Valuation: The Time Value of Money 37. Last year, you deposited $25,000 into a retirement savings account at a fixed rate of 7.5 percent. Today, you could earn a fixed rate of 8 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 8 percent and still have the same amount as you currently will when you retire 40 years from today? a. $1,218.46 less b. $1,666.67 less c. $2,408.28 less d. $3,628.09 less E. $4,331.30 less Future value = $25,000 (1 + .075)41 = $484,938.92; Present value = $484,938.92 [1 (1 + .08)41] = $20,668.70; Difference = $25,000.00 $20,668.70 = $4,331.30 AACSB TOPIC: ANALYTIC SECTION: 5.2 AND 5.3 TOPIC: PRESENT VALUE AND TIME CHANGES TYPE: PROBLEMS 5-19 Chapter 005 Introduction to Valuation: The Time Value of Money 38. When you retire 36 years from now, you want to have $2 million. You think you can earn an average of 11.5 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 3 years from today. How much more will you have to deposit as a lump sum if you wait for 3 years before making the deposit? A. $15,344.14 b. $15,677.78 c. $16,208.11 d. $17,021.12 e. $19,407.78 Present value = $2,000,000 [1 / (1 + .115)36] = $39,731.48; Present value = $2,000,000 [1 / (1 + .115)33] = $55,075.62; Difference = $55,075.62 $39,731.48 = $15,344.14 AACSB TOPIC: ANALYTIC SECTION: 5.2 AND 5.3 TOPIC: PRESENT VALUE AND RATE CHANGES TYPE: PROBLEMS 5-20 Chapter 005 Introduction to Valuation: The Time Value of Money 39. Marie needs $26,000 as a down payment for a house 4 years from now. She earns 5.25 percent on her savings. Marie can either deposit one lump sum today for this purpose or she can wait a year and deposit a lump sum. How much additional money must Marie deposit if she waits for one year rather than making the deposit today? a. $878.98 b. $911.13 C. $1,112.36 d. $1,348.03 e. $1,420.18 Present value = $26,000 [1 / (1 + .0525)4] = $21,187.75; Present value = $26,000 [1 / (1 + .0525)3] = $22,300.11; Difference = $22,300.11 $21,187.75 = $1,112.36 AACSB TOPIC: ANALYTIC SECTION: 5.2 AND 5.3 TOPIC: PRESENT VALUE AND TIME CHANGES TYPE: PROBLEMS 5-21 Chapter 005 Introduction to Valuation: The Time Value of Money 40. Wexter and Daughter invested $165,000 to help fund a company expansion project planned for 3 years from now. How much additional money will the firm have saved 3 years from now if it can earn 7 percent rather than 5 percent on this money? a. $7,940.09 b. $8,218.07 C. $11,123.97 d. $12,648.18 e. $13,211.21 Future value = $165,000 (1 + .07)3 = $202,132.10; Future value = $165,000 (1 + .05)3 = $191,008.13; Difference = $202,132.10 $191,008.13 = $11,123.97 AACSB TOPIC: ANALYTIC SECTION: 5.1 AND 5.3 TOPIC: FUTURE VALUE AND RATE CHANGES TYPE: PROBLEMS 5-22 Chapter 005 Introduction to Valuation: The Time Value of Money 41. You just received $278,000 from an insurance settlement. You have decided to set this money aside and invest it for your retirement. Currently, your goal is to retire 38 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 9.5 percent rather than just 9.0 percent? a. $794,014 B. $1,396,036 c. $1,611,408 d. $1,818,342 e. $2,033,333 Future value = $278,000 (1 + .095)38 = $8,745,433.15; Future value = $278,000 (1 + 09)38 = $7,349,397.17; Difference = $8,745,433.15 $7,349,397.17 = $1,396,036 AACSB TOPIC: ANALYTIC SECTION: 5.1 AND 5.3 TOPIC: FUTURE VALUE AND RATE CHANGES TYPE: PROBLEMS 5-23 Chapter 005 Introduction to Valuation: The Time Value of Money 42. You will be receiving $2,500 from your family as a graduation present. You have decided to save this money for your retirement. You plan to retire 40 years after graduation. How much additional money will you have at that time if you can earn an average of 12.5 percent on your investment instead of just 12 percent? A. $45,370.08 b. $51,400.62 c. $53,018.97 d. $58,811.99 e. $64,367.48 Future value = $2,500 (1 + .125)40 = $277,997.51; Future value = $2,500 (1 + .12)40 = $232,627.43; Difference = $277,997.51 $232,627.43 = $45,370.08 AACSB TOPIC: ANALYTIC SECTION: 5.1 AND 5.3 TOPIC: FUTURE VALUE AND RATE CHANGES TYPE: PROBLEMS 5-24 Chapter 005 Introduction to Valuation: The Time Value of Money 43. You deposit $1,000 in a retirement account today at 8.5 percent interest. How much more money will you have if you leave the money invested for 40 years rather than 35 years? a. $7,714.91 b. $7,799.08 c. $7,839.73 d. $7,846.52 E. $8,753.38 Future value = $1,000 (1 + .085)40 = $26,133.02; Future value = $1,000 (1 + .085)35 = $17,379.64; Difference = $26,133.02 $17,379.64 = $8,753.38 AACSB TOPIC: ANALYTIC SECTION: 5.1 AND 5.3 TOPIC: FUTURE VALUE AND TIME CHANGES TYPE: PROBLEMS 5-25 Chapter 005 Introduction to Valuation: The Time Value of Money 44. You collect old model trains. One particular model increases in value at a rate of 6.5 percent per year. Today, the model is worth $1,670. How much additional money can you make if you wait 4 years to sell the model rather than selling it 2 years from now? a. $196.67 b. $208.04 c. $241.79 D. $254.24 e. $280.15 Future value = $1,670 (1 + .065)4 = $2,148.40; Future value = $1,670 (1 + .065)2 = $1,894.16; Difference = $2,148.40 $1,894.16 = $254.24 AACSB TOPIC: ANALYTIC SECTION: 5.1 AND 5.3 TOPIC: FUTURE VALUE AND TIME CHANGES TYPE: PROBLEMS Essay Questions 45. Write a sentence explaining why present values decrease as the discount rate increases. Student answers will vary but should present the idea that when you can earn more interest, you need less of your own money to reach the same future dollar amount AACSB TOPIC: REFLECTIVE THINKING SECTION: 5.2 TOPIC: PRESENT VALUE AND DISCOUNTING 5-26 Chapter 005 Introduction to Valuation: The Time Value of Money 46. Explain the relationship between compound interest and time. Interest compounds exponentially over time. The longer the time period, the greater the annual interest earnings AACSB TOPIC: REFLECTIVE THINKING SECTION: 5.1 TOPIC: COMPOUNDING 47. Draw a graph illustrating the future value of $1, using five different interest rates (including 0 percent) and maturities ranging from today to 10 years from now. Plot time to maturity on the horizontal axis and dollars on the vertical axis. (Note: You do not need to do any calculations. Just draw the graph using your intuition.) Graphs should illustrate 3 concepts: 1) At zero percent interest, the $1 will not increase in value. 2) The higher the interest rate, the higher the future value of $1. 3) The future values should illustrate exponential growth AACSB TOPIC: REFLECTIVE THINKING SECTION: 5.1 TOPIC: FUTURE VALUES 48. You are considering two lottery payment streams, choice A pays $1,000 today and choice B pays $1,500 at the end of five years. Using a discount rate of 5 percent, based on present values, which would you choose? Using the same discount rate of 5 percent, based on future values five years from now, which would you choose? What do your results suggest as a general rule for approaching such problems? (Make your choices based purely on the time value of money.) PV of A = $1,000; PV of B = $1,175; FV of A = $1,276; FV of B = $1,500. Based on both present values and future values, B is the better choice. The student should recognize that finding present values and finding future values are simply reverse processes of one another, and that choosing between two lump sums based on PV will always give the same result as choosing between the same two lump sums based on FV AACSB TOPIC: REFLECTIVE THINKING SECTION: 5.1 AND 5.2 TOPIC: COMPARING LUMP SUMS 5-27 Chapter 005 Introduction to Valuation: The Time Value of Money 49. At an interest rate of 10 percent and using the Rule of 72, how long will it take to double the value of a lump sum invested today? How long will it take after that until the account grows to four times the initial investment? Given the power of compounding, shouldn't it take less time for the money to double the second time? It will take 7.2 years to double the initial investment, then another 7.2 years to double it again That is, it takes 14.4 years for the value to reach four times the initial investment. Compounding doesn't affect the amount of time it takes for an investment to double the second time, but note that during the first 7.2 years, the interest earned is equal to 100 percent of the initial investment. During the second 7.2 years, the interest earned is equal to 200 percent of the initial investment. That is the power of compounding AACSB TOPIC: REFLECTIVE THINKING SECTION: 5.3 TOPIC: RULE OF 72 AND COMPOUNDING 50. Some financial advisors recommend you increase the amount of federal income taxes withheld from your paycheck each month so that you will get a larger refund come April 15th That is, you take home less today but get a bigger lump sum when you get your refund. Based on your knowledge of the time value of money, what do you think of this idea? Explain. Some students may slip in a discussion about the benefits of forced savings, etc., but these issues are based on preferences, not the time value of money. Based on the time value of money, the student should recommend just the opposite. That is, withhold as little as possible, while still avoiding tax penalties for under withholding, and pay the remaining tax bill when it comes due the following year as next year's dollars are cheaper than this year's dollars AACSB TOPIC: REFLECTIVE THINKING SECTION: 5.3 TOPIC: THE TIME VALUE OF MONEY 5-28