Chapter 4Introduction to Valuation: The Time Value of Money I. DEFINITIONSTopic: FUTURE VALUE1. The amount an investment is worth after one or more periods of time is the ___________.A) future valueB) present valueC) principal valueD) compound interest rateE) simple interest rateAnswer: ATopic: COMPOUNDING2. The process of accumulating interest on an investment over time to earn more interest is called:A) Growth.B) Compounding.C) Aggregation.D) Accumulation.Answer: BTopic: INTEREST ON INTEREST3. Interest earned on the reinvestment of previous interest payments is called _____________ .A) free interestB) annual interestC) simple interestD) interest on interestE) compound interestAnswer: DTopic: COMPOUND INTEREST4. Interest earned on both the initial principal and the interest reinvested from prior periods is called_______________ .A) free interestB) annual interestC) simple interestD) interest on interestE) compound interestAnswer: ETopic: SIMPLE INTEREST5. Interest earned only on the original principal amount invested is called _____________.A) free interestB) annual interestC) simple interestD) interest on interestE) compound interestAnswer: CTopic: FUTURE VALUE INTEREST FACTOR6. The future value interest factor is calculated as:A) (1 + r)tB) (1 + rt)C) (1 + r)(t)D) 1 + r – tE) None of the above are correctAnswer: ATopic: PRESENT VALUE7. The current value of future cash flows discounted at the appropriate discount rate is called the:A) Principal value.B) Future value.C) Present value.D) Simple interest rate.E) Compound interest rate.Answer: CTopic: DISCOUNTING8. The process of finding the present value of some future amount is often called ______________.A) growthB) discountingC) accumulationD) compoundingE) reductionAnswer: BTopic: PRESENT VALUE INTEREST FACTOR9. The present value interest factor is calculated as:A) 1/(1 + r – t)B) 1/(1 + rt)C) 1/(1 + r)(t)D) 1/(1 + r)tE) 1 + r + tAnswer: DTopic: DISCOUNT RATE10. The interest rate used to calculate the present value of future cash flows is called the _________rate.A) free interestB) annual interestC) compound interestD) simple interestE) discountAnswer: EII CONCEPTSTopic: PRESENT VALUE FACTORS11. Suppose you are trying to find the present value of two different cash flows using the same interestrate for each. One cash flow is $1,000 ten years from now, the other $800 seven years from now.Which of the following is true about the discount factors used in these valuations?A) The discount factor for the cash flow ten years away is always less than or equal to the discountfactor for the cash flow that is received seven years from now.B) Both discount factors are greater than one.C) Regardless of the interest rate, the discount factors are such that the present value of the $1,000will always be greater than the present value of the $800.D) Since the payments are different, no statement can be made regarding the discount factors.E) You should factor in the time differential and choose the payment that arrives the soonest.Answer: ATopic: SIMPLE VS. COMPOUND INTEREST12. You are choosing between investments offered by two different banks. One promises a return of10% for three years using simple interest while the other offers a return of 10% for three years using compound interest. You should:A) Choose the simple interest option because both have the same basic interest rate.B) Choose the compound interest option because it provides a higher return.C) Choose the compound interest option only if the compounding is for monthly periods.D) Choose the simple interest option only if compounding occurs more than once a year.E) Choose the compound interest option only if you are investing less than $5,000.Answer: BTopic: TIME VALUE FACTORS13. Given r and t greater than zero and assuming a lump sum payment:I. Present value interest factors are less than one.II. Future value interest factors are greater than one.III. Present value interest factors are greater than future value interest factors.IV. Present value interest factors grow as t grows, provided r is held constant.A) I onlyB) I and II onlyC) I and IV onlyD) II and III onlyE) II and IV onlyAnswer: BTopic: PRESENT VALUE14. Which of the following statements is/are FALSE, all else the same?I. Present values increase as the discount rate increases.II. Present values increase the further away in time the future value.III. Present values are smaller than future values when both r and t are positive.A) I onlyB) I and II onlyC) II onlyD) III onlyE) II and III onlyAnswer: BIII. PROBLEMSTopic: PRESENT VALUE LUMP SUM15. Fresh out of college, you are negotiating with your prospective new employer. They offer you asigning bonus of $1,000,000 today or a lump sum payment of $1,250,000 three years from now. If you can earn 7% on your invested funds, which of the following is true?A) Take the signing bonus because it has the lower present value.B) Take the signing bonus because it has the higher future value.C) Take the lump sum because it has the higher present value.D) Take the lump sum because it has the lower future value.E) Based on these numbers, you are indifferent between the two.Answer: CResponse:FV of bonus = $1,000,000(1.07)3 = $1,225,043;PV of lump sum = $1,250,00 / (1.07)3 = $1,020,37216. You received a $1 savings account earning 6% on your 1st birthday. How much will you have inthe account on your 30th birthday if you don't withdraw any money before then?A) $3.56B) $4.90C) $5.42D) $5.90E) $6.13Answer: CResponse: FV = $1(1.06)29 = $5.42Topic: FUTURE VALUE LUMP SUM17. What is the future value of $15,000 received today if it is invested at 7.5% compounded annuallyfor five years?A) $15,133.35B) $17,476.42C) $21,534.44D) $24,521.75E) $28,374.89Answer: CResponse: FV = $15,000(1.075)5 = $21,534.44Topic: PRESENT VALUE LUMP SUM18. How much would you have to invest today at 9% compounded annually to have $35,000 availablefor the purchase of a car five years from now?A) $20,267.26B) $22,747.60C) $24,147.25D) $26,370.10E) $28,149.57Answer: BResponse: PV = $35,000 / (1.09)5 = $22,747.60Topic: PRESENT VALUE LUMP SUM19. You will receive a $250,000 inheritance in 25 years. You can invest that money today at 8%compounded annually. What is the present value of your inheritance?A) $ 17,491.53B) $ 29,767.15C) $ 36,504.48D) $ 65,492.34E) $100,000.00Answer: CResponse: PV = $250,000 / (1.08)25 = $36,504.4820. You just won the lottery and want to put some money away for your child's college education.College will cost $75,000 in 15 years. You can earn 7% compounded annually. How much do you need to invest today?A) $19,828.18B) $21,763.07C) $23,690.82D) $25,258.17E) $27,183.45Answer: EResponse: PV = $75,000 / (1.07)15 = $27,183.45Topic: PRESENT VALUE LUMP SUM21. You are supposed to receive $3,000 four years from now. At an interest rate of 8%, what is that$3,000 worth today?A) $1,591.97B) $1,892.43C) $2,205.09D) $2,497.91E) $2,699.01Answer: CResponse: PV = $3,000 / (1.08)4 = $2,205.09Topic: INTEREST RATE22. Your grandfather placed $5,000 in a trust fund for you. In 12 years the fund will be worth $10,000.What is the rate of return on the trust fund?A) 3.70%B) 4.16%C) 5.95%D) 6.90%E) 8.42%Answer: CResponse: r = ($10,000 / 5,000)1/12 1 = 5.95%Topic: NUMBER OF PERIODS23. You need $3,000 to buy a new stereo for your car. If you have $1,200 to invest at 6% compoundedannually, how long will you have to wait to buy the stereo?A) 6.58 yearsB) 8.42 yearsC) 11.60 yearsD) 14.58 yearsE) 15.73 yearsAnswer: EResponse: t = ln($3,000 / 1,200) / ln (1.06) = 15.73 years24. Your parents agree to pay half of the purchase price of a new car when you graduate from college.You will graduate and buy the car two years from now. You have $9,000 to invest today and canearn 12% on invested funds. If your parents match the amount of money you have in two years, what is the maximum you can spend on the new car?A) $ 7,260B) $11,290C) $15,000D) $19,250E) $22,579Answer: EResponse: FV = $9,000(1.12)2 = $11,289.60; $11,289.60 x 2 = $22,579Topic: RULE OF 7225. Granny puts $25,000 into a bank account earning 6%. You can't withdraw the money until thebalance has doubled. How long will you have to leave the money in the account?A) 6 yearsB) 9 yearsC) 12 yearsD) 14 yearsE) 20 yearsAnswer: CResponse: t = 72 / 6 = 12 yearsTopic: COMPOUNDING26. Many economists view a 3% annual inflation rate as "acceptable". Assuming a 3% annual increasein the price of automobiles, how much will a new BMW cost you 8 years from now, if today's priceis $40,000?A) $38,779B) $42,110C) $45,575D) $47,813E) $50,671Answer: EResponse: FV = $40,000(1.03)8 = $50,671Topic: COMPOUNDING27. In 1889, Vincent Van Gogh's painting "Sunflowers" sold for $125. One hundred years later it soldfor $36 million. Had the painting been purchased by your great-grandfather and passed on to you,what annual return on investment would your family have earned on the painting?A) 9.11%B) 10.09%C) 11.88%D) 11.99%E) 13.40%Answer: EResponse: r = ($36,000,000 / 125)1/100 -1 = 13.40%28. An insurance company promises to pay Jane $1 million on her 65th birthday in return for aone-time payment of $125,000 today. (Jane just turned 30.) At what rate of interest would Jane be indifferent between accepting the company's offer and investing the premium on her own?A) 3.4%B) 4.5%C) 5.1%D) 6.1%E) 7.2%Answer: DResponse: r = ($1,000,000 / 125,000)1/35 -1 = 6.12%Topic: PRESENT VALUE LUMP SUM29. Homer promises Bart that he will give him $8,000 upon his graduation from college at SpringfieldU. How much must Homer invest today to make good on his promise, if Bart is expected tograduate in 13 years and Homer can earn 6% on his money?A) $3,235.32B) $3,750.71C) $4,881.11D) $5,012.88E) $5,979.28Answer: BResponse: PV = $8,000 / (1.06)13 = $3,750.71Topic: ANNUAL INTEREST30. You have $200 in an account which pays 5% compound interest. How much additional dollars ofinterest would you earn over 6 years if you moved the money to an account earning 6%?A) $11.89B) $15.68C) $18.93D) $22.88E) $24.94Answer: BResponse:current: FV = $200(1.05)6 = $268.02; proposed:FV = $200(1.06)6 = $283.70difference = $283.70 - 268.02 = $15.68Topic: INVESTMENT RETURNS31. An account was opened with an investment of $2,000 10 years ago. The ending balance in theaccount is $3,500. If interest was compounded annually, what rate was earned on the account?A) 2.66%B) 3.22%C) 3.95%D) 4.81%E) 5.76%Answer: EResponse: r = ($3,500 / 2,000)1/10 -1 = 5.76%Topic: INTEREST ON INTEREST32. An account was opened with $2,000 10 years ago. Today, the account balance is $3,500. If theaccount paid interest compounded annually, how much interest on interest was earned?A) $ 348B) $ 521C) $ 706D) $1,152E) $1,500Answer: AResponse:r = ($3,500 / 2,000)1/10 1 = 5.76%;annual simple interest = $2,000 x .0576 = $115.20total simple interest = $115.20 x 10 = $1,152;difference = $1,500 - 1,152 = $348Topic: SIMPLE INTEREST33. An account paying annual compound interest was opened with $2,000 10 years ago. Today, theaccount balance is $3,500. If the same interest rate is offered on an account paying simple interest, how much income would be earned over the same time period?A) $ 576B) $ 862C) $1,152D) $1,500E) $1,719Answer: CResponse:r = ($3,500 / 2,000)1/10 -1 = 5.76%;annual simple interest = $2,000 x .0576 = $115.20total simple interest = $115.20 x 10 = $1,152Topic: SIMPLE INTEREST34. An account paying annual compound interest was opened with $2,000 10 years ago. Today, theaccount balance is $3,500. If the same interest rate is offered on an account paying simple interest, how much income would be earned each year over the same time period?A) $ 56.90B) $ 80.40C) $ 92.60D) $115.20E) $150.00Answer: DResponse: r = ($3,500 / 2,000)1/10 -1 = 5.76%;annual simple interest = $2,000 x .0576 = $115.20Topic: SIMPLE INTEREST35. An account was opened with $1,000 three years ago. Today, the account balance is $1,157.63. Ifthe account earns a fixed annual interest rate, how long will it take until the account has earned a total of $225 in simple interest?A) Less than one more year.B) Between one and two more years.C) Between two and three more years.D) Between three and four more years.E) Between four and five more years.Answer: BResponse:r = (1,157.63/1,000)1/3 - 1= 5%;1,000 x .05 = 50/yr total of 4.5 yr 3yr to date = 1.5 remainingUse the following to answer questions 36-40:In a growing midwestern town, the number of eating establishments at the end of each of the last five years are as follows:Year 1 = 273;Year 2 = 279;Year 3 = 302;Year 4 = 320;Year 5 = 344Topic: GROWTH RATES36. From the end of year 1 to the end of year 5, the number of eating establishments grew at a rate of_______ compounded annually.A) 3.45%B) 4.15%C) 5.95%D) 6.75%E) 8.25%Answer: CResponse: r = (344 / 273)1/4 -1 = 5.95%Chapter 4: Introduction to Valuation: The Time Value of MoneyTopic: FUTURE VALUE37. If, over the next five years, eating establishments are expected to grow at the same rate as they didduring year 5, forecast the number of eating establishments at the end of year 10.A) 494B) 510C) 534D) 555E) 629Answer: AResponse: r = (344 / 320)1/1 -1 = 7.5%; FV = 344(1.075)5 = 494Topic: FUTURE VALUE38. If the number of eating establishments are expected to grow in year 6 at the same rate as thepercentage increase in year 5, how many new eating establishments will be added in year 6?A) 15B) 22C) 26D) 28E) 31Answer: CResponse: r = (344 / 320)1/1 -1 = 7.5%; new restaurants = 344 x .075 = 26Topic: PRESENT VALUE39. If the town's population was 90,000 at the end of year 5, and the population grew at the same annualrate as the number of eating establishments between the end of year 1 and the end of year 5, what was the town's population at the end of year 1?A) 71,423B) 61,433C) 51,223D) 41,333E) 31,723Answer: AResponse: r = (344 / 273)1/4 -1 = 5.95%; PV = 90,000 / (1.0595)4 = 71,423Topic: GROWTH RATES40. Between the end of year 2 and the end of year 3, the number of eating establishments grew at a rateof _________ compounded annually.A) 5.2%B) 6.7%C) 7.6%D) 8.2%E) 9.3%Answer: DResponse: r = (302 / 279)1/1 -1 = 8.24%Chapter 4: Introduction to Valuation: The Time Value of MoneyIV. ESSAYSTopic: PRESENT VALUE AND DISCOUNTING41. Explain intuitively why it is that present values decrease as the discount rate increases.Answer:Intuitively, a dollar today is worth more than a dollar tomorrow. As a practical matter, the discount rate is an opportunity cost, and the higher the rate, the higher the cost.Topic: COMPOUNDING42. Explain what compounding is and the relationship between compound interest earned and thenumber of years over which an investment is compounded.Answer:Compounding is earning interest on interest. Compounding is not significant over short timeperiods, but increases in importance the longer the time period considered.Topic: FUTURE VALUES43. Draw a picture illustrating the future value of $1, using five different interest rates (including 0%)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 need not make any calculations, draw the figure using your intuition.)Answer:The student should basically replicate Figure 4.2.Topic: COMPARING LUMP SUMS44. You are considering two lottery payment streams, choice A pays $1,000 today and choice B pays$1,750 at the end of five years from now. Using a discount rate of 5%, based on present values, which would you choose? Using the same discount rate of 5%, based on future values, 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.)Answer:PV of A = $1,000; PV of B = $1,371; FV of A = $1,276; FV of B = $1,750. 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.Topic: RULE OF 72 AND COMPOUNDING45. At an interest rate of 10% and using the Rule of 72, how long will it take to double the value of alump 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?Chapter 4: Introduction to Valuation: The Time Value of MoneyAnswer:It will take approximately 7.2 years to double the initial investment, then another 7.2 years todouble it again. That is, it takes approximately 14.4 years for the value to reach four times the initialinvestment. 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% of theinitial investment. During the second 7.2 years, the interest earned is equal to 200% of the initialinvestment. That is the power of compounding.Topic: THE TIME VALUE OF MONEY46. Some financial advisors recommend you increase the amount of federal income taxes withheldfrom your paycheck each month so that you will get a larger refund come April 15th. That is, youtake home less today but get a bigger lump sum when you get your refund. Based on yourknowledge of the time value of money, what do you think of this idea? Explain.Answer: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, thestudents should recommend the opposite strategy, that is, withhold as little as possible and pay thetax bill when it comes the following year. This is the usual dollar today versus a dollar tomorrowargument. Of course, the astute student will note the potential tax complications of this strategy,namely the IRS penalty for insufficient withholding, but the basic argument still applies.。
