Chapter 9 Discussion Questions 9-1 How is the future value(Appendix a)related to the present value of a single sum(Appendix B)? The future value represents the expected worth of a single amount, whereas the present value represents the current worth FV=PV(1+r future value Present va lu (1+i 9-2 How is the present value of a single sum(Appendix b)related to the present value of an annuity(Appendix D)? The present value of a single amount is the discounted value for one future payment, whereas the present value of an annuity represents the discounted value of a series of consecutive payments of equal amount why does money have a time value? Money has a time value because funds received today can be reinvested to reach a greater value in the future. a person would rather receive $1 today than I in ten years, because a dollar received today, invested at 6 percent, is worth $.791 after ten years Does inflation have anything to do with making a dollar today worth more than a dollar tomorrow? Inflation makes a dollar tod ay worth more than a dollar in the future. Because inflation tends to erode the purchasing power of money, funds received today will be worth more than the same amount received in the future S-305 Copyright C2005 by The McGra-Hill Companies, Inc
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-305 Chapter 9 Discussion Questions 9-1. How is the future value (Appendix A) related to the present value of a single sum (Appendix B)? The future value represents the expected worth of a single amount, whereas the present value represents the current worth. FV = PV (1 + I) n future value ( ) Present va lue 1 1 PV FV + = n i 9-2. How is the present value of a single sum (Appendix B) related to the present value of an annuity (Appendix D)? The present value of a single amount is the discounted value for one future payment, whereas the present value of an annuity represents the discounted value of a series of consecutive payments of equal amount. 9-3. Why does money have a time value? Money has a time value because funds received today can be reinvested to reach a greater value in the future. A person would rather receive $1 today than $1 in ten years, because a dollar received today, invested at 6 percent, is worth $1.791 after ten years. 9-4. Does inflation have anything to do with making a dollar today worth more than a dollar tomorrow? Inflation makes a dollar today worth more than a dollar in the future. Because inflation tends to erode the purchasing power of money, funds received today will be worth more than the same amount received in the future
Adjust the annual formula for a future value of a single amount at 12 percent for 10 years to a semiannual compound ing formula. What are the interest factors(FVIE) before and after? Why are they different? FV=PVxFⅤ( Appendix A) i=12%,n=10 3.106 Annual 6%.n=20 3.207 Semiannual The more frequent compounding under the semiannual compound ing assumption increases the future value 9-6 If, as an investor, you had a choice of daily ly, or quarterly compounding, which would you choose? Why? The greater the number of compounding periods, the larger the future value The investor should choose daily compounding over monthly or quarterly What is a deferred annuity? a deferred annuity is an annuity in which the equal payments will begin at some future point in time 9-8 List five different financial applications of the time value of money Different financial applications of the time value of money Equipment purchase or new product decision, Present value of a contract providing future payments, Future worth of an investment Regular payment necessary to provide a future sum, Regular payment necessary to amortize a loan Determination of return on an investment Determination of the value of a bond CopyrightC 2005 by The McGray-Hill Companies, Inc
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-306 9-5. Adjust the annual formula for a future value of a single amount at 12 percent for 10 years to a semiannual compounding formula. What are the interest factors (FVIF) before and after? Why are they different? ( ) i 6%, n 20 3.207 Semiannual i 12%, n 10 3.106 Annual FV PV FVIF Appendix A = = = = = The more frequent compounding under the semiannual compounding assumption increases the future value. 9-6. If, as an investor, you had a choice of daily, monthly, or quarterly compounding, which would you choose? Why? The greater the number of compounding periods, the larger the future value. The investor should choose daily compounding over monthly or quarterly. 9-7. What is a deferred annuity? A deferred annuity is an annuity in which the equal payments will begin at some future point in time. 9-8. List five different financial applications of the time value of money. Different financial applications of the time value of money: Equipment purchase or new product decision, Present value of a contract providing future payments, Future worth of an investment, Regular payment necessary to provide a future sum, Regular payment necessary to amortize a loan, Determination of return on an investment, Determination of the value of a bond
Problems You invest $3,000 a year for 3 years at 12 percent a. What is the value of your investment after one year? Multiply $3,000x 1.12 b. What is the value of your investment after two years? multiply your answer to part a by 1.12 c. What is the value of your investment after three years? Multiply your answer to part b by 1. 12. This gives you your final answer d. Confirm that your final answer is correct by going to Appendix A(future value ofa $1), and looking up the future value for n=3, and i=12 percent Multiply this tabular value by $3,000 and compare your answer to the answer in part c. There may be a slight difference due to round ing Solution: a.$3.000x1.12 $3.360.00 b.$3,360x1.12=$3.763.20 C.$3,763.20x1.12=$4,214.78 d.$3,000X1.405=$4,21500( Appendix a) 9-2 What is the present value of a. $9,000 in 7 years at 8 percent? b. $20,000 in 5 years at 10 percent? c. $10,000 in 25 years at 6 percent? d. $1,000 in 50 years at 16 percent? Solution: appendix b PⅤ=FⅤⅹPv a.$9,000x583=$5,247 b.$20000X.621=$12420 C.$10,000×233=$2,30 d.$1000x.001=$1 -307 oyrightC2005 by The McGraw-Hill Companies, Inc
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-307 Problems 9-1. You invest $3,000 a year for 3 years at 12 percent. a. What is the value of your investment after one year? Multiply $3,000 x 1.12. b. What is the value of your investment after two years? Multiply your answer to part a by 1.12. c. What is the value of your investment after three years? Multiply your answer to part b by 1.12. This gives you your final answer. d. Confirm that your final answer is correct by going to Appendix A (future value of a $1), and looking up the future value for n = 3, and i = 12 percent. Multiply this tabular value by $3,000 and compare your answer to the answer in part c. There may be a slight difference due to rounding. Solution: a. $3,000 x 1.12 = $3,360.00 b. $3,360 x 1.12 = $3,763.20 c. $3,763.20 x 1.12 = $4,214.78 d. $3,000 x 1.405 = $4,215.00 (Appendix A) 9-2. What is the present value of: a. $9,000 in 7 years at 8 percent? b. $20,000 in 5 years at 10 percent? c. $10,000 in 25 years at 6 percent? d. $1,000 in 50 years at 16 percent? Solution: Appendix B PV = FV x PVIF a. $ 9,000 x .583 = $5,247 b. $20,000 x .621 = $12,420 c. $10,000 x .233 = $2,330 d. $ 1,000 x .001 = $1
If you invest $9, 000 today, how much will you have a. In 2 years at 9 percent? b. In 7 years at 12 percent? c. In 25 years at 14 percent? d. In 25 years at 14 percent(compounded semiannually )? Solution: ppendix a FVEPVXFVIE a.$9.000x1.188=$10692 b.$9,000x2211=$19899 c.$9,000X26462=$238,158 d.$9000x29457=$265,113(7%,50 periods Your uncle offers you a choice of $30,000 in 50 years or $95 today. If money discounted at 12 percent, which should you choose? Solution: appendix b PV=FVX PVIF(12%, 50 periods) PV=$30.000X.003=$90 Choose $95 today CopyrightC 2005 by The McGray-Hill Companies, Inc. -308
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-308 9-3. If you invest $9,000 today, how much will you have: a. In 2 years at 9 percent? b. In 7 years at 12 percent? c. In 25 years at 14 percent? d. In 25 years at 14 percent (compounded semiannually)? Solution: Appendix A FV = PV x FVIF a. $9,000 x 1.188 = $ 10,692 b. $9,000 x 2.211 = $ 19,899 c. $9,000 x 26.462 = $238,158 d. $9,000 x 29.457 = $265,113 (7%, 50 periods) 9-4. Your uncle offers you a choice of $30,000 in 50 years or $95 today. If money is discounted at 12 percent, which should you choose? Solution: Appendix B PV = FV x PVIF (12%, 50 periods) PV = $30,000 x .003 = $90 Choose $95 today
How much would you have to invest today to receive a. $15,000 in 8 years at 10 percent? b.$20,000in12 at 13 c. $6,000 each year for 10 years at 9 percent? d. $50,000 each year for 50 years at 7 percent? Solution: ppendix b(a and b) PⅤ=FⅤ X PVIE a.$15000X467=$7005 b.$20,000x.231=$4620 Appendix d(c and d) C.$6000x6418=$38508 d.$50,000x13.801=$690050 If you invest $2,000 a year in a retirement account, how much would you have a. In 5 years at 6 percent? b. In 20 years at 10 perc c. In 40 years at 12 percent? Solution: Appendix c FVA=AxFⅤ a.$2,000X5637=$11,274 b.$2,000x57275=$114,550 C.$2.000x76709=$1.534180 -309 Copyright C2005 by The McGra-Hill Companies, Inc
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-309 9-5. How much would you have to invest today to receive: a. $15,000 in 8 years at 10 percent? b. $20,000 in 12 years at 13 percent? c. $6,000 each year for 10 years at 9 percent? d. $50,000 each year for 50 years at 7 percent? Solution: Appendix B (a and b) PV = FV x PVIF a. $15,000 x .467 = $7,005 b. $20,000 x .231 = $4,620 Appendix D (c and d) c. $ 6,000 x 6.418 = $ 38,508 d. $50,000 x 13.801 = $690,050 9-6. If you invest $2,000 a year in a retirement account, how much would you have: a. In 5 years at 6 percent? b. In 20 years at 10 percent? c. In 40 years at 12 percent? Solution: Appendix C FVA = A x FVIFA a. $2,000 x 5.637 = $ 11,274 b. $2,000 x 57.275 = $ 114,550 c. $2,000 x 767.09 = $1,534,180
years you take the proceeds and invest them for 12 years at 15 percent. How J You invest a single amount of $10,000 for 5 years at 10 percent. At the end much will you have after 17 years? Solution Appendix a FV≡PVxFⅤIF $l0000x1611=$16,110 Appendix a FV≡PⅤxFVF $16,110x5350=$86,188 9-8 Jean Splicing will receive $8, 500 a year for the next 15 years from her trust. If a 7 percent interest rate is applied, what is the current value of the future payments? Solution: ppendix d PVA=AX PVIFA(7%, 15 periods) =$8,500X9.108=$77,418 99 Phil Goode will receive $175,000 in 50 years. His friends are very jealous of him. If the funds are discounted back at a rate of 14 percent, what is the present value of his future"pot of gold"? Solution: Appendix b PV=FVX PVIF(14%0, 50 periods) $175,000x001=$175 CopyrightC 2005 by The McGray-Hill Companies, Inc. S-310
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-310 9-7. You invest a single amount of $10,000 for 5 years at 10 percent. At the end of 5 years you take the proceeds and invest them for 12 years at 15 percent. How much will you have after 17 years? Solution: Appendix A FV = PV x FVIF $10,000 x 1.611 = $16,110 Appendix A FV = PV x FVIF $16,110 x 5.350 = $86,188 9-8. Jean Splicing will receive $8,500 a year for the next 15 years from her trust. If a 7 percent interest rate is applied, what is the current value of the future payments? Solution: Appendix D PVA = A x PVIFA (7%, 15 periods) = $8,500 x 9.108 = $77,418 9-9. Phil Goode will receive $175,000 in 50 years. His friends are very jealous of him. If the funds are discounted back at a rate of 14 percent, what is the present value of his future "pot of gold"? Solution: Appendix B PV = FV x PVIF (14%, 50 periods) = $175,000 x .001 = $175
Polly graham will receive $12,000 a year for the next 15 years as a result of her patent. If a 9 percent rate is applied, should she be willing to sell out her future rights now for $100.000? Solution Appendix D PVA=AX PVIFA (9%, 15 periods) $12000x8.061=$96,732 Yes, the present value of the annuity is worth less than $100,000 9-11 Carrie Tune will receive $19, 500 for the next 20 years as a payment for a new song she has written. If a 10 percent rate is applied, should she be willing to sell out her future rights now for $160,000 Solution: Appendix d PVA=AX PVIFA(10%0, 20 periods) PVA=$19,500x8.514=$166023 o, the present value of the annuity is worth more than $160,000 9-12 The Clearinghouse Sweepstakes has just informed you that you have won $1 million. The amount is to be paid out at the rate of $20,000 a year for the next 50 years. With a discount rate of 10 percent, what is the present value of your winnings? Solution: Appendix d PVA=AXPVIFA(10%0, 50 periods) PVA=$20,000x9.915=$198,300 S-311 Copyright o 2005 by The McGraw-Hill Companies, Inc
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-311 9-10. Polly Graham will receive $12,000 a year for the next 15 years as a result of her patent. If a 9 percent rate is applied, should she be willing to sell out her future rights now for $100,000? Solution: Appendix D PVA = A x PVIFA (9%, 15 periods) = $12,000 x 8.061 = $96,732 Yes, the present value of the annuity is worth less than $100,000. 9-11. Carrie Tune will receive $19,500 for the next 20 years as a payment for a new song she has written. If a 10 percent rate is applied, should she be willing to sell out her future rights now for $160,000. Solution: Appendix D PVA = A x PVIFA (10%, 20 periods) PVA = $19,500 x 8.514 = $166,023 No, the present value of the annuity is worth more than $160,000. 9-12. The Clearinghouse Sweepstakes has just informed you that you have won $1 million. The amount is to be paid out at the rate of $20,000 a year for the next 50 years. With a discount rate of 10 percent, what is the present value of your winnings? Solution: Appendix D PVA = A x PVIFA (10%, 50 periods) PVA = $20,000 x 9.915 = $198,300
Al Rosen invests $25,000 in a mint condition 1952 Mickey Mantle Topps baseball card. He expects the card to increase in value 12 percent per year for the next 10 years How much will his card be worth after 10 years' Solution Appendix a FV=PVX FVIF (12%, 10 periods) $25000x3.106=$77,650 9-14 Dr. Ruth has been secretly depositing $2, 500 in her savings account every December starting in 1995. Her account earns 5 percent compounded annuall How much will she have in december of 2004?(Assume that a deposit is made in 2004.) Make sure to carefully count the years Solution: ppendⅸxC FVA=AX FVIEA (5%0, 10 periods FVA=$2,500x12.578=$31,445 9-15 At a growth(interest)rate of 9 percent annually, how long will it take for a sum to double? To triple? Select the year that is closest to the correct answer Solution: Appendix a If the sum is doubling, then the interest factor must equal 2 s In Appendix A, looking down the 9% column, we find the factor closest to 2(1.993)on the 8-year row. The factor closest to 3 (3.066)is on the 13-year row CopyrightC 2005 by The McGray-Hill Companies, Inc. S-312
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-312 9-13. Al Rosen invests $25,000 in a mint condition 1952 Mickey Mantle Topps baseball card. He expects the card to increase in value 12 percent per year for the next 10 years. How much will his card be worth after 10 years? Solution: Appendix A FV = PV x FVIF (12%, 10 periods) = $25,000 x 3.106 = $77,650 9-14. Dr. Ruth has been secretly depositing $2,500 in her savings account every December starting in 1995. Her account earns 5 percent compounded annually. How much will she have in December of 2004? (Assume that a deposit is made in 2004.) Make sure to carefully count the years. Solution: Appendix C FVA = A x FVIFA (5%, 10 periods) FVA = $2,500 x 12.578 = $31,445 9-15. At a growth (interest) rate of 9 percent annually, how long will it take for a sum to double? To triple? Select the year that is closest to the correct answer. Solution: Appendix A If the sum is doubling, then the interest factor must equal 2. * In Appendix A, looking down the 9% column, we find the factor closest to 2 (1.993) on the 8-year row. The factor closest to 3 (3.066) is on the 13-year row
cred itor accept in payment immed iately if she could earn 12 percent on herour If you owe $40,000 payable at the end of seven years, what amount should mone Solution Appendix b PⅤ=FⅤxPVF(12%,7 periods) PV=$40.000X452=$18,080 9-17 Jack Hammer invests in a stock that will pay dividends of $2.00 at the end of the first year; $2. 20 at the end of the second year; and $2.40 at the end of the third year. Also, he believes that at the end of the third year he will be able to ll the stock for $33. What is the present value of all future benefits if a discount rate of 1 l percent is applied?(Round all values to two places to the right of the decimal point. Solution: ppendix B PⅤ=FⅤI Discount rate =11% $200x.901=$1.80 2.20x.802 79 2.40x.731=175 33.00x.731=24.12 $2946 -313 CopyrightC2005 by The McGraw-Hill Companies, Inc
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-313 9-16. If you owe $40,000 payable at the end of seven years, what amount should your creditor accept in payment immediately if she could earn 12 percent on her money? Solution: Appendix B PV = FV x PVIF (12%, 7 periods) PV = $40,000 x .452 = $18,080 9-17. Jack Hammer invests in a stock that will pay dividends of $2.00 at the end of the first year; $2.20 at the end of the second year; and $2.40 at the end of the third year. Also, he believes that at the end of the third year he will be able to sell the stock for $33. What is the present value of all future benefits if a discount rate of 11 percent is applied? (Round all values to two places to the right of the decimal point.) Solution: Appendix B PV = FVIF Discount rate = 11% $ 2.00 x .901 = $ 1.80 2.20 x .802 = 1.79 2.40 x .731 = 1.75 33.00 x .731 = 24.12 $29.46
Les moore retired as president of Good man Snack Foods Company but is currently on a consulting contract for $35,000 per year for the next 10 years a. If Mr. Moore's opportunity cost(potential return)is 10 percent, what is the present value of his consulting contract? b. Assume Mr Moore will not retire for two more years and will not start to receive his 10 payments until the end of the third year, what would be the value of his deferred annuity? Solution: ppendix d a. PVA=AX PVIFA (10%, 10 periods) PVA=$35000X6.145=$215,075 b Deferred annuity-Appendix d PVA=AX PVIFA (I=10%, 10 periods) PVA=$35,000×6145=$215075 Now, discount back this value for 2 periods PⅤ=FⅤxPⅤF(1=10%,2 periods) Appendix e $215075X826=$177,652 OR A dix D PⅤA=$35,000(6814-1.7360 where n=12,n=2andi=10%) $35,000(5078)=$177,730 The answer is slightly different from the answer above due to rounding in the tables CopyrightC 2005 by The McGray-Hill Companies, Inc. -314
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-314 9-18. Les Moore retired as president of Goodman Snack Foods Company but is currently on a consulting contract for $35,000 per year for the next 10 years. a. If Mr. Moore's opportunity cost (potential return) is 10 percent, what is the present value of his consulting contract? b. Assume Mr. Moore will not retire for two more years and will not start to receive his 10 payments until the end of the third year, what would be the value of his deferred annuity? Solution: Appendix D a. PVA = A x PVIFA (10%, 10 periods) PVA = $35,000 x 6.145 = $215,075 b. Deferred annuity—Appendix D PVA = A x PVIFA (i = 10%, 10 periods) PVA = $35,000 x 6.145 = $215,075 Now, discount back this value for 2 periods PV = FV x PVIF (i = 10%, 2 periods) Appendix B $215,075 x .826 = $177,652 OR Appendix D PVA = $35,000 (6.814 – 1.7360 where n = 12, n = 2 and i = 10%) = $35,000 (5.078) = $177,730 The answer is slightly different from the answer above due to rounding in the tables