Chapter 5 Bank Intro to TMV

Chapter 5 Bank Intro to TMV - .afié& 53 mag/gm war/age...

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Unformatted text preview: _.afié& 53% -. mag/gm war/age W; (aw—3%) amt-Lat: CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY CRITICAL THINKING AND CONCEPTS REVIEW 4.1 Compounding. What is compounding? What is discounting? Answer: Compounding refers to the process of determining the future value of an investment. Discounting is the process of determining the value today of an amount to be received in the future. 4.2 Compounding and Periods. As you increase the length of time involved, what happens to future values? What happens to present values? Answer: Future values grow (assuming a positive rate of return); present values shrink. 4.3 Compounding and Interest Rates. What happens to a future value if you increase the rate r? What happens to a present value? Answer: The future value rises (assuming a positive rate of return); the present value falls. 4.4 Future Values. Suppose you deposit a large sum in an account that earns a low interest rate and simultaneously deposit a small sum in an account with a high interest rate. Which account will have the larger future value? Answer: It depends. The large deposit will have a larger future value for some period, but after time, the smaller deposit with the larger interest rate will eventually become larger. The length of time for the smaller deposit to overtake the larger deposit depends on the amount deposited in each account and the interest rates. 4.5 Ethical Considerations. Take a look back at Example 4.6. Is it deceptive advertising? Is it unethical to advertise a future value like this without a disclaimer? CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-1 933‘ -. mag/gm wax/egg We (aw—3%) Erma: To answer the next five questions, refer to the TMCC security we discussed to open the chapter. Answer: It would appear to be both deceptive and unethical to run such an ad without a disclaimer or explanation. 4.6 Time Value of Money. Why would TMCC be willing to accept such a small amount today ($1,163) in exchange for a promise to repay about 9 times that amount ($10,000) in the future? Answer: It’s a reflection of the time value of money. TMCC gets to use the $1,163. If TMCC uses it wisely, it will be worth more than $10,000 in thirty years. 4.7 Call Provisions. TMCC has the right to buy back the securities on the anniversary date at a price established when the securities were issued (this feature is a term of this particular deal).What impact does this feature have on the desirability of this security as an investment? Answer: This will probably make the security less desirable. TMCC will only repurchase the security prior to maturity if it to its advantage, i.e. interest rates decline. Given the drop in interest rates needed to make this viable for TMCC, it is likely the company will repurchase the security. This is an example of a “call” feature. Such features are discussed at length in a later chapter. 4.8 Time Value of Money. Would you be willing to pay $1,163 today in exchange for $10,000 in 30 years? What would be the key considerations in answering yes or no? Would your answer depend on who is making the promise to repay? Answer: The key considerations would be: (1) Is the rate of return implicit in the offer attractive relative to other, similar risk investments? and (2) How risky is the investment; i.e., how certain are we that we will actually get the $10,000? Thus, our answer does depend on who is making the promise to repay. 4.9 Investment Comparison. Suppose that when TMCC offered the security for $1,163, the US. Treasury had offered an essentially identical security. Do you think it would CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-2 _ r is fit "a? #5. 93% -. «437.42% war/egg We (aw—55%) emit: have had a higher or lower price? Why? Answer: The Treasury security would have a somewhat higher price because the Treasury is the strongest of all borrowers. 4.10 Length of Investment. The TMCC security is bought and sold on the New York Stock Exchange. If you looked at the price today, do you think the price would exceed the $1,163 original price? Why? If you looked in the year 2015, do you think the price would be higher or lower than today's price? Why? Answer: The price would be higher because, as time passes, the price of the security will tend to rise toward $10,000. This rise is just a reflection of the time value of money. As time passes, the time until receipt of the $10,000 grows shorter, and the present value rises. In 2015, the price will probably be higher for the same reason. We cannot be sure, however, because interest rates could be much higher, or TMCC’s financial position could deteriorate. Either event would tend to depress the security’s price. Solutions to Questions and Problems Basic (Questions 1-13) 1. Simple Interest versus Compound Interest. First City Bank pays 8 percent simple interest on its savings account balances, whereas Second City Bank pays 8 percent interest compounded annually. If you made a $6,000 deposit in each bank, how much more money would you earn flom your Second City Bank account at the end of 10 years? Solution: The simple interest per year is: $6,000 X 0.08 = $480 So, after 10 years, you will have: $480 X 10 = $4,800 in interest. The total balance will be $6,000 + 4,800 = $10,800 With compound interest, we use the future value formula: FV = PV(1 +1»)t FV = $6,000(1.08)1° = $12,953.55 CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-3 93% 4 «2167,4264» fiéfléfi 7568—6 (Vi—6%) 66242:: The difference is: $12,953.55 — 10,800 = $2,153.55 2. Calculating Future Values. For each of the following, compute the future value: Present Value Years Interest Rate Future Value $ 3 , 1 5 0 5 1 8% 8,453 10 6 89,305 17 11 227,382 20 5 Solution: To find the FV of a lump sum, we use: FV = PV(1 + r)’ FV = $3,150(1.18) = $ 7,206.44 FV = $8,453(1.06)10 = $ 15,138.04 FV = $89,305(1.11)17 = $526,461.25 FV = $227,382(1.05)20 = $603,312.14 3. Calculating Present Values. For each of the following, compute the present value: Present Value Years Interest Rate Future Value $ 15,451 5 1,557 886,073 901,450 Solution: To find the PV of a lump sum, we use: PV=FV/(1+r)t PV = $15,451 /(1.04)12 = $ 9,650.65 PV = $51,557 / (1.09)4 = $36,524.28 PV = $886,073 / (1.17)16 = $71,861.41 PV = $901,450 / (1.20)21 = $19,594.56 4. Calculating Interest Rates. Solve for the unknown interest rate in each of the CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-4 914‘ 4 (Z‘flfifim 2352152666 5666 (Vi—6%) 4162112 following: Present Value Years Interest Rate Future Value $ 307 905 143,625 255,810 Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)‘ Solving for r, we get: r=(FV/PV)1”— 1 FV = $307 = $221(1 + 06 r = ($307 / $221)“6 — 1 = 0.0563 or 5.63% FV = $905 = $425(1 + r)7 r = ($905 / $425)“7 — 1 = 0.1140 or 11.40% FV = $143,625 = $25,000(1 + r)18 r = ($143,625 / $25,000)“18 — 1 = 0.1020 or 10.20% FV = $255,810 = $40,200(1 + r)21 r = ($255,810 / $40,200)“21 — 1 = 0.0921 or 9.21% 5. Calculating the Number of Periods. Solve for the unknown number of years in each of the following: Present Value Years Interest Rate Future Value $ 1 , 1 05 3,700 387,120 198,212 Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-5 =7 g: 553‘ é K. 93% -. «gee/a?» war/egg W; (Vi—3%) $62112 FV = PV(1 + r)t Solving for t, we get: t= 1n(FV / PV) / ln(1 + r) FV = $1,105 = $250 (1.08)‘ t= ln($1,105 /$250) / In 1.08 = 19.31 years FV = $3,700 = $1,941(1.05)t t= ln($3,700 / $1,941) / In 1.05 = 13.22 years FV = $387,120 = $32,805(1.14)t t= 1n($387,120 / $32,805) / In 1.14 = 18.84 years FV = $198,212 = $32,500(1.24)t t= 1n($198,212 / $32,500) / In 1.24 = 8.41 years 6. Calculating Interest Rates. Assume the total cost of a college education will be $290,000 when your child enters college in 18 years. You presently have $35,000 to invest. What annual rate of interest must you earn on your investment to cover the cost of your child's college education? Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r=(FV/PV)1”— 1 r = ($290,000 / $35,000)“18 — 1 = 0.1247 or 12.47% 7. Calculating the Number of Periods. At 9 percent interest, how long does it take to double your money? To quadruple it? Solution: To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for t, we get: CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-6 1 g: 553‘ é fit 93% a mafia war/egg We (Vi—3%) $62112 t= 1n(FV / PV) / 1n(1 + r) The length of time to double your money is: FV = $2 = $1(1.09)t t= ln 2 / In 1.09 = 8.04 years The length of time to quadruple your money is: FV = $4 = $1(1.09)t t = ln 4 / In 1.09 = 16.09 years Notice that the length of time to quadruple your money is twice as long as the time needed to double your money (the slight difference in these answers is due to rounding). This is an important concept of time value of money. 8. Calculating Rates of Return. In 2006, a 50-cent piece issued in 1904 sold for $1,300. What was the rate of return on this investment? Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r=(FV/PV)1”— 1 r = ($1,300 / $0.50)“102 — 1 = 0.0801 or 8.01% 9. Calculating the Number of Periods. You're trying to save to buy a new $160,000 Ferrari. You have $30,000 today that can be invested at your bank. The bank pays 4.7 percent annual interest on its accounts. How long will it be before you have enough to buy the car? Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for t, we get: t= 1n(FV / PV) / 1n(1 + r) t= ln($160,000 / $30,000) / ln 1.047= 36.45 years CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-7 _ __ r is fit "a? ti. 933‘ 9 «21671421915» figfing 9% (Vi—56%) 616292 10. Calculating Present Values. Imprudential, Inc., has an unfunded pension liability of $800 million that must be paid in 20 years. To assess the value of the firm's stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 8 percent, what is the present value of this liability? Solution: To find the PV of a lump sum, we use: PV=FV/(1+r)t PV = $800,000,000 / (1.08)20 = $171,638,566 11. Calculating Present Values. You have just received notification that you have won the $2 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday ( assuming you're around to collect),80 years fiom now. What is the present value of your windfall if the appropriate discount rate is 11 percent? Solution: To find the PV of a lump sum, we use: PV=FV/(1+r)t PV = $2,000,000 /(1.11)80 = $473.36 12. Calculating Future Values. Your coin collection contains 50 1952 silver dollars. If your grandparents purchased them for their face value when they were new, how much will your collection be worth when you retire in 2060, assuming they appreciate at a 5.7 percent annual rate? Solution: To find the FV of a lump sum, we use: FV = PV(1 + r)t FV = $50(1.057)108 = $19,909.88 13. Calculating Interest Rates and Future Values. In 1895, the first US. Open Golf Championship was held. The winner's prize money was $150. In 2006, the winner's check was $1,225,000. What was the annual percentage increase in the winner's check over this period? If the winner's prize increases at the same rate, what will it be in 2040? Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-8 14. 15. CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 337,”? Mist" Gutters sf Esau-nu“ fififi’fi. «21373150 fiéfifinfi 9%.? (Vi-3%) $132101 is: FV = PV(1 + r)t Solving for r, we get: r=(FV/PV)1”— 1 r = ($1,225,000 / $150)“111 — 1 = 0.0845 or 8.45% To find what the check will be in 2040, we use the FV of a lump sum, so: FV = PV(1 + r)t FV = $1,225,000(1.0845)34 = $19,317,058.93 Calculating Rates of return. In 2006, a gold $3 coin minted in 1879 was auctioned for $9,000. For this to have been true, what was the annual increase in the value of the coin? Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r=(FV/PV)1”— 1 r = ($9,000 / $3)“127 — 1 = 0.0651 or 6.51% Calculating Rates of Return. Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2003, Sotheby's sold the Edgar Degas bronze sculpture Petite danseuse de quartorze ans at auction for a price of $10,311,500. Unfortunately for the previous owner, he had purchased it in 1999 at a price of $12,377,500. What was his annual rate of return on this sculpture? Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r=(FV/PV)1”— 1 r = ($10,311,500 / $12,377,500)“4 — 1 = —0.0446 or 4.46% 1 g: 553‘ é Ii 93% -. «2.137142% war/egg We (Vi—3%) $6212 Intermediate ( Question] 6-25 ) 16. Calculating Rates of Return. Referring to the TMCC security we discussed at the very beginning of the chapter: a. Based on the $1,163 price, what rate was TMCC paying to borrow money? b. Suppose that, on April 21, 2015, this security's price is $2,500. If an investor had purchased it for $1,163 at the offering and sold it on this day, what annual rate of return would she have earned? c. If an investor had purchased the security at market on April 21, 2015, and held it until it matured, what annual rate of return would she have earned? Solution: a. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r=(FV/PV)1”— 1 r = ($10,000 / $1,163)1 ’30 — 1 = 0.0744 or 7.44% b. Using the FV formula and solving for the interest rate, we get: r=(FV/PV)“‘— 1 r = ($2,500 / $1,163)1 ’9 — 1 = 0.0888 or 8.88% c. Using the FV formula and solving for the interest rate, we get: r=(FV/PV)“‘— 1 r = ($10,000 / $2,500)1 ’21 — 1 = 0.0682 or 6.82% 17. Calculating Present Values. Suppose you are still committed to owning a $160,000 Ferrari (see Question 9). If you believe your mutual fund can achieve a 10.75 percent annual rate of return, and you want to buy the car in 10 years on the day you turn 30, how much must you invest today? Solution: To find the PV of a lump sum, we use: PV=FV/(1+r)t PV = $160,000 / (1.1075)10 = $57,634.51 18. Calculating Future Values. You have just made your first $5,000 contribution to CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-10 933‘ -. «2.137142% gazes We (Vi-5%) $13212 your individual retirement account. Assuming you earn an 11 percent rate of return and make no additional contributions, what will your account be worth when you retire in 45 years? What if you wait 10 years before contributing? (Does this suggest an investment strategy?) Solution: To find the FV of a lump sum, we use: FV = PV(1 + r)t FV = $5,000(1.11)45 = $547,651.21 If you wait 10 years, the value of your deposit at your retirement will be: FV = $5,000(1.11)35 = $192,874.26 Better start early! 19. Calculating Future Values. You are scheduled to receive $15,000 in two years. When you receive it, you will invest it for six more years at 8 percent per year. How much will you have in eight years? Solution: Even though we need to calculate the value in eight years, we will only have the money for six years, so we need to use six years as the number of periods. To find the FV of a lump sum, we use: FV = PV(1 + r)t FV = $15,000(1.08)6 = $23,803.11 20. Calculating the Number of Periods. You expect to receive $30,000 at graduation in two years. You plan on investing it at 9 percent until you have $160,000. How long will you wait fiom now? (Better than the situation in Question 9, but still no Ferrari.) Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t $160,000 = $30,000(1.09)t t= ln($160,000 / $30,000) / In 1.09 = 19.42 years From now, you’ll wait 2 + 19.42 = 21.42 years 21. Calculating Future Values. You have $7,000 to deposit. Regency Bank offers 12 CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 4-11 22. 23. CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 337,”? fififi’fi. l-Qur" Gutters sf Esau-nu“ male/54> fiéfifinfi 7%? (Vi-3%) areal-Fa percent per year compounded monthly (1 percent per month), while King Bank offers 12 percent but will only compound annually. How much will your investment be worth in 20 years at each bank? Solution: To find the FV of a lump sum, we use: FV = PV(1 + r)t In Regency Bank, you will have: FV = $7,000(1.01)240 = $76,247.88 And in King Bank, you will have: FV = $7,000(1.12)20 = $67,524.05 Calculating Interest Rates. An investment offers to triple your money in 24 months (don't believe it). What rate per three months are you being offered? Solution: To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. We also need to be careful about the number of periods. Since the length of the compounding is three months and we have 24 months, there are eight compounding periods. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r=(FV/PV)1”— 1 r = ($3 /$1)“8 — 1 = 0.1472 or 14.72% Calculating the Number of Periods. You can earn 0.45 percent per month at your bank. If you deposit $1,500, how long must you wait until your account has grown to $3,600? Solution: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t $3,600 = $1,500(1.0045)t 4-12 =7 7%. 553‘ é fit 93% a mag/gm wax/egg W; (Vi-5%) Erma t= 1n($3,600 / $1,500) /1n 1.0045 = 194.99 months 24. Calculating Present Values. You need $75,000 in 10 years. If you can earn 0.55 percent per month, how much will you have to deposit today? Solution: To find the PV of a lump sum, we use: PV=FV/(1+r)t PV = $75,000 / (1.0055)120 PV = $38,834.01 25. Calculating Present Values. You have decided that you want to be a millionaire when you retire in 45 years. If you can earn a 12 percent a...
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