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53% . mag/gm war/age W; (aw—3%) amtLat: 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 41 933‘ . 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 42 _ 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 113) 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 43 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 44 914‘ 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 45 =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 46 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 50cent 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 47 _ __ 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 48 14. 15. CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 337,”? Mist" Gutters sf Esaunu“ fiﬁﬁ’ﬁ. «21373150 ﬁéﬁﬁnﬁ 9%.? (Vi3%) $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] 625 ) 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 410 933‘ . «2.137142% gazes We (Vi5%) $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 411 22. 23. CHAPTER 4 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 337,”? fiﬁﬁ’ﬁ. lQur" Gutters sf Esaunu“ male/54> ﬁéﬁﬁnﬁ 7%? (Vi3%) arealFa 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 412 =7 7%. 553‘ é ﬁt
93% a mag/gm wax/egg W; (Vi5%) 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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