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study guide

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Department
Finance
Course Code
FIN 300
Professor
Scott Anderson

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CHAPTER 5
INTRODUCTION TO VALUATION: THE TIME VALUE
OF MONEY
Answers to Concepts Review and Critical Thinking Questions
1. The four parts are the present value (PV), the future value (FV), the discount rate (r), and the life of
the investment (t).
2. Compounding refers to the growth of a dollar amount through time via reinvestment of interest
earned. It is also 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.
3. Future values grow (assuming a positive rate of return); present values shrink.
4. The future value rises (assuming it’s positive); the present value falls.
5. It’s a reflection of the time value of money. ScotiaMcLeod gets to use the $29.19 immediately. If
Scotia uses it wisely, it will be worth more than $100 in twenty years.
6. 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.
7. The Government of Canada security would have a somewhat higher price because the Government of
Canada is the strongest of all borrowers.
8. The price would be higher because, as time passes, the price of the security will tend to rise toward
$100. This rise is just a reflection of the time value of money. As time passes, the time until receipt of
the $100 grows shorter, and the present value rises. In 2007, the price will probably be higher for the
same reason. We cannot be sure, however, because interest rates could be much higher, or Canadas
financial position could deteriorate. Either event would tend to depress the security’s price.
Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1. The simple interest per year is:
$5,000 × .07 = $350
So after 10 years you will have:
$350 × 10 = $3,500 in interest.
45
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The total balance will be $5,000 + 3,500 = $8,500
With compound interest we use the future value formula:
FV = PV(1 +r)t
FV = $5,000(1.07)10 = $9,835.76
The difference is:
$9,835.76 – 8,500 = $1,335.76
2. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $2,250(1.10)19 = $ 13,760.80
FV = $9,310(1.08)13 = $ 25,319.70
FV = $76,355(1.22)4= $169,151.87
FV = $183,796(1.07)8= $315,795.75
3. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $15,451 / (1.05)6= $11,529.77
PV = $51,557 / (1.11)9= $20,154.91
PV = $886,073 / (1.16)23 = $29,169.95
PV = $550,164 / (1.19)18 = $24,024.09
4. 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 / t – 1
FV = $307 = $265(1 + r)2;r = ($307 / $265)1/2 – 1 = 7.63%
FV = $896 = $360(1 + r)9;r = ($896 / $360)1/9 – 1 = 10.66%
FV = $162,181 = $39,000(1 + r)15; r = ($162,181 / $39,000)1/15 – 1 = 9.97%
FV = $483,500 = $46,523(1 + r)30;r = ($483,500 / $46,523)1/30 – 1= 8.12%
5. 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 = ln(FV / PV) / ln(1 + r)
46
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FV = $1,284 = $625(1.08)t;t = ln($1,284/ $625) / ln 1.08 = 9.36 yrs
FV = $4,341 = $810(1.07)t;t = ln($4,341/ $810) / ln 1.07 = 24.81 yrs
FV = $402,662 = $18,400(1.21)t;t = ln($402,662 / $18,400) / ln 1.21 = 16.19 yrs
FV = $173,439 = $21,500(1.29)t;t = ln($173,439 / $21,500) / ln 1.29 = 8.20 yrs
6. 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 / t – 1
r = ($80,000 / $15,000)1/18 – 1 = 9.75%
7. 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:
t = ln(FV / PV) / ln(1 + r)
The length of time to double your money is:
FV = $2 = $1(1.07)t
t = ln 2 / ln 1.07 = 10.24 years
The length of time to quadruple your money is:
FV = $4 = $1(1.07)t
t = ln 4 / ln 1.07 = 20.49 years
Notice that the length of time to quadruple your money is twice as long as the time needed to double
your money (the difference in these answers is due to rounding). This is an important concept of time
value of money.
8. 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 / t – 1
r = ($28,835 / $21,608)1/5 – 1 = 5.94%
9. 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:
47
www.notesolution.com

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Description
CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount rate (r), and the life of the investment (t). 2. Compounding refers to the growth of a dollar amount through time via reinvestment of interest earned. It is also 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. 3. Future values grow (assuming a positive rate of return); present values shrink. 4. The future value rises (assuming it’s positive); the present value falls. 5. It’s a reflection of the time value of money. ScotiaMcLeod gets to use the $29.19 immediately. If Scotia uses it wisely, it will be worth more than $100 in twenty years. 6. 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. 7. The Government of Canada security would have a somewhat higher price because the Government of Canada is the strongest of all borrowers. 8. The price would be higher because, as time passes, the price of the security will tend to rise toward $100. This rise is just a reflection of the time value of money. As time passes, the time until receipt of the $100 grows shorter, and the present value rises. In 2007, the price will probably be higher for the same reason. We cannot be sure, however, because interest rates could be much higher, or Canada’s financial position could deteriorate. Either event would tend to depress the security’s price. Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. The simple interest per year is: $5,000 × .07 = $350 So after 10 years you will have: $350 × 10 = $3,500 in interest. 45 www.notesolution.com The total balance will be $5,000 + 3,500 = $8,500 With compound interest we use the future value formula: FV = PV(1 +r) t 10 FV = $5,000(1.07) = $9,835.76 The difference is: $9,835.76 – 8,500 = $1,335.76 2. To find the FV of a lump sum, we use: FV = PV(1 + r) t 19 FV = $2,250(1.10) = $ 13,760.80 FV = $9,310(1.08) 13 = $ 25,319.70 4 FV = $76,355(1.22) = $169,151.87 FV = $183,796(1.07) 8 = $315,795.75 3. To find the PV of a lump sum, we use: t PV = FV / (1 + r) 6 PV = $15,451 / (1.05) = $11,529.77 PV = $51,557 / (1.11) 9 = $20,154.91 23 PV = $886,073 / (1.16) = $29,169.95 PV = $550,164 / (1.19) 18 = $24,024.09 4. 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: 1 / t r = (FV / PV) – 1 FV = $307 = $265(1 + r) ; 2 r = ($307 / $265) – 12 = 7.63% 9 1/9 FV = $896 = $360(1 + r) ; r = ($896 / $360) – 1 = 10.66% FV = $162,181 = $39,000(1 + r) ; 15 r = ($162,181 / $39,000) 1/1– 1 = 9.97% 30 1/30 FV = $483,500 = $46,523(1 + r) ; r = ($483,500 / $46,523) – 1= 8.12% 5. 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: t FV = PV(1 + r) Solving for t, we get: t = ln(FV / PV) / ln(1 + r) 46 www.notesolution.com FV = $1,284 = $625(1.08); t t = ln($1,284/ $625) / ln 1.08 = 9.36 yrs FV = $4,341 = $810(1.07); t t = ln($4,341/ $810) / ln 1.07 = 24.81 yrs t FV = $402,662 = $18,400(1.21); t = ln($402,662 / $18,400) / ln 1.21 = 16.19 yrs FV = $173,439 = $21,500(1.29); t t = ln($173,439 / $21,500) / ln 1.29 = 8.20 yrs 6. 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: t FV = PV(1 + r) Solving for r, we get: r = (FV / PV) 1 /– 1 1/18 r = ($80,000 / $15,000) – 1 = 9.75% 7. 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: t = ln(FV / PV) / ln(1 + r) The length of time to double your money is: t FV = $2 = $1(1.07) t = ln 2 / ln 1.07 = 10.24 years The length of time to quadruple your money is: FV = $4 = $1(1.07) t t = ln 4 / ln 1.07 = 20.49 years Notice that the length of time to quadruple your money is twice as long as the time needed to double your money (the difference in these answers is due to rounding). This is an important concept of time value of money. 8. 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: t FV = PV(1 + r) Solving for r, we get: r = (FV / PV) 1 /– 1 1/5 r = ($28,835 / $21,608) – 1 = 5.94% 9. 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: 47 www.notesolution.com t FV = PV(1 + r) Solving for t, we get: t = ln(FV / PV) / ln(1 + r) t = ln ($150,000 / $40,000) / ln 1.055 = 24.69 years 10. To find the PV of a lump sum, we use: t PV = FV / (1 + r) 20 PV = $800,000,000 / (1.095) = $130,258,959.12 11. To find the PV of a lump sum, we use: t PV = FV / (1 + r) PV = $1M / (1.10) = $488.19 12. To find the FV of a lump sum, we use: t FV = PV(1 + r) FV = $50(1.05) 102 = $7,249.01 13. 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 = ($1,080,000 / $150) 1/108– 1 = 8.57% To find the FV of the first prize, we use: FV = PV(1 + r) t FV = $1,080,000(1.0857) = $22,642,130.85 (as mentioned initially, a more exact rate was used to obtain this solution than the one displayed) 14. To find the PV of a lump sum, we use: PV = FV / (1 + r) t PV = $350,000 / (1.2609) = $0.10 15. 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 48 www.notesolution.com r = ($10,311,500 / $12,377,500) – 1 = – 4.46% Notice that the interest rate is negative. This occurs when the FV is less than the PV. Intermediate 16. a. PV = $100/ (1 + r) = $29.19; r = 6.35% b. PV = $35 / (1 + r) = $29.19; r = 19.90% c. PV = $100 / (1 + r) = $35; r = 5.68% 17. To find the PV of a lump sum, we
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