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study guide

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Department
Finance
Course Code
FIN 300
Professor
Scott Anderson

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CHAPTER 6
DISCOUNTED CASH FLOW VALUATION
Answers to Concepts Review and Critical Thinking Questions
1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and the number of
payments, or the life of the annuity, t.
2. Assuming positive cash flows, both the present and the future values will rise.
3. Assuming positive cash flows, the present value will fall and the future value will rise.
4. It’s deceptive, but very common. The basic concept of time value of money is that a dollar today is not worth
the same as a dollar tomorrow. The deception is particularly irritating given that such lotteries are usually
government sponsored!
5. If the total money is fixed, you want as much as possible as soon as possible. The team (or, more accurately,
the team owner) wants just the opposite.
6. The better deal is the one with equal installments.
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. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we
use:
PV = FV / (1 + r)t
PV@10% = $1,200 / 1.10 + $600 / 1.102 + $855 / 1.103 + $1,480 / 1.104 = $3,240.01
PV@18% = $1,200 / 1.18 + $600 / 1.182 + $855 / 1.183 + $1,480 / 1.184 = $2,731.61
PV@24% = $1,200 / 1.24 + $600 / 1.242 + $855 / 1.243 + $1,480 / 1.244 = $2,432.40
2. To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r)]t } / r )
At a 5 percent interest rate:
X@5%: PVA = $4,000{[1 – (1/1.05)9 ] / .05 } = $28,431.29
Y@5%: PVA = $6,000{[1 – (1/1.05)5 ] / .05 } = $25,976.86
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And at a 22 percent interest rate:
X@22%: PVA = $4,000{[1 – (1/1.22)9 ] / .22 } = $15,145.14
Y@22%:PVA = $6,000{[1 – (1/1.22)5 ] / .22 } = $17,181.84
Notice that the PV of Cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a 22 percent
interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is
more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more
valuable since it has larger cash flows. At the higher interest rate, these bigger cash flows early are more
important since the cost of waiting (the interest rate) is so much greater.
3. To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a lump sum, we
use:
FV = PV(1 + r)t
FV@8% = $800(1.08)3 + $900(1.08)2 + $1,000(1.08) + $1,100 = $4,237.53
FV@11% = $800(1.11)3 + $900(1.11)2 + $1,000(1.11) + $1,100 = $4,412.99
FV@24% = $800(1.24)3 + $900(1.24)2 + $1,000(1.24) + $1,100 = $5,249.14
Notice we are finding the value at Year 4, the cash flow at Year 4 is simply added to the FV of the other cash
flows. In other words, we do not need to compound this cash flow.
4. To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r)]t } / r )
PVA@15 yrs: PVA = $3,600{[1 – (1/1.10)15 ] / .10} = $27,381.89
PVA@40 yrs: PVA = $3,600{[1 – (1/1.10)40 ] / .10} = $35,204.58
PVA@75 yrs: PVA = $3,600{[1 – (1/1.10)75 ] / .10} = $35,971.70
To find the PV of a perpetuity, we use the equation:
PV = C / r
PV = $3,600 / .10 = $36,000.00
Notice that as the length of the annuity payments increases, the present value of the annuity approaches the
present value of the perpetuity. The present value of the 75 year annuity and the present value of the perpetuity
imply that the value today of all perpetuity payments beyond 75 years is only $28.30.
5. Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity
payment. Using the PVA equation:
PVA = C({1 – [1/(1 + r)]t } / r )
PVA = $28,000 = $C{[1 – (1/1.0765)14 ] / .0765}
We can now solve this equation for the annuity payment. Doing so, we get:
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C = $28,000 / 8.4145 = $3,327.58
6. To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r)]t } / r )
PVA = $80,000{[1 – (1/1.082)8 ] / .082} = $456,262.25
7. Here we need to find the FVA. The equation to find the FVA is:
FVA = C{[(1 + r)t – 1] / r}
FVA for 20 years = $2,000[(1.10520 – 1) / .105] = $121,261.62
FVA for 40 years = $2,000[(1.10540 – 1) / .105] = $1,014,503.16
Notice that doubling the number of periods does not double the FVA.
8. Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity
payment. Using the FVA equation:
FVA = C{[(1 + r)t – 1] / r}
$80,000 = $C[(1.05810 – 1) / .058]
We can now solve this equation for the annuity payment. Doing so, we get:
C = $80,000 / 13.05765 = $6,126.68
9. Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity
payment. Using the PVA equation:
PVA = C({1 – [1/(1 + r)]t } / r)
$40,000 = C{[1 – (1/1.09)7 ] / .09}
We can now solve this equation for the annuity payment. Doing so, we get:
C = $40,000 / 5.03295 = $7,947.62
10. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
PV = C / r
PV = $15,000 / .08 = $187,500.00
11. Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows.
Using the PV of a perpetuity equation:
PV = C / r
$195,000 = $15,000 / r
We can now solve for the interest rate as follows:
r = $15,000 / $195,000 = 7.69%
53
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Description
CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and the number of payments, or the life of the annuity, t. 2. Assuming positive cash flows, both the present and the future values will rise. 3. Assuming positive cash flows, the present value will fall and the future value will rise. 4. Its deceptive, but very common. The basic concept of time value of money is that a dollar today is not worth the same as a dollar tomorrow. The deception is particularly irritating given that such lotteries are usually government sponsored! 5. If the total money is fixed, you want as much as possible as soon as possible. The team (or, more accurately, the team owner) wants just the opposite. 6. The better deal is the one with equal installments. 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. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we use: PV = FV / (1 + r)t [email protected]% = $1,200 / 1.10 + $600 / 1.10 + $855 / 1.10 + $1,480 / 1.10 = $3,240.01 2 3 4 [email protected]% = $1,200 / 1.18 + $600 / 1.18 + $855 / 1.18 + $1,480 / 1.18 = $2,731.61 [email protected]% = $1,200 / 1.24 + $600 / 1.24 + $855 / 1.24 + $1,480 / 1.24 = $2,432.40 2. To find the PVA, we use the equation: t PVA = C({1 [1/(1 + r)] } / r ) At a 5 percent interest rate: 9 [email protected]%: PVA = $4,000{[1 (1/1.05) ] / .05 } = $28,431.29 5 [email protected]%: PVA = $6,000{[1 (1/1.05) ] / .05 } = $25,976.86 51 www.notesolution.com And at a 22 percent interest rate: [email protected]%: PVA = $4,000{[1 (1/1.22) ] / .22 } = $15,145.14 5 [email protected]%: PVA = $6,000{[1 (1/1.22) ] / .22 } = $17,181.84 Notice that the PV of Cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a 22 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more valuable since it has larger cash flows. At the higher interest rate, these bigger cash flows early are more important since the cost of waiting (the interest rate) is so much greater. 3. To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a lump sum, we use: FV = PV(1 + r) t 3 2 [email protected]% = $800(1.08) + $900(1.08) + $1,000(1.08) + $1,100 = $4,237.53 3 2 [email protected]% = $800(1.11) + $900(1.11) + $1,000(1.11) + $1,100 = $4,412.99 [email protected]% = $800(1.24) + $900(1.24) + $1,000(1.24) + $1,100 = $5,249.14 Notice we are finding the value at Year 4, the cash flow at Year 4 is simply added to the FV of the other cash flows. In other words, we do not need to compound this cash flow. 4. To find the PVA, we use the equation: PVA = C({1 [1/(1 + r)] } / r ) [email protected] yrs: PVA = $3,600{[1 (1/1.10) ] / .10} = $27,381.89 40 [email protected] yrs: PVA = $3,600{[1 (1/1.10) ] / .10} = $35,204.58 [email protected] yrs: PVA = $3,600{[1 (1/1.10) ] / .10} = $35,971.70 To find the PV of a perpetuity, we use the equation: PV = C / r PV = $3,600 / .10 = $36,000.00 Notice that as the length of the annuity payments increases, the present value of the annuity approaches the present value of the perpetuity. The present value of the 75 year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years is only $28.30. 5. Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the PVA equation: PVA = C({1 [1/(1 + r)] } / r ) PVA = $28,000 = $C{[1 (1/1.0765) ] / .0765} We can now solve this equation for the annuity payment. Doing so, we get: 52 www.notesolution.com C = $28,000 / 8.4145 = $3,327.58 6. To find the PVA, we use the equation: PVA = C({1 [1/(1 + r)] } / r ) 8 PVA = $80,000{[1 (1/1.082) ] / .082} = $456,262.25 7. Here we need to find the FVA. The equation to find the FVA is: FVA = C{[(1 + r) 1] / r} 20 FVA for 20 years = $2,000[(1.105 1) / .105] = $121,261.62 FVA for 40 years = $2,000[(1.105 1) / .105] = $1,014,503.16 Notice that doubling the number of periods does not double the FVA. 8. Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the FVA equation: t FVA = C{[(1 + r) 1] / r} $80,000 = $C[(1.058 1) / .058] We can now solve this equation for the annuity payment. Doing so, we get: C = $80,000 / 13.05765 = $6,126.68 9. Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the PVA equation: t PVA = C({1 [1/(1 + r)] } / r) $40,000 = C{[1 (1/1.09) ] / .09} We can now solve this equation for the annuity payment. Doing so, we get: C = $40,000 / 5.03295 = $7,947.62 10. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation: PV = C / r PV = $15,000 / .08 = $187,500.00 11. Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using the PV of a perpetuity equation: PV = C / r $195,000 = $15,000 / r We can now solve for the interest rate as follows: r = $15,000 / $195,000 = 7.69% 53 www.notesolution.com 12. For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m)] 1 EAR = [1 + (.11 / 4)] 1 = 11.46% 12 EAR = [1 + (.07 / 12)] 1 = 7.23% 365 EAR = [1 + (.09 / 365)] 1 = 9.42% To find the EAR with continuous compounding, we use the equation: q EAR = e 1 EAR = e 1 = 18.53% 13. Here we are given the EAR and need to find the APR. Using the equation for discrete compounding: m EAR = [1 + (APR / m)] 1 We can now solve for the APR. Doing so, we get: APR = m[(1 + EAR) 1/m 1] EAR = .081 = [1 + (APR / 2)] 1 APR = 2[(1.081) 1]2 = 7.94% 12 1/12 EAR = .076 = [1 + (APR / 12)] 1 APR = 12[(1.076) 1] = 7.35% 52 1/52 EAR = .168 = [1 + (APR / 52)] 1 APR = 52[(1.168) 1] = 15.55% Solving the continuous compounding EAR equation: EAR = e 1 We get: APR = ln(1 + EAR) APR = ln(1 + .262) APR = 23.27% 14. For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m)] 1 So, for each bank, the EAR is: 12 First National: EAR = [1 + (.122 / 12)] 1 = 12.91% First United: EAR = [1 + (.124 / 2)] 1 = 12.78% Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding periods within a year will also affect the EAR. 15. The reported rate is the APR, so we need to convert the EAR to an APR as follows: 54 www.notesolution.com
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