Business Finance Solution
Content
CHAPTER 5
INTRODUCTION TO VALUATION: THE
TIME VALUE OF MONEY
Solutions to Questions and Problems
1.The difference is:
$10,794.62 – 9,000 = $1,794.62
2.
3.
FV = PV(1 + r)t
FV = $2,250(1.10)11
FV = $8,752(1.08)7
FV = $76,355(1.17)14
FV = $183,796(1.07)8
= $ 6,419.51
= $ 14,999.39
= $687,764.17
= $315,795.75
To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $15,451 / (1.07)6
= $ 10,295.65
7
PV = $51,557 / (1.13)
= $ 21,914.85
PV = $886,073 / (1.14)23
= $ 43,516.90
PV = $550,164 / (1.09)18
= $116,631.32
4.
r = ($297 / $240)1/2 – 1 = 11.24%
r = ($1,080 / $360)1/10 – 1 = 11.61%
r = ($185,382 / $39,000)1/15 – 1
= 10.95%
r = ($531,618 / $38,261)1/30 – 1
= 9.17%
5.
t = ln($1,284/ $560) / ln 1.09 = 9.63 years
t = ln($4,341/ $810) / ln 1.10 = 17.61 years
t = ln($364,518 / $18,400) / ln 1.17 = 19.02 years
t = ln($173,439 / $21,500) / ln 1.15 = 14.94 years
6.
r = 9.68%
7.
t = 10.24 years
t = 20.49 years
8.
r =.0666 or 6.66%
9.
t = 28.02 years
10.
PV = $650,000,000 / (1.074)20 = $155,893,400.13
11.
PV = $1,000,000 / (1.10)80 = $488.19
12.
FV $5,083.71
13. r =.0840 or 8.40%
FV = $1,260,000(1.0840)33 = $18,056,409.94
14. r =.0990 or 9.90%
15.
r =– 4.46%
Notice that the interest rate is negative. This occurs when the FV is less than the PV.
16. a. r = 4.86%
b. r = 3.93%
c. r = 5.48%
17.
PV =$61,303.70
18. FV = $438,120.97
FV = $4,000(1.11)35 = $154,299.40
19.
FV = $32,449.33
20.
t = 19.31
2 years + 19.31 years = 21.31 years
CHAPTER 6
DISCOUNTED CASH FLOW VALUATION
1.
PV@10%=$3,306.37
PV@18% = $2,794.22
PV@24% = $2,489.88
2.
X@5%:
PVA = $6,000{[1 – (1/1.05)9 ] / .05 } = $42,646.93
Y@5%:
PVA = $8,000{[1 – (1/1.05)6 ] / .05 } = $40,605.54
And at a 15 percent interest rate:
X@15%: PVA =$28,629.50
Y@15%: PVA = $30,275.86
Notice that the PV of cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a
15 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.
FV@8%
=$5,307.71
FV@11% = $5,520.96
FV@24% = $6,534.81
4.
PVA@15 yrs:
= $48,271.94
PVA@40 yrs:
= $70,658.06
PVA@75 yrs:
PVA = $5,300{[1 – (1/1.07)75 ] / .07} = $75,240.70
PV = $5,300 / .07 = $75,714.29
5. C = $3,887.72
6.
PVA =$411,660.36
7.
FVA for 20 years = $262,781.16
FVA for 40 years =$2,459,072.63
8.
C = $6,575.77
9.
C = $9,440.02
10. PV = $347,222.22
11. r =.0667 or 6.67%
12. EAR = [1 + (.08 / 4)]4 – 1
EAR = [1 + (.16 / 12)]12 – 1
= .0824 or 8.24%
= .1723 or 17.23%
EAR = [1 + (.12 / 365)]365 – 1 = .1275 or 12.75%
To find the EAR with continuous compounding, we use the equation:
EAR = eq – 1
EAR = e.15 – 1 = .1618 or 16.18%
13.
APR= .0842 or 8.42%
APR = .1820 or 18.20%
APR = .0899 or 8.99%
Solving the continuous compounding EAR equation:
EAR = eq – 1
We get:
APR = ln(1 + EAR)
APR = ln(1 + .1650)
APR = .1527 or 15.27%
14. For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)]m – 1
So, for each bank, the EAR is:
First National: EAR = [1 + (.1420 / 12)]12 – 1 = .1516 or 15.16%
First United:
EAR = [1 + (.1450 / 2)]2 – 1 = .1503 or 15.03%
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.
APR = 365[(1.16)1/365 – 1] = .1485 or 14.85%.
16. FV = $8,505.93
17. FV in 5 years = $7,163.64
FV in 10 years = $11,403.94
FV in 20 years = $28,899.97
18. PV = $28,804.71
19. = 2,229.81%
20. C = $1,353.15
EAR = 7.12%
21. t = 48.86 months
22. r = 33.33% per week
The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks
in a year, so:
APR = (52)33.33% = 1,733.33%
And using the equation to find the EAR:
EAR = [1 + (APR / m)]m – 1
EAR = [1 + .3333]52 – 1 = 313,916,515.69%
23. r = 1.89% per month
APR = (12)1.89% = 22.74%
And using the equation to find an EAR:
EAR = [1 + (APR / m)]m – 1
EAR = [1 + .0189]12 – 1 = 25.26%
24. FVA =$678,146.38
25. $647,623.45
26.$34,843.71
27. = $2,320.36
28. $6,570.86
29.
r = 5.45%
30. Here we need to convert an EAR into interest rates for different compounding periods. Using the
equation for the EAR, we get:
EAR = [1 + (APR / m)]m – 1
31.
32.
33.
34.
35.
36.
EAR = .17 = (1 + r)2 – 1;
r = (1.17)1/2 – 1
= .0817 or 8.17% per six months
EAR = .17 = (1 + r)4 – 1;
r = (1.17)1/4 – 1
= .0400 or 4.00% per quarter
EAR = .17 = (1 + r)12 – 1;
r = (1.17)1/12 – 1
= .0132 or 1.32% per month
FV = $5,037.62
FV = $5,508.35
Interest = $508.35
Stock account: FVA = $700[{[1 + (.11/12) ]360 – 1} / (.11/12)] = $1,963,163.82
Bond account: FVA = $300[{[1 + (.06/12) ]360 – 1} / (.06/12)] = $301,354.51
So, the total amount saved at retirement is = $2,264,518.33
PVA = $2,264,518.33
C = $19,003.763 withdrawal per month
FV in one year = $1.15
FV in two years = $1.32
EAR = .1498 or 14.98%
FV in one year = $1.15
FV in two years = $1.32
Starting today:
Starting in 10 years:
FVA = $85.00
FVA = $286.13
Starting in 20 years:
FVA = $1,010.86
FV = $3
r = .3161 or 31.61%
Since we have an APR compounded monthly and an annual payment, we must first convert the
interest rate to an EAR so that the compounding period is the same as the cash flows.
EAR = .104713 or 10.4713%
PVA1 = $163,839.09
PVA2 = $165,723.54
37. PV = $19,016,563.18
38. Next year’s salary = $52,000
Next year’s deposit = $2,600
PV = $34,399.45
FV = $2,235,994.31
39. The relationship between the PVA and the interest rate is:
PVA@10% = $68,454.72
PVA@5% = $93,416.92
PVA@15% = $52,626.33
40. FVA = $20,000
Solving for t, we get:
t = 51.69 payments
41. PVA = $73,000
APR = 7.13%
42. PVA = $184,817.42
The monthly payments of $1,150 will amount to a principal payment of $184,817.42. The amount
of principal you will still owe is:
$55,182.58
Balloon payment = $368,936.54
43. PV of Year 1 CF: = $1,545.45
PV of Year 3 CF: = $1,577.76
PV of Year 4 CF: = $1,912.44
So, the PV of the missing CF is: = $1,514.35
The value of the missing CF is: = $1,832.36
44. PV = $22,812,873.40
45. Amount borrowed = $2,320,000
Using the PVA equation:
PVA = $2,320,000
APR = 6.72%
And the EAR is:
EAR = (1 + .00560)12 – 1 = .0693 or 6.93%
46. PV = $101,197.59
Profit = $7,197.59
r = .1510 or 15.10%
47. PVA = $32,805.65
PV = $16,834.48
48. PVA2 = $110,021.35
PV = $51,120.33
PVA1 = $87,604.36
The value of the cash flows today is the sum of these two cash flows, so:
PV = $138,724.68
49. FVA = $434,143.62
PV = $130,761.55
50. PV = $56,451.61
PV = $37,051.41
51. PVA = $25,000
APR = 28.33%
And the EAR is:
EAR = .3231 or 32.31%
52. Monthly rate = .00833
Semiannual rate =.0511 or 5.11%
The PV of the annuity is:
PVA @ year 8: = $53,776.72
PV @ year 5: = $39,888.33
EAR =.1047 or 10.47%
PV @ year 5: = $39,888.33
PV @ year 3: = $32,684.88
PV @ year 3: = $32,684.88
PV @ year 0: = $24,243.67
PV @ year 0: = $24,243.67
53. a. PVA = $36,958.97
PVAdue = $41,024.46
FVA = $62,278.01
FVAdue = $69,128.60
54. We need to use the PVA due equation, that is:
C = $1,364.99
55. C = $10,519.17
Total interest over life of the loan = $10,595.86
56. Total interest over life of the loan = $10,080
57. APR = .0957 or 9.57%
APR = .0678 or 6.78%
PVA = $2,885,496.45
PV = $165,824.26
So, at retirement, he needs: $3,051,320.71
FVA = $499,659.64
After he purchases the cabin, the amount he will have left is: $119,659.64
FV = $805,010.23
So, when he is ready to retire, based on his current savings, he will be short: $2,246,310.48
FVA = $2,246,310.48 C = $3,127.44
58. PV = $14,672.91
PV = $18,654.82
The PV of the decision to purchase is: $13,345.18
PV of resale price = $17,327.09
Breakeven resale price = $21,363.01
59. EAR = 5.65%
PV = $38,610,482.57
PV = $40,010,482.57
$31,010,482.57
Effective quarterly rate =.01384 or 1.384%
PVA = $31,010,482.57
C = $1,527,463.76
60. r = .1765 or 17.65%
61. APR = .0772 or 7.72%
FVA = $48,699.39
FV = $52,595.34
FVA = $51,807.86
PVA = $227,539.14
Award = $451,942.34
b.
INTRODUCTION TO VALUATION: THE
TIME VALUE OF MONEY
Solutions to Questions and Problems
1.The difference is:
$10,794.62 – 9,000 = $1,794.62
2.
3.
FV = PV(1 + r)t
FV = $2,250(1.10)11
FV = $8,752(1.08)7
FV = $76,355(1.17)14
FV = $183,796(1.07)8
= $ 6,419.51
= $ 14,999.39
= $687,764.17
= $315,795.75
To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $15,451 / (1.07)6
= $ 10,295.65
7
PV = $51,557 / (1.13)
= $ 21,914.85
PV = $886,073 / (1.14)23
= $ 43,516.90
PV = $550,164 / (1.09)18
= $116,631.32
4.
r = ($297 / $240)1/2 – 1 = 11.24%
r = ($1,080 / $360)1/10 – 1 = 11.61%
r = ($185,382 / $39,000)1/15 – 1
= 10.95%
r = ($531,618 / $38,261)1/30 – 1
= 9.17%
5.
t = ln($1,284/ $560) / ln 1.09 = 9.63 years
t = ln($4,341/ $810) / ln 1.10 = 17.61 years
t = ln($364,518 / $18,400) / ln 1.17 = 19.02 years
t = ln($173,439 / $21,500) / ln 1.15 = 14.94 years
6.
r = 9.68%
7.
t = 10.24 years
t = 20.49 years
8.
r =.0666 or 6.66%
9.
t = 28.02 years
10.
PV = $650,000,000 / (1.074)20 = $155,893,400.13
11.
PV = $1,000,000 / (1.10)80 = $488.19
12.
FV $5,083.71
13. r =.0840 or 8.40%
FV = $1,260,000(1.0840)33 = $18,056,409.94
14. r =.0990 or 9.90%
15.
r =– 4.46%
Notice that the interest rate is negative. This occurs when the FV is less than the PV.
16. a. r = 4.86%
b. r = 3.93%
c. r = 5.48%
17.
PV =$61,303.70
18. FV = $438,120.97
FV = $4,000(1.11)35 = $154,299.40
19.
FV = $32,449.33
20.
t = 19.31
2 years + 19.31 years = 21.31 years
CHAPTER 6
DISCOUNTED CASH FLOW VALUATION
1.
PV@10%=$3,306.37
PV@18% = $2,794.22
PV@24% = $2,489.88
2.
X@5%:
PVA = $6,000{[1 – (1/1.05)9 ] / .05 } = $42,646.93
Y@5%:
PVA = $8,000{[1 – (1/1.05)6 ] / .05 } = $40,605.54
And at a 15 percent interest rate:
X@15%: PVA =$28,629.50
Y@15%: PVA = $30,275.86
Notice that the PV of cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a
15 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.
FV@8%
=$5,307.71
FV@11% = $5,520.96
FV@24% = $6,534.81
4.
PVA@15 yrs:
= $48,271.94
PVA@40 yrs:
= $70,658.06
PVA@75 yrs:
PVA = $5,300{[1 – (1/1.07)75 ] / .07} = $75,240.70
PV = $5,300 / .07 = $75,714.29
5. C = $3,887.72
6.
PVA =$411,660.36
7.
FVA for 20 years = $262,781.16
FVA for 40 years =$2,459,072.63
8.
C = $6,575.77
9.
C = $9,440.02
10. PV = $347,222.22
11. r =.0667 or 6.67%
12. EAR = [1 + (.08 / 4)]4 – 1
EAR = [1 + (.16 / 12)]12 – 1
= .0824 or 8.24%
= .1723 or 17.23%
EAR = [1 + (.12 / 365)]365 – 1 = .1275 or 12.75%
To find the EAR with continuous compounding, we use the equation:
EAR = eq – 1
EAR = e.15 – 1 = .1618 or 16.18%
13.
APR= .0842 or 8.42%
APR = .1820 or 18.20%
APR = .0899 or 8.99%
Solving the continuous compounding EAR equation:
EAR = eq – 1
We get:
APR = ln(1 + EAR)
APR = ln(1 + .1650)
APR = .1527 or 15.27%
14. For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)]m – 1
So, for each bank, the EAR is:
First National: EAR = [1 + (.1420 / 12)]12 – 1 = .1516 or 15.16%
First United:
EAR = [1 + (.1450 / 2)]2 – 1 = .1503 or 15.03%
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.
APR = 365[(1.16)1/365 – 1] = .1485 or 14.85%.
16. FV = $8,505.93
17. FV in 5 years = $7,163.64
FV in 10 years = $11,403.94
FV in 20 years = $28,899.97
18. PV = $28,804.71
19. = 2,229.81%
20. C = $1,353.15
EAR = 7.12%
21. t = 48.86 months
22. r = 33.33% per week
The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks
in a year, so:
APR = (52)33.33% = 1,733.33%
And using the equation to find the EAR:
EAR = [1 + (APR / m)]m – 1
EAR = [1 + .3333]52 – 1 = 313,916,515.69%
23. r = 1.89% per month
APR = (12)1.89% = 22.74%
And using the equation to find an EAR:
EAR = [1 + (APR / m)]m – 1
EAR = [1 + .0189]12 – 1 = 25.26%
24. FVA =$678,146.38
25. $647,623.45
26.$34,843.71
27. = $2,320.36
28. $6,570.86
29.
r = 5.45%
30. Here we need to convert an EAR into interest rates for different compounding periods. Using the
equation for the EAR, we get:
EAR = [1 + (APR / m)]m – 1
31.
32.
33.
34.
35.
36.
EAR = .17 = (1 + r)2 – 1;
r = (1.17)1/2 – 1
= .0817 or 8.17% per six months
EAR = .17 = (1 + r)4 – 1;
r = (1.17)1/4 – 1
= .0400 or 4.00% per quarter
EAR = .17 = (1 + r)12 – 1;
r = (1.17)1/12 – 1
= .0132 or 1.32% per month
FV = $5,037.62
FV = $5,508.35
Interest = $508.35
Stock account: FVA = $700[{[1 + (.11/12) ]360 – 1} / (.11/12)] = $1,963,163.82
Bond account: FVA = $300[{[1 + (.06/12) ]360 – 1} / (.06/12)] = $301,354.51
So, the total amount saved at retirement is = $2,264,518.33
PVA = $2,264,518.33
C = $19,003.763 withdrawal per month
FV in one year = $1.15
FV in two years = $1.32
EAR = .1498 or 14.98%
FV in one year = $1.15
FV in two years = $1.32
Starting today:
Starting in 10 years:
FVA = $85.00
FVA = $286.13
Starting in 20 years:
FVA = $1,010.86
FV = $3
r = .3161 or 31.61%
Since we have an APR compounded monthly and an annual payment, we must first convert the
interest rate to an EAR so that the compounding period is the same as the cash flows.
EAR = .104713 or 10.4713%
PVA1 = $163,839.09
PVA2 = $165,723.54
37. PV = $19,016,563.18
38. Next year’s salary = $52,000
Next year’s deposit = $2,600
PV = $34,399.45
FV = $2,235,994.31
39. The relationship between the PVA and the interest rate is:
PVA@10% = $68,454.72
PVA@5% = $93,416.92
PVA@15% = $52,626.33
40. FVA = $20,000
Solving for t, we get:
t = 51.69 payments
41. PVA = $73,000
APR = 7.13%
42. PVA = $184,817.42
The monthly payments of $1,150 will amount to a principal payment of $184,817.42. The amount
of principal you will still owe is:
$55,182.58
Balloon payment = $368,936.54
43. PV of Year 1 CF: = $1,545.45
PV of Year 3 CF: = $1,577.76
PV of Year 4 CF: = $1,912.44
So, the PV of the missing CF is: = $1,514.35
The value of the missing CF is: = $1,832.36
44. PV = $22,812,873.40
45. Amount borrowed = $2,320,000
Using the PVA equation:
PVA = $2,320,000
APR = 6.72%
And the EAR is:
EAR = (1 + .00560)12 – 1 = .0693 or 6.93%
46. PV = $101,197.59
Profit = $7,197.59
r = .1510 or 15.10%
47. PVA = $32,805.65
PV = $16,834.48
48. PVA2 = $110,021.35
PV = $51,120.33
PVA1 = $87,604.36
The value of the cash flows today is the sum of these two cash flows, so:
PV = $138,724.68
49. FVA = $434,143.62
PV = $130,761.55
50. PV = $56,451.61
PV = $37,051.41
51. PVA = $25,000
APR = 28.33%
And the EAR is:
EAR = .3231 or 32.31%
52. Monthly rate = .00833
Semiannual rate =.0511 or 5.11%
The PV of the annuity is:
PVA @ year 8: = $53,776.72
PV @ year 5: = $39,888.33
EAR =.1047 or 10.47%
PV @ year 5: = $39,888.33
PV @ year 3: = $32,684.88
PV @ year 3: = $32,684.88
PV @ year 0: = $24,243.67
PV @ year 0: = $24,243.67
53. a. PVA = $36,958.97
PVAdue = $41,024.46
FVA = $62,278.01
FVAdue = $69,128.60
54. We need to use the PVA due equation, that is:
C = $1,364.99
55. C = $10,519.17
Total interest over life of the loan = $10,595.86
56. Total interest over life of the loan = $10,080
57. APR = .0957 or 9.57%
APR = .0678 or 6.78%
PVA = $2,885,496.45
PV = $165,824.26
So, at retirement, he needs: $3,051,320.71
FVA = $499,659.64
After he purchases the cabin, the amount he will have left is: $119,659.64
FV = $805,010.23
So, when he is ready to retire, based on his current savings, he will be short: $2,246,310.48
FVA = $2,246,310.48 C = $3,127.44
58. PV = $14,672.91
PV = $18,654.82
The PV of the decision to purchase is: $13,345.18
PV of resale price = $17,327.09
Breakeven resale price = $21,363.01
59. EAR = 5.65%
PV = $38,610,482.57
PV = $40,010,482.57
$31,010,482.57
Effective quarterly rate =.01384 or 1.384%
PVA = $31,010,482.57
C = $1,527,463.76
60. r = .1765 or 17.65%
61. APR = .0772 or 7.72%
FVA = $48,699.39
FV = $52,595.34
FVA = $51,807.86
PVA = $227,539.14
Award = $451,942.34
b.
