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CHAPTER 4 -
21
42. The time line is:
0
3
PV
$165,000
The profit the firm earns is the PV of the sales price minus the cost to produce the asset. We find the
43. The time line is:
0
1
…
5
6
…
25
$8,500
$8,500
$8,500
$8,500
44. The time line for the annuity is:
0
1
…
180
$1,940
$1,940
$1,940
$1,940
$1,940
$1,940
$1,940
$1,940
$1,940
This question is asking for the present value of an annuity, but the interest rate changes during the
45. The time line for the annuity is:
0
1
…
180
$1,175
$1,175
$1,175
$1,175
$1,175
$1,175
$1,175
$1,175
$1,175
FV
Here, we are trying to find the dollar amount invested today that will equal the FVA with a known
46. The time line is:
0
1
…
7
…
14
15
…
∞
PV
$2,350
$2,350
$2,350
$2,350
To find the value of the perpetuity at t = 7, we first need to use the PV of a perpetuity equation. Using
47. The time line is:
0
1
…
12
–$26,000
$2,519.83
$2,519.83
$2,519.83
$2,519.83
$2,519.83
$2,519.83
$2,519.83
$2,519.83
$2,519.83
To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the interest
rate quoted in the problem is only relevant to determine the total interest under the terms given. The
interest rate for the cash flows of the loan is:
48. The time line is:
0
1
…
…
18
19
…
28
$5,230
$5,230
$5,230
$5,230
The cash flows in this problem are semiannual, so we need the effective semiannual rate. The
interest rate given is the APR, so the monthly interest rate is:
49. a. The time line for the ordinary annuity is:
0
1
2
3
4
5
PV
$13,250
$13,250
$13,250
$13,250
$13,250
CHAPTER 4 -
25
If the payments are in the form of an ordinary annuity, the present value will be:
b. The time line for the ordinary annuity is:
0
1
2
3
4
5
FV
$13,250
$13,250
$13,250
$13,250
$13,250
We can find the future value of the ordinary annuity as:
FVA = C{[(1 + r)t – 1]/r}
c. Assuming a positive interest rate, the present value of an annuity due will always be larger than
the present value of an ordinary annuity. Each cash flow in an annuity due is received one period
50. The time line is:
0
1
…
59
60
–$84,000
C
C
C
C
C
C
C
C
C
51. The time line is:
0
1
…
23
24
–$3,350
C
C
C
C
C
C
C
C
C
52. The time line is:
0
1
…
15
16
17
18
19
20
$72,000
$72,000
$72,000
$72,000
$72,000
$72,000
$72,000
$72,000
C
C
C
C
First, we will calculate the present value of the college expenses for each child. The expenses are an
CHAPTER 4 -
27
PVA = $72,000({1 – [1/(1 + .079)]4}/.079)
PVA = $239,004.91
This is the cost of each child’s college expenses one year before they enter college. So, the cost of the
oldest child’s college expenses today will be:
53. The salary is a growing annuity, so we use the equation for the present value of a growing annuity.
The salary growth rate is 3.8 percent and the discount rate is 7.1 percent, so the value of the salary
offer today is:
PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}
PV = $71,000{[1/(.071 – .038)] – [1/(.071 – .038)] × [(1 + .038)/(1 + .071)]25}
54. Here, we need to compare two options. In order to do so, we must get the value of the two cash flow
streams to the same time, so we will find the value of each today. We must also make sure to use the
aftertax cash flows, since they are more relevant. For Option A, the aftertax cash flows are:
So, the cash flows are:
0
1
…
30
31
PV
$180,000
$180,000
$180,000
$180,000
$180,000
$180,000
$180,000
$180,000
$180,000
The aftertax cash flows from Option A are in the form of an annuity due, so the present value of the
cash flow today is:
The cash flows are:
0
1
…
29
30
PV
$530,000
$144,000
$144,000
$144,000
$144,000
$144,000
$144,000
$144,000
$144,000
$144,000
The aftertax cash flows from Option B are an ordinary annuity, plus the cash flow today, so the present
value is:
55. We need to find the first payment into the retirement account. The present value of the desired amount
at retirement is:
PV = FV/(1 + r)t
PV = $2,500,000/(1 + .094)30
PV = $168,818.62
56. Since she put $1,500 down, the amount borrowed will be:
Amount borrowed = $17,000 – 1,500
Amount borrowed = $15,500
So, the monthly payments will be:
PVA = C({1 – [1/(1 + r)]t}/r)
57. The time line is:
0
1
…
120
…
360
361
…
600
–$1,800
–$1,800
$17,500
$17,500
$350,000
C
C
C
$1,500,000
The cash flows for this problem occur monthly, and the interest rate given is the EAR. Since the cash
PV = $321,822.31
So, at retirement, he needs:
$2,136,360.28 + 321,822.31 = $2,458,182.59
He will be saving $1,800 per month for the next 10 years until he purchases the cabin. The value of
his savings after 10 years will be:
58. To answer this question, we should find the PV of both options, and compare them. Since we are
purchasing the car, the lowest PV is the best option. The PV of the leasing option is the PV of the lease
payments, plus the $2,400. The interest rate we would use for the leasing option is the same as the
interest rate of the loan. The PV of leasing is:
0
1
…
36
59. To find the quarterly salary for the player, we first need to find the PV of the current contract. The
cash flows for the contract are annual, and we are given a daily interest rate. We need to find the EAR
so the interest compounding is the same as the timing of the cash flows. The EAR is:
EAR = [1 + (.057/365)]365 – 1
EAR = .0587, or 5.87%
60. The time line for the cash flows is:
0
1
–$16,720
$20,000
19.62 percent, not 16.4 percent.
CHAPTER 4 -
33
–24
–23
…
–12
–11
…
0
1
…
60
$3,250
$3,250
$3,583.33
$3,583.33
$3,916.67
$3,916.67
$3,916.67
$150,000
$25,000
Here, we have cash flows that would have occurred in the past and cash flows that would occur in the
Notice we found the FV of the annuity with the effective monthly rate, and then found the FV of the
lump sum with the EAR. Alternatively, we could have found the FV of the lump sum with the effective
monthly rate as long as we used 12 periods. The answer would be the same either way.
Now, we need to find the value today of last year’s back pay:
FVA = ($43,000/12)[{[ 1 + (.0716/12)]12 – 1}/(.0716/12)]
62. To find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is in
the form of a lump sum, the amount you will repay is the FV of the principal amount, which will be:
Loan repayment amount = $10,000(1.08)
Loan repayment amount = $10,800
The amount you will receive today is the principal amount of the loan times one minus the points.
63. First, we will find the APR and EAR for the loan with the refundable fee. Remember, we need to use
the actual cash flows of the loan to find the interest rate. With the $2,900 application fee, you will need
to borrow $302,900 to have $300,000 after deducting the fee. The time line is:
0
1
…
360
APR = 5.39%
EAR = (1 + .004489)12 – 1
APR = 5.30%
64. The time line is:
0
1
…
36
–$1,000
$46.11
$46.11
$46.11
$46.11
$46.11
$46.11
$46.11
$46.11
$46.11
Be careful of interest rate quotations. The actual interest rate of a loan is determined by the cash flows.
Here, we are told that the PV of the loan is $1,000, and the payments are $46.11 per month for three
years, so the interest rate on the loan is:
CHAPTER 4 -
36
65. We will calculate the number of periods necessary to repay the balance with no fee first. We need to
use the PVA equation and solve for the number of payments.
Without fee and annual rate = 18.6%:
PVA = $12,000 = $250{[1 – (1/1.0155)t]/.0155} where .0155 = .186/12
Without fee and annual rate = 8.2%:
PVA = $12,000 = $250{[1 – (1/1.006833)t ]/.006833} where .006833 = .082/12
You will pay off your account:
88.59 – 58.37 = 30.22 months quicker
Note that we do not need to calculate the time necessary to repay your current credit card with a fee
since no fee will be incurred. The time to repay the new card with a transfer fee is:
You will pay off your account:
66. We need to find the FV of the premiums to compare with the cash payment promised at age 65. We
have to find the value of the premiums at Year 6 first since the interest rate changes at that time. So:
FV1 = $500(1.11)5 = $842.53
FV2 = $600(1.11)4 = $910.84
Finding the FV of this lump sum at the child’s 65th birthday:
FV = $5,695.39(1.07)59
FV = $308,437.08
The policy is not worth buying; the future value of the policy is $308,437.08, but the policy contract
will pay off $300,000. The premiums are worth $8,437.08 more than the policy payoff.
Note, we could also compare the PV of the two cash flows. The PV of the premiums is:
The premiums still have the higher cash flow. At Year 0, the difference is $83.29. When you are
comparing two or more cash flow streams, the cash flow with the highest value at one time will have
the highest value at any other time.
Here is a question for you: Suppose you invest $83.29, the difference in the cash flows at time zero,
for 6 years at an 11 percent interest rate, and then for 59 years at a 7 percent interest rate. How much
will it be worth? Without doing calculations, you know it will be worth $8,437.08, the difference in
67. Since the payments occur at six month intervals, we need to get the effective six-month interest rate.
We can calculate the daily interest rate since we have an APR compounded daily, so the effective six-
month interest rate is:
Effective six-month rate = (1 + Daily rate)180 – 1
Effective six-month rate = (1 + .09/360)180 – 1
Effective six-month rate = .0460, or 4.60%
Now, we can use the PVA equation to find the present value of the semi-annual payments. Doing so,
we find:
68. Here, we need to find the interest rate that makes the PVA, the college costs, equal to the FVA, the
savings. The PV of the college costs is:
PVA = $20,000[{1 – [1/(1 + r)]4}/r]
r = 8.07%
CHAPTER 4 -
39
69. The time line is:
0
1
…
10
…
∞
–$25,000
–$25,000
–$25,000
–$25,000
–$25,000
–$25,000
–$25,000
–$25,000
$51,000
$51,000
$51,000
70. The time line is:
0
1
3
…
∞
$50,000
$50,000
$50,000
$50,000
$50,000
The cash flows in this problem occur every two years, so we need to find the effective two year rate.
One way to find the effective two year rate is to use an equation similar to the EAR, except use the
number of days in two years as the exponent. (We use the number of days in two years since it is daily
compounding; if monthly compounding was assumed, we would use the number of months in two
years.) So, the effective two-year interest rate is:
71. To solve for the PVA due:
PVA =
) (1
....
) (1
) (1 2t
r
C
r
C
r
C
+
++
+
+
+
72. a. The APR is the interest rate per week times 52 weeks in a year, so:
APR = 52(7.1%)
APR = 369.20%
b. In a discount loan, the amount you receive is lowered by the discount, and you repay the full
principal. With a discount of 7.1 percent, you would receive $9.29 for every $10 in principal, so
