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51. The payment for a loan repaid with equal payments is the annuity payment with the loan value as the
PV of the annuity. So, the loan payment will be:
PVA = C({1 – [1/(1 + r)]t }/r)
$57,000 = C{[1 – 1/(1 + .09)3]/.09}
C = $20,147.79
The interest payment is the beginning balance times the interest rate for the period, and the principal
payment is the total payment minus the interest payment. The ending balance is the beginning balance
minus the principal payment. The ending balance for a period is the beginning balance for the next
period. The amortization table for an equal payment is:
Year
Beginning
Balance
Total
Payment
Interest
Payment
Principal
Payment
Ending
Balance
1
$51,000.00
$20,147.79
$4,590.00
$15,557.79
$35,442.21
2
35,442.21
20,147.79
3,189.80
16,957.99
18,484.21
3
18,484.21
20,147.79
1,663.58
18,484.21
0
In the third year, $1,663.58 of interest is paid.
Total interest over life of the loan = $4,590 + 3,189.80 + 1,663.58
Total interest over life of the loan = $9,443.38
52. This amortization table calls for equal principal payments of $17,000 per year. The interest payment
is the beginning balance times the interest rate for the period, and the total payment is the principal
payment plus the interest payment. The ending balance for a period is the beginning balance for the
next period. The amortization table for an equal principal reduction is:
Year
Beginning
Balance
Total
Payment
Interest
Payment
Principal
Payment
Ending
Balance
1
$51,000.00
$21,590.00
$4,590.00
$17,000.00
$34,000.00
2
34,000.00
20,060.00
3,060.00
17,000.00
17,000.00
3
17,000.00
18,530.00
1,530.00
17,000.00
-
In the third year, $1,530 of interest is paid.
Challenge
53. The time line is:
0
1
…
23
24
–$3,100
C
C
C
C
C
C
C
C
C
The monthly interest rate is the annual interest rate divided by 12, or:
Monthly interest rate = .097/12
Monthly interest rate = .00808
Now we can set the present value of the lease payments equal to the cost of the equipment, or $3,100.
The lease payments are in the form of an annuity due, so:
PVAdue = (1 + r)C({1 – [1/(1 + r)t]}/r )
$3,100 = (1 + .00808)C({1 – [1/(1 + .00808)]24 }/.00808 )
C = $141.48
54. The time line is:
0
1
…
15
16
17
18
19
20
$55,000
$55,000
$55,000
$55,000
$55,000
$55,000
$55,000
$55,000
C
C
C
C
First, we will calculate the present value of the college expenses for each child. The expenses are an
annuity, so the present value of the college expenses is:
PV = $43,391.44
Therefore, the total cost today of your children’s college expenses is:
This is the present value of your annual savings, which are an annuity. So, the amount you must save
each year will be:
PVA = C({1 – [1/(1 + r)]t }/r )
$95,134.17 = C({1 – [1/(1 + .092)15]}/.092)
C = $11,941.94
55. The salary is a growing annuity, so we use the equation for the present value of a growing annuity.
The salary growth rate is 3.2 percent and the discount rate is 9 percent, so the value of the salary offer
today is:
PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}
PV = $65,000{[1/(.09 – .032)] – [1/(.09 – .032)] × [(1 + .032)/(1 + .09)]35}
PV = $955,361.30
The yearly bonuses are 10 percent of the annual salary. This means that next year’s bonus will be:
Next year’s bonus = .10($65,000)
of the bonus will always be the same percentage of the present value of the salary as the bonus
percentage. So, the total value of the offer is:
PV = PV(Salary) + PV(Bonus) + Bonus paid today
PV = $955,361.30 + 95,536.13 + 12,000
PV = $1,062,897.43
56. 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 it is more relevant. For Option A, the aftertax cash flows are:
Aftertax cash flows = Pretax cash flows(1 – tax rate)
Aftertax cash flows = $400,000(1 – .36)
Aftertax cash flows = $256,000
So, the cash flows are:
0
1
…
30
31
PV
$256,000
$256,000
$256,000
$256,000
$256,000
$256,000
$256,000
$256,000
$256,000
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:
PVAdue = (1 + r) C({1 – [1/(1 + r)t]}/r )
PVAdue = (1 + .045)$256,000({1 – [1/(1 + .045)]31 }/.045 )
PVAdue = $4,425,955.47
For Option B, the aftertax cash flows are:
Aftertax cash flows = Pretax cash flows(1 – tax rate)
Aftertax cash flows = $325,000(1 – .36)
This is the value today. Since the savings are in the form of a growing annuity, we can use the growing
$136,843.40 = C{[1/(.097 – .03)] – [1/(.097 – .03)] × [(1 + .03)/(1 + .097)]30}
C = $10,798.92
This is the amount you need to save next year. So, the percentage of your salary is:
Percentage of salary = $10,798.92/$70,000
Percentage of salary = .1543, or 15.43%
PV = C({1 – [1/(1 + r)t]}/r ) + C0
PV = $654.16({1 – [1/(1 + .058/12)34]}/(.058/12))
C = $20,464.53
She must also pay a one percent prepayment penalty and the payment due on November 1, 2016, so
the total amount of the payment is:
Total payment = Balloon payment(1 + Prepayment penalty) + Current payment
Total payment = $20,464.53(1 + .01) + $654.16
Total payment = $21,323.33
59. The time line is:
0
1
…
120
…
360
361
…
600
–$2,200
–$2,200
$25,000
$25,000
$320,000
C
C
C
$1,000,000
EAR = .07 = [1 + (APR/12)]12 – 1; APR = 12[(1.07)1/12 – 1] = .0678, or 6.78%
First, we will calculate how much he needs at retirement. The amount needed at retirement is the PV
of the monthly spending plus the PV of the inheritance. The PV of these two cash flows is:
PVA = $25,000{1 – [1/(1 + .0678/12)12(20)]}/(.0678/12) = $3,278,926.11
He still has 20 years until retirement. When he is ready to retire, this amount will have grown to:
FV = $123,298.72[1 + (.1048/12)]12(20) = $994,072.72
So, when he is ready to retire, based on his current savings, he will be short:
$3,537,345.11 – 994,072.72 = $2,543,272.39
This amount is the FV of the monthly savings he must make between Years 10 and 30. So, finding the
annuity payment using the FVA equation, we find his monthly savings will need to be:
FVA = $2,543,272.39 = C[{[1 + (.1048/12)]12(20) – 1}/(.1048/12)]
C = $3,145.49
60. 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 $2,500. 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
$2,500
$425
$425
$425
$425
$425
$425
$425
$425
$425
PV = $7,000,000 + $6,100,000/1.0513 + $6,900,000/1.05132 + $7,600,000/1.05133
+ $8,200,000/1.05134 + $9,500,000/1.05135 + $8,400,000/1.05136
PV = $45,922,807.29
$48,422,807.29 – 9,000,000 = $39,422,807.29
To find the quarterly payments, first realize that the interest rate we need is the effective quarterly rate.
Using the daily interest rate, we can find the quarterly interest rate using the EAR equation, with the
number of days being 91.25, the number of days in a quarter (365/4). The effective quarterly rate is:
Effective quarterly rate = [1 + (.05/365)]91.25 – 1 = .01258 or 1.258%
Now, we have the interest rate, the length of the annuity, and the PV. Using the PVA equation and
solving for the payment, we get:
PVA = $39,422,807.29 = C{[1 – (1/1.01258)24]/.01258}
C = $1,913,224.07
$3,500
$3,500
$3,750
$3,750
$4,083.33
$4,083.33
$4,083.33
$150,000
$25,000
Here, we have cash flows that would have occurred in the past and cash flows that would occur in the
future. We need to bring both cash flows to today. Before we calculate the value of the cash flows
today, we must adjust the interest rate, so we have the effective monthly interest rate. Finding the APR
with monthly compounding and dividing by 12 will give us the effective monthly rate. The APR with
monthly compounding is:
APR = 12[(1.09)1/12 – 1] = 8.65%
To find the value today of the back pay from two years ago, we will find the FV of the annuity (salary),
and then find the FV of the lump sum value of the salary. Doing so gives us:
FV = ($42,000/12)[{[1 + (.0865/12)]12 – 1}/(.0865/12)](1 + .09) = $47,639.05
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 = ($45,000/12)[{[1 + (.0865/12)]12 – 1}/(.0865/12)] = $46,827.37
Next, we find the value today of the five year’s future salary:
PVA = ($49,000/12){[{1 – {1/[1 + (.0865/12)]12(5)}]/(.0865/12)} = $198,332.55
The value today of the jury award is the sum of salaries, plus the compensation for pain and suffering,
and court costs. The award should be for the amount of:
Award = $47,639.05 + 46,827.37 + 198,332.55 + 150,000 + 25,000
Award = $467,798.97
As the plaintiff, you would prefer a lower interest rate. In this problem, we are calculating both the PV
and FV of annuities. A lower interest rate will decrease the FVA, but increase the PVA. So, by a lower
interest rate, we are lowering the value of the back pay. But, we are also increasing the PV of the future
salary. Since the future salary is larger and has a longer time, this is the more important cash flow to
the plaintiff.
64. To find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is in
Amount received = $10,000(1 – .02) = $9,800
So, the time line is:
0
9
–$9,800
$11,250
65. This is the same question as before, with different values. Assuming a $10,000 face value loan, the
time line is:
0
9
–$9,700
$10,900
Loan repayment amount = $10,000(1.09) = $10,900
Amount received = $10,000(1 – .03) = $9,700
$10,900 = $9,700(1 + r)
r = ($10,900/$9,700) – 1
r = .1237, or 12.37%
The effective rate is not affected by the loan amount, since it drops out when solving for r.
