55. 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 = $42,000 = C {[1 – 1 / (1 + .08)5] / .08}
C = $10,519.17
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:
30 1 $42,000.00 $10,519.17 $3,360.00 $7,159.17 $34,840.83
2 34,840.83 10,519.17 2,787.27 7,731.90 27,108.92
3 27,108.92 10,519.17 2,168.71 8,350.46 18,758.47
4 18,758.47 10,519.17 1,500.68 9,018.49 9,739.97
5 9,739.97 10,519.17 779.20 9,739.97 0.00
In the third year, $2,168.71 of interest is paid.
Total interest over life of the loan = $3,360 + 2,787.27 + 2,168.71 + 1,500.68 + 779.20 Total interest over life of the loan = $10,595.86
56. This amortization table calls for equal principal payments of $8,400 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 1 $42,000.00 $11,760.00 $3,360.00 $8,400.00 $33,600.00
2 33,600.00 11,088.00 2,688.00 8,400.00 25,200.00
3 25,200.00 10,416.00 2,016.00 8,400.00 16,800.00
4 16,800.00 9,744.00 1,344.00 8,400.00 8,400.00
5 8,400.00 9,072.00 672.00 8,400.00 0.00
In the third year, $2,016 of interest is paid.
Total interest over life of the loan = $3,360 + 2,688 + 2,016 + 1,344 + 672 = $10,080
Notice that the total payments for the equal principal reduction loan are lower. This is because more principal is repaid early in the loan, which reduces the total interest expense over the life of the loan.
Challenge
57. The cash flows for this problem occur monthly, and the interest rate given is the EAR. Since the cash flows occur monthly, we must get the effective monthly rate. One way to do this is to find the APR based on monthly compounding, and then divide by 12. So, the pre-retirement APR is:
EAR = .10 = [1 + (APR / 12)]12 – 1; APR = 12[(1.10)1/12 – 1] = .0957 or 9.57%
And the post-retirement APR is:
EAR = .07 = [1 + (APR / 12)]12 – 1; APR = 12[(1.07)1/12 – 1] = .0678 or 6.78%
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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 = $20,000{1 – [1 / (1 + .0678/12)12(25)]} / (.0678/12) = $2,885,496.45 PV = $900,000 / [1 + (.0678/12)]300 = $165,824.26
So, at retirement, he needs:
$2,885,496.45 + 165,824.26 = $3,051,320.71
He will be saving $2,500 per month for the next 10 years until he purchases the cabin. The value of his savings after 10 years will be:
FVA = $2,500[{[ 1 + (.0957/12)]12(10) – 1} / (.0957/12)] = $499,659.64 After he purchases the cabin, the amount he will have left is:
$499,659.64 – 380,000 = $119,659.64
He still has 20 years until retirement. When he is ready to retire, this amount will have grown to:
FV = $119,659.64[1 + (.0957/12)]12(20) = $805,010.23
So, when he is ready to retire, based on his current savings, he will be short:
$3,051,320.71 – 805,010.23 = $2,246,310.48
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 = $2,246,310.48 = C[{[ 1 + (.1048/12)]12(20) – 1} / (.1048/12)]
C = $3,127.44
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 is simply the PV of the lease payments, plus the $99. 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:
PV = $99 + $450{1 – [1 / (1 + .07/12)12(3)]} / (.07/12) = $14,672.91
The PV of purchasing the car is the current price of the car minus the PV of the resale price. The PV of the resale price is:
PV = $23,000 / [1 + (.07/12)]12(3) = $18,654.82 The PV of the decision to purchase is:
$32,000 – 18,654.82 = $13,345.18
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In this case, it is cheaper to buy the car than leasing it since the PV of the purchase cash flows is lower. To find the breakeven resale price, we need to find the resale price that makes the PV of the two options the same.
In other words, the PV of the decision to buy should be:
$32,000 – PV of resale price = $14,672.91 PV of resale price = $17,327.09
The resale price that would make the PV of the lease versus buy decision is the FV of this value, so:
Breakeven resale price = $17,327.09[1 + (.07/12)]12(3) = $21,363.01
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 + (.055/365)]365 – 1 = 5.65%
The PV of the current contract offer is the sum of the PV of the cash flows. So, the PV is:
PV = $7,000,000 + $4,500,000/1.0565 + $5,000,000/1.05652 + $6,000,000/1.05653 + $6,800,000/1.05654 + $7,900,000/1.05655 + $8,800,000/1.05656
PV = $38,610,482.57
The player wants the contract increased in value by $1,400,000, so the PV of the new contract will be:
PV = $38,610,482.57 + 1,400,000 = $40,010,482.57
The player has also requested a signing bonus payable today in the amount of $9 million. We can simply subtract this amount from the PV of the new contract. The remaining amount will be the PV of the future quarterly paychecks.
$40,010,482.57 – 9,000,000 = $31,010,482.57
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 + (.055/365)]91.25 – 1 = .01384 or 1.384%
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 = $31,010,482.57 = C{[1 – (1/1.01384)24] / .01384}
C = $1,527,463.76
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60. 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 cash flows of the loan are the $25,000 you must repay in one year, and the $21,250 you borrow today. The interest rate of the loan is:
$25,000 = $21,250(1 + r)
r = ($25,000 / 21,250) – 1 = .1765 or 17.65%
Because of the discount, you only get the use of $21,250, and the interest you pay on that amount is 17.65%, not 15%.