DISCOUNTED CASH FLOW VALUATION
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 = $1,000,000/(1 + .10)30 PV = $57,308.55
This is the value today. Since the savings are in the form of a growing annuity, we can use the growing annuity equation and solve for the payment. Doing so, we get:
PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}
$57,308.55 = C{[1/(.10 – .03)] – [1/(.10 – .03)] × [(1 + .03)/(1 + .10)]30} C = $4,659.79
This is the amount you need to save next year. So, the percentage of your salary is:
Percentage of salary = $4,659.79/$55,000 Percentage of salary = .0847 or 8.47%
Note that this is the percentage of your salary you must save each year. Since your salary is increasing at 3 percent, and the savings are increasing at 3 percent, the percentage of salary will remain constant.
56. Since she put MXN1,000 down, the amount borrowed will be:
Amount borrowed = MXN15,000 – 1,000 Amount borrowed = MXN14,000
So, the monthly payments will be:
PVA = C({1 – [1/(1 + r)]t } / r )
MXN14,000 = C[{1 – [1/(1 + .096/12)]60 } / (.096/12)]
C = MXN294.71
The amount remaining on the loan is the present value of the remaining payments. Since the first payment was made on October 1, 2004, and she made a payment on October 1, 2006, there are 35 payments remaining, with the first payment due immediately. So, we can find the present value of the remaining 34 payments after November 1, 2006, and add the payment made on this date. So the remaining principal owed on the loan is:
PV = C({1 – [1/(1 + r)]t } / r ) + C0
PV = MXN294.71[{1 – [1/(1 + .096/12)]34 } / (.096/12)] + MXN294.71 C = MXN9,037.33
She must also pay a one percent prepayment penalty, so the total amount of the payment is:
Total payment = Amount due(1 + Prepayment penalty) Total payment = MXN9,037.33(1 + .01)
Total payment = MXN9,127.71
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 = .1011 = [1 + (APR / 12)]12 – 1; APR = 12[(1.11)1/12 – 1] = 10.48%
And the post-retirement APR is:
EAR = .08 = [1 + (APR / 12)]12 – 1; APR = 12[(1.08)1/12 – 1] = 7.72%
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 + .0772/12)12(20)]} / (.0772/12) = $3,051,943.26 PV = $750,000 / [1 + (.0772/12)]240 = $160,911.16
So, at retirement, he needs:
$3,051,943.26 + 160,911.16 = $3,212,854.42
He will be saving $2,100 per month for the next 10 years until he purchases the cabin. The value of his savings after 10 years will be:
FVA = $2,100[{[ 1 + (.1048/12)]12(10) – 1} / (.1048/12)] = $442,239.69 After he purchases the cabin, the amount he will have left is:
$442,239.69 – 350,000 = $92,239.69
He still has 20 years until retirement. When he is ready to retire, this amount will have grown to:
FV = $92,239.69[1 + (.1048/12)]12(20) = $743,665.12
So, when he is ready to retire, based on his current savings, he will be short:
$3,212,854.41 – 743,665.12 = $2,469,189.29
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 = $2,469,189.29 = C[{[ 1 + (.1048/12)]12(20) – 1} / (.1048/12)]
C = $3,053.87
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 €1. 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 = €1 + €450{1 – [1 / (1 + .08/12)12(3)]} / (.08/12) = €14,361.31
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 + (.08/12)]12(3) = €18,106.86 The PV of the decision to purchase is:
€35,000 – €18,106.86 = €16,893.14
In this case, it is cheaper to lease the car than buy it since the PV of the leasing 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:
€35,000 – PV of resale price = €14,361.31 PV of resale price = €20,638.69
The resale price that would make the PV of the lease versus buy decision is the FV of this value, so:
Breakeven resale price = €20,638.69[1 + (.08/12)]12(3) = €26,216.03
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 + (.045/365)]365 – 1 = 4.60%
The PV of the current contract offer is the sum of the PV of the cash flows. So, the PV is:
PV = ¥8,000,000 + ¥4,000,000/1.046 + ¥4,800,000/1.0462 + ¥5,700,000/1.0463 + ¥6,400,000/1.0464 + ¥7,000,000/1.0465 + ¥7,500,000/1.0466
PV = ¥37,852,037.91
The player wants the contract increased in value by ¥750,000, so the PV of the new contract will be:
PV = ¥37,852,037.91 + 750,000 = ¥38,602,037.91
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.
¥38,602,037.91 – 9,000,000 = ¥29,602,037.91
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 + (.045/365)]91.25 – 1 = 1.131%
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 = ¥29,602,037.91 = C{[1 – (1/1.01131)24] / .01131}
C = ¥1,415,348.37
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 Rs.20,000 you must repay in one year, and the Rs.17,600 you borrow today. The interest rate of the loan is:
Rs.20,000 = Rs.17,600(1 + r)
r = (Rs.20,000 – 17,600) – 1 = 13.64%
Because of the discount, you only get the use of Rs.17,600, and the interest you pay on that amount is 13.64%, not 12%.
61. 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, and then find the FV of the lump sum. Doing so gives us:
FVA = ($40,000/12) [{[ 1 + (.0865/12)]12 – 1} / (.0865/12)] = $41,624.33 FV = $41,624.33(1.09) = $45,370.52
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 + (.0865/12)]12 – 1} / (.0865/12)] = $44,746.15 Next, we find the value today of the five year’s future salary:
PVA = ($45,000/12){[{1 – {1 / [1 + (.0865/12)]12(5)}] / (.0865/12)}= $182,142.14
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 = $45,370.52 + 44,746.15 + 182,142.14 + 100,000 + 20,000 Award = $392,258.81
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.
62. Again, 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 = ₩1,000,000(1.10) = ₩1,100,000
The amount you will receive today is the principal amount of the loan times one minus the points.
Amount received = ₩1,000,000(1 – .03) = ₩970,000 Now, we simply find the interest rate for this PV and FV.
₩1,100,000 = ₩970,000(1 + r)
r = (₩1,100,000 / ₩970,000) – 1 = 13.40%
63. This is the same question as before, with different values. So:
Loan repayment amount = ₩1,000,000(1.13) = ₩1,130,000 Amount received = ₩1,000,000(1 – .02) = ₩980,000
₩1,130,000 = ₩980,000(1 + r)
r = (₩1,130,000 / ₩980,000) – 1 = 15.31%
The effective rate is not affected by the loan amount, since it drops out when solving for r.
64. 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 $1,500 application fee, you will need to borrow $201,500 to have $200,000 after deducting the fee. Solving for the payment under these circumstances, we get:
PVA = $201,500 = C {[1 – 1/(1.00625)360]/.00625} where .00625 = .075/12 C = $1,408.92
We can now use this amount in the PVA equation with the original amount we wished to borrow,
$200,000. Solving for r, we find:
PVA = $200,000 = $1,408.92[{1 – [1 / (1 + r)]360}/ r]
Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:
r = 0.6314% per month APR = 12(0.6314%) = 7.58%
EAR = (1 + .006314)12 – 1 = 7.85%
With the nonrefundable fee, the APR of the loan is simply the quoted APR since the fee is not considered part of the loan. So:
APR = 7.50%
EAR = [1 + (.075/12)]12 – 1 = 7.76%
65. 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 £42.25 per month for three years, so the interest rate on the loan is:
PVA = £1,000 = £42.25[ {1 – [1 / (1 + r)]36 } / r ]
Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:
r = 2.47% per month APR = 12(2.47%) = 29.63%
EAR = (1 + .0247)12 – 1 = 34.00%
It’s called add-on interest because the interest amount of the loan is added to the principal amount of the loan before the loan payments are calculated.
66. Here, we are solving a two-step time value of money problem. Each question asks for a different possible cash flow to fund the same retirement plan. Each savings possibility has the same FV, that is, the PV of the retirement spending when your friend is ready to retire. The amount needed when your friend is ready to retire is:
PVA = $90,000{[1 – (1/1.08)15] / .08} = $770,353.08 This amount is the same for all three parts of this question.
a. If your friend makes equal annual deposits into the account, this is an annuity with the FVA equal to the amount needed in retirement. The required savings each year will be:
FVA = $770,353.08 = C[(1.0830 – 1) / .08]
C = $6,800.24
b. Here we need to find a lump sum savings amount. Using the FV for a lump sum equation, we get:
FV = $770,353.08 = PV(1.08)30 PV = $76,555.63
c. In this problem, we have a lump sum savings in addition to an annual deposit. Since we already know the value needed at retirement, we can subtract the value of the lump sum savings at retirement to find out how much your friend is short. Doing so gives us:
FV of trust fund deposit = $25,000(1.08)10 = $53,973.12 So, the amount your friend still needs at retirement is:
FV = $770,353.08 – 53,973.12 = $716,379.96
Using the FVA equation, and solving for the payment, we get:
$716,379.96 = C[(1.08 30 – 1) / .08]
C = $6,323.80
This is the total annual contribution, but your friend’s employer will contribute $1,500 per year, so your friend must contribute:
Friend's contribution = $6,323.80 – 1,500 = $4,823.80
67. We will calculate the number of periods necessary to repay the balance with no fee first. We simply