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[(1.08)1/12 – 1] = .0772 or 7.72%
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 = ($47,000/12) [{[ 1 + (.0772/12)]12 – 1} / (.0772/12)] = $48,699.39 FV = $48,699.39(1.08) = $52,595.34
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 = ($50,000/12) [{[ 1 + (.0772/12)]12 – 1} / (.0772/12)] = $51,807.86 Next, we find the value today of the five year’s future salary:
PVA = ($55,000/12){[{1 – {1 / [1 + (.0772/12)]12(5)}] / (.0772/12)}= $227,539.14
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 = $52,595.34 + 51,807.86 + 227,539.14 + 100,000 + 20,000 = $451,942.34
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.
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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 = $10,000(1.08) = $10,800
The amount you will receive today is the principal amount of the loan times one minus the points.
Amount received = $10,000(1 – .03) = $9,700
Now, we simply find the interest rate for this PV and FV.
$10,800 = $9,700(1 + r)
r = ($10,800 / $9,700) – 1 = .1134 or 11.34%
63. This is the same question as before, with different values. So:
Loan repayment amount = $10,000(1.11) = $11,100 Amount received = $10,000(1 – .02) = $9,800
$11,100 = $9,800(1 + r)
r = ($11,100 / $9,800) – 1 = .1327 or 13.27%
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 $2,300 application fee, you will need to borrow
$242,300 to have $240,000 after deducting the fee. Solving for the payment under these circumstances, we get:
PVA = $242,300 = C {[1 – 1/(1.005667)360]/.005667} where .005667 = .068/12 C = $1,579.61
We can now use this amount in the PVA equation with the original amount we wished to borrow, $240,000.
Solving for r, we find:
PVA = $240,000 = $1,579.61[{1 – [1 / (1 + r)]360}/ r]
Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:
r = 0.5745% per month APR = 12(0.5745%) = 6.89%
EAR = (1 + .005745)12 – 1 = 7.12%
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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 = 6.80%
EAR = [1 + (.068/12)]12 – 1 = 7.02%
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 $41.15 per month for three years, so the interest rate on the loan is:
PVA = $1,000 = $41.15[{1 – [1 / (1 + r)]36 } / r ]
Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:
r = 2.30% per month APR = 12(2.30%) = 27.61%
EAR = (1 + .0230)12 – 1 = 31.39%
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 = $105,000{[1 – (1/1.07)20] / .07} = $1,112,371.50 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 = $1,112,371.50 = C[(1.0730 – 1) / .07]
C = $11,776.01
b. Here we need to find a lump sum savings amount. Using the FV for a lump sum equation, we get:
FV = $1,112,371.50 = PV(1.07)30 PV = $146,129.04
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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 = $150,000(1.07)10 = $295,072.70 So, the amount your friend still needs at retirement is:
FV = $1,112,371.50 – 295,072.70 = $817,298.80
Using the FVA equation, and solving for the payment, we get:
$817,298.80 = C[(1.07 30 – 1) / .07]
C = $8,652.25
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 = $8,652.25 – 1,500 = $7,152.25
67. We will calculate the number of periods necessary to repay the balance with no fee first. We simply need to