Unformatted text preview: Tutorial Answers (Oct.1): Ch.6 #59, 63, 68, 72, 77
59. (LO1) 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 = .11 = [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 = $20,000{1 – [1 / (1 + .0772/12)12(20)]} / (.0772/12) = $2,441,554.61 PV = $750,000 / [1 + (.0772/12)]240 = $160,911.16 So, at retirement, he needs: $2,441,544.61 + 160,911.16 = $2,602,465.76 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 = $2,000[{[ 1 + (.1048/12)]12(10) – 1} / (.1048/12)] = $421,180.66 After he purchases the cabin, the amount he will have left is: $421,180.66 – 325,000 = $96,180.66 He still has 20 years until retirement. When he is ready to retire, this amount will have grown to: FV = $96,180.66[1 + (.1048/12)]12(20) = $775,438.43 So, when he is ready to retire, based on his current savings, he will be short: $2,602,465.76 – 775,438.43 = $1,827,027.33 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 = $1,827,027.33 = C[{[ 1 + (.1048/12)]12(20) – 1} / (.1048/12)] C = $2,259.65 63. (LO4) 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 $20,000 you must repay in one year, and the $17,200 you borrow today. The interest rate of the loan is: $20,000 = $17,200(1 + r) r = ($20,000 – 17,200) – 1 = .1628 or 16.28% Because of the discount, you only get the use of $17,200, and the interest you pay on that amount is 16.28%, not 14%. 68. (LO4) 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 $40.08 per month for three years, so the interest rate on the loan is: PVA = $1,000 = $40.08[ {1 – [1 / (1 + r)]36 } / r ] Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives: r = 2.13% per month APR = 12(2.13%) = 25.60% EAR = (1 + .0213)12 – 1 = 28.83% 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. 72. (LO2) The monthly payments with a balloon payment loan are calculated assuming a longer amortization schedule, in this case, 30 years. The payments based on a 30‐year repayment schedule would be: PVA = $450,000 = C({1 – [1 / (1 + .085/12)]360} / (.085/12)) C = $3,460.11 Now, at time = 8, we need to find the PV of the payments which have not been made. The balloon payment will be: PVA = $3,460.11({1 – [1 / (1 + .085/12)]12(22)} / (.085/12)) PVA = $412,701.01 77. (LO1) 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 – ((1 + g)/(1 + r))t ] / (r – g)} $57,308.55 = C{[1 – ((1 + .03)/(1 + .10))30 ] / (.10 – .03)} 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. ...
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 Spring '10
 jillian

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