Bilbo Baggins wants to save money to meet three objectives. First, he would like
ID: 2723952 • Letter: B
Question
Bilbo Baggins wants to save money to meet three objectives. First, he would like to be able to retire 30 years from now with retirement income of $29,500 per month for 25 years, with the first payment received 30 years and 1 month from now. Second, he would like to purchase a cabin in Rivendell in 10 years at an estimated cost of $395,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $1,475,000 to his nephew Frodo. He can afford to save $3,600 per month for the next 10 years. If he can earn an EAR of 10 percent before he retires and an EAR of 7 percent after he retires, how much will he have to save each month in years 11 through 30
Explanation / Answer
1.
Savings per month for 10 years = $3,600
EAR = 10%
Monthly interest rate = 0.10/12 = 0.00833
No. of deposits = 10 years *12 = 120
Future value of annuity = Annuity * {(1+r)n – 1 }/r
Future value of monthly savings after 10 years = $3,600 * (1.00833120 – 1)/0.00833 = $3,600 * 204.7981 = $737,273.16
Amount used to purchase cabin = $395,000
Remaining balance = $737,273.16 - $395,000 = $342,273.16
Future value of remaining balance at the time of retirement (after 20 years) = $342,273.16 * 1.00833240 = $342,273.16 * 7.3223 = $2,506,226.76
2.
Amount required at the time of retirement = Present value of annuity of withdrawals + Present value of inheritance
Monthly withdrawals after retirement = $29,500
No. of withdrawals = 25 years *12 = 300
Monthly interest rate = 0.07/12 = 0.00583
Present value of annuity = Annuity * {1 – (1+r)-n}/r
Present value of withdrawals = $29,500 * (1 – 1.00583-300)/0.00583 = $29,500 * 141.538 = $4,175,370.98
Present value of inheritance to be left = $1,475,000 / 1.00583300 = $257,880.66
Total amount required at the time of retirement = $257,880.66 + $4,175,370.98 = $4,433,251.64
3.
Value of current savings at the time of retirement = $2,506,226.76
Additional amount to be invested = $4,433,251.64 - $2,506,226.76 = $1,927,024.88
Monthly deposit = ($1,927,024.88 * 0.00833) / ( 1 – 1.00833-240) = $18,591.10
Hence, he will have to save $18,591 per month
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