Bilbo Baggins wants to save money to meet three objectives. First, he would like
ID: 2623904 • 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 $32,000 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 $420,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $1,350,000 to his nephew Frodo. He can afford to save $4,100 per month for the next 10 years. If he can earn a 10 percent EAR before he retires and a 7 percent EAR after he retires, how much will he have to save each month in years 11 through 30?
Explanation / Answer
Monthly rate = (1+7%)^(1/12)-1 = 0.5654%
Present value of retirement income as of 30 years from now = 32,000 * (1-1/(1+0.5654%)^(25*12)) / (0.5654%) = 4616794
Present value of inheritance as of 30 years from now = 1,350,000 / (1+0.5654%)^(25*12) = 248736
So total value as of 30 years from now = 4616794 + 248736 = 4,865,531
Effective rate before retirement = (1+10%)^(1/12)-1 = 0.7974%
Savings till year 10 as of year 10 = 4100 * ((1+0.7974%)^(12*10) - 1) / (0.7974%) = 819,442
After deducting 420,000 for the cabin, total savings as of year 10 = 819,442 - 420,000 = 399,442
Future value of this 399,442 as of 30 years from now = 399,442 * (1+0.7974%)^(12*20) = 2,687,245
As we can see, current savings is worth 2,687,245 as of 30 years from now, while we need 4,865,531 as of 30 years from now.
The difference, ie. 4,865,531 - 2,687,245 = 2,178,286 needs to be funded by savings between years 11 and 30.
Let this monthly saving be X
Then X * ((1+0.7974%)^(12*20) - 1) / (0.7974%) = 2,178,286
Solving, we get X = monthly saving between years 11 and 30 = 3,033
Answer: $3,033
Hope this helped ! Let me know in case of any queries.
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