Bilbo Baggins wants to save money to meet three objectives. First, he would like
ID: 2680365 • 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 $24,000 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 $340,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $1,500,000 to his nephew Frodo. He can afford to save $2,500 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
First let us calculate how much Bilbo needs to have at the time of retirement.
Monthly rate = 1.07^(1/12) - 1 = 0.56541%
No of periods = 25*12 = 300
Present value at time of retirement of 25 years of monthly retirement income of 24,000 is equal to 24,000 * (1-1/(1+0.56541%)^300) / (0.56541%) = 3,462,595.74
Present value at time of retirement of Frodo's inheritance = 1,500,000 / (1+7%)^25 = 276,373.77
So total savings at time of retirement = 3,462,595.74 + 276,373.77 = 3,738,969.51
Now let us calculate total savings for the next 30 years before Bilbo retires
Monthly rate = 1.1^(1/12) - 1 = 0.79741%
Value of next 10 years savings after 10 years = 2500 * (((1+0.79741%)^120)-1) / 0.79741% = 499,659.64
After buying the cabin in Rivendell, net savings after 10 years = 499,659.64 - 340,000 = 159,659.64
Let monthly amount saved from years 11 to 30 be X.
So 159,659.64 * (1+10%)^20 + X * (((1+0.79741%)^240)-1) / 0.79741% = total savings at retirement = 3,738,969.51
Solving, we get X * (((1+0.79741%)^240)-1) / 0.79741% = 2,664,859.28
Or, X = 3,710.16
Answer: Monthly saving between years 11-30 = $ 3,710.16
Hope this helped ! Let me know in case of any queries.
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