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 $28,000 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 $286,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $550,000 to his nephew Frodo. He can afford to save $1,700 per month for the next 10 years. Required: If he can earn a 11 percent EAR before he retires and a 9 percent EAR after he retires, how much will he have to save each month in years 11 through 30?

$3,608.55

$4,237.47

$3,680.72

$3,536.38

$4,177.33

Please show steps, thanks!

Answer 1

Solution :-

In case EAR = 9%

One month Rate = ( 1 + 0.09 )^1/12 - 1 = 0.007207 = 0.7207%

In case EAR = 11%

One month Rate = ( 1 + 0.11 )^1/12 - 1 = 0.0087346 = 0.87346%

Amount Needed after retirement = Amount for inheritance and Amount of Annuity after retirement

Present Value of amount of inheritance at the time of Retirement = $550,000 / ( 1 + 0.09 )²⁵

= $550,000 * 0.11597 = $63,782.31

Present value at the time of Retirement of 25 years ( 300 months ) monthly annuity after Retirement

= $28,000 * PVAF ( 0.7207% , 300 )

= $28,000 * 122.6614

= $3,434,520.31

Now the Value at Retirement of the Amount Required After Retirement = $63,782.31 + $3,434,520.31

= $3,498,302.62

Now the Future Value of Monthly Saving after 10 Years ( 10 * 12 ) = 120 months = $1,700 * FVAF ( 0.87346% , 120 )

= $1,700 * 210.59

= $358,003.71

Cost of Cabin = $286,000

Net Value of Saving at the end of 10 Years = $358,003.71 - $286,000 = $72,003.71

Now the Value of $161,576.72 at the time of Retirement =$72,003.71 * ( 1 + 0.11 )²⁰ = $580,516.34

Now Future More amount Required = $3,498,302.62 - $580,516.34 = $2,917,786.28

Let amount of Monthly Deposit for 20 Years be X

Now X * FVAF ( 0.87346% , 240 ) = $2,917,786.28

X * 808.546 = $2,917,786.28

X = $3,608.55

Therefore Correct Answer is (A)

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