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 $26,000 per month for 20 years, with the first payment received 30 years and 1 month from now. Second, he would like to purchase a cabin in Rivendell in 15 years at an estimated cost of $675,000. Third, after he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $850,000 to his nephew Frodo. He can afford to save $2,300 per month for the next 15 years. If he can earn a 9 percent EAR before he retires and a 8 percent EAR after he retires, how much will he have to save each month in Years 16 through 30?

$6,573.08

Explanation:

first we need to determine the present value (in 20 years) of Bilbo's retirement distributions:

present value = monthly distribution x annuity factor (PV, 0.6667%, 240 periods) = $26,000 x 119.51872 = $3,107,486.72

now we need to find the present value (in 20 years) of Frodo's inheritance:

present value = $850,000 / (1 + 8%)²⁰ = $182,365.98

this means that in 20 years, Bilbo must have saved $3,107,486.72 + $182,365.98 = $3,289,852.70

but that is not all, he also needs $675,000 in 15 years:

in 15 years, Bilbo's savings account will have monthly contribution x annuity factor (FV, 0.75%, 180 periods) = $2,300 x 378.40577 = $870,333.27

that means that Bilbo will have $870,333.27 - $675,000 = $220,333.27

in 15 more years, the $220,333.27 will be worth: $220,333.27 x (1 + 9%)¹⁵ = $802,560.07

so Bilbo is $3,289,852.70 - $802,560.07 = $2,487,292.63 short

his monthly contribution to his savings plan should be:

$2,487,292.63 = monthly payment x annuity factor (FV, 0.75%, 180)

$2,487,292.63 = monthly payment x 378.40577

monthly payment = $2,487,292.63 / 378.40577 = $6,573.08

answer; /// i do believe that the above is (true)