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 $26,000 per month for 30 years, with the first payment received 30 years and 1 month from now. Second, he would like to purchase a cabin in Rivendell in 20 years at an estimated cost of $977,000. Third, after he passes on at the end of the 30 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 20 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 21 through 30?

Answer 1

EAR = [1 + (APR/m)]^m - 1

APR for working period is 9.57%

APR for retirement period is 6.78%

First, we need to calculate the money Bilbo need when he retires

Present Value = Present Value of Annuity + Present Value of Inheritance

= $26,000 x PVIFA + $550,000 x PVIF

= $26,000 x [{1 - (1 + 0.0678/12)^-360} / (0.0678/12)] + [$550,000 / (1 + 0.0678/12)³⁶⁰]

= $3,996,350.918 + $72,359.212 = $4,068,710.13

Second, we need to calculate the money that bilbo save

Future Value = Future Value of Annuity

= $1,700 x FVIFA

= $1,700 x [{(1+0.0957/12)²⁴⁰ - 1} / (0.0957/12)]

= $1,700 x 718.35

= $1,221,202.615

Bilbo has saved $1,221,202.615. He pays $977,000 for the house.

That means that he has $244,202.615 left in his bank account.

Now , we can calculate how much will he have to save each month in years 21 through 30

Future Value = $244,202.615 x FVIF

= $244,202.615 x [ 1 + 0.0957/12]¹²⁰

= $244,202.615 x 2.59

= $633,463.512

•Bilbo will have $633,463.512

•Bilbo needs $4,068,710.13

•There Bilbo needs to save $4,068,710.13- $633,463.512

•Needed savings $3,435,246.62

Future Value Annuity = PMT x FVIFA

$3,435,246.62 = PMT x [{(1 + 0.0957/12)¹²⁰ - 1} / (0.0957/12)]

PMT = $3,435,246.62 / 718.35

PMT = $4,782.105