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 20 years at an estimated cost of $1,322,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $900,000 to his nephew Frodo. He can afford to save $2,300 per month for the next 20 years.

  

Required:

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

Solution :-

Monthly Rate if EAR 7% = ( 1 + 0.07 )^1/12 - 1 = 0.5654%

Monthly Rate if EAR 10 % = ( 1 + 0.10 )^1/12- 1 = 0.7974%

Number of monthly Payments Required after retirement = 25 * 12 = 300

Present Value at retirement = $28,000 * PVAF ( 0.5654% , 300 ) =

= $28,000 * 144.277 = $4,039,759.34

Present Value of Amount of Inheritance at Retirement = $900,000 / ( 1 + 0.07 )²⁵

= $900,000 * 0.184249

= $165,824.26

Now Value of Savings of $2300 per month after 20 Years = $2,300 * FVAF ( 007974% , 240 )

= $2300 * 718.2433

= $1,651,959.59

Cost of Car after 20 years = $1,322,000

Now Amount Remaining of Saving = $1,651,959.59 - $1,322,000 = $329,959.59

Now value of Monthly deposits of 20 Years at retirement means after 10 years = $329,959.59 * ( 1 + 0.10 )¹⁰

= $329,959.59 * 2.5937

= $855,816.19

Now Net Amount required = $4,039,759.34 + $165,824.26 - $855,816.19

= $3,349,767.41

Now Amount of monthly Deposit Required for 10 Years =  

= $3,349,767.41 = X * FVAF ( 0.7974% , 120 )

= $3,349,767.41 = X * 199.862

X = $16,760.40

Therefore the amount need  to save each month in years 21 through 30 = $16,760.40

If there is any doubt please ask in comments