Question

Problem 4-59 Calculating Annuity Values

Bilbo Baggins wants to save money to meet three objectives. First, he would like to be able to retire 30 years from now with a retirement income of $29,000 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 10 years at an estimated cost of $370,000. Third, after he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $1,250,000 to his nephew Frodo. He can afford to save $3,100 per month for the next 10 years. If he can earn an EAR of 10 percent before he retires and an EAR of 7 percent after he retires, how much will he have to save each month in Years 11 through 30? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Monthly savings           $ _______

Answer 1

Consumption: Post-Retirement withdrawals each worth $ 29000 per month for 20 years or 240 months, Cabin Purchase = $ 370000 after 10 years and Inheritance Worth = $1250000 at the end of retirement period of 20 years

Post-Retirement Interest Rate = 7 % and Pre-Retirement Interest Rate = 10%

Applicable Monthly Rate Post-Retirement = (1.07)^(1/12) - 1 = 0.00565 or 0.565 % and Applicable Monthly Rate Pre-Retirement = (1.1)^(1/12) - 1 = 0.00797 or 0.797 %

If the current time is assumed to be t=0, Total Present Value of Consumptions at the end of Year 30 (t=30) = 370000 x (1.1)^(20) + 29000 x (1/0.00565) x [1-{1/(1.00565)^(240)}] + 1250000 / (1.00565)^(240) = $ 6617550.545

Deposits between Year 1 and Year 10 = $ 3100 per month

Therefore, Total Future Value of these deposits at the end of Year 30 = [3100 x (1.00797)^() + ...............+ 3100] x (1.1)^(20) = $ 4167031.376

Let the required deposits between Year 11 and Year 30 be $ k
Therefore, k x (1.00797)^(239) + ...........+ k = 6617550.545 - 4167031.376 = $ 2450519.169

k x [{(1.00797)^(240)-1}/{(1.00797)-1}] = 2450519.169

k x 717.8006 = 2450519.169

k = 2450519.169 / 717.8006 = $ 3413.927 ~ $ 3413.93