Problem 6-57 Calculating Annuity Values [Lo1] Bilbo Baggins wants to save money
ID: 2817877 • Letter: P
Question
Problem 6-57 Calculating Annuity Values [Lo1] 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 $380,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $1,700,000 to his nephew Frodo. He can afford to save $3,300 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 savingsExplanation / Answer
Given:
EAR before retirement=10%
APR with monthly compounding=((1+10%)^(1/12)-1)*12=9.569%
Future Value of savings till year 10
=3300/(9.569%/12)*((1+9.569%/12)^120-1)
=659550.73
Alternatively, using financial calculator
PMT=3300
I/Y=9.569%/12
N=10*12=120
CPT FV=659550.73
Amount used for purchase of cabin=380000
Amount left=659550.73-380000=279550.73
EAR after retirement=7%
APR with monthly compounding=((1+7%)^(1/12)-1)*12=6.785%
At the end of 30 years, Present value of withdrawals and inheritance left for Frodo=28000/(6.785%/12)*(1-1/(1+6.785%/12)^(12*25))+1700000/(1+6.785%/12)^300=4352907.23
Alternatively, Using financial calculator:
N=12*25
I/Y=6.785%/12
PMT=28000
FV=1700000
CPT PV=4352907.23
Now:
The amount required to be deposited from years 11-30:
=(4352907.23-279550.73*(1+9.569%/12)^(12*20))*(9.569%/12)/((1+9.569%/12)^(12*20)-1)
=2930.32
Alternatively, Using financial calculator:
FV=-4352907.23
I/Y=9.569%/12
N=12*20
PV=279550.73
CPT PMT=3441.94
Hence, Bilbo needs to save $3441.94 per year from years 11-30