Question

Problem 4-57 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 $31,000 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 $390,000. Third, after he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $900,000 to his nephew Frodo. He can afford to save $3,500 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

First let's calculate the PV of retirement corpus required at the age of 60
First we should calculate the PV of all future withdrawls
Annual withdrals	=31000*12
	372,000.00
Interest rate	7%
Years	20.00
P = PMT x (((1-(1 + r) ^- n)) / i)
Where:
P = the present value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the effective interest rate (also known as the discount rate)
i=nominal Interest rate
n = the number of periods in which payments will be made
Funds required at 60 for pension	=Annual withdrawl*(((1-(1 + 7%) ^-20)) /7%)
Funds required at 60 for pension	=Annual withdrawl*10.59
Funds required at 60 for pension	=372000*10.59
Funds required at 60 for pension	3,939,480
PV of inheritance fund	=900000*(1/(1+7%)^20)
	232,577
Total Funds required at retirmenet age	=3939480+232577
	4,172,057
Now we should assess the position at year 10 after puchasing the cabin
Future value of his monthly saving @ 3500 per month at year 10
Annual saving	=3500*12	42000
FV of annuity
P = PMT x ((((1 + r) ^ n) - 1) / i)
Where:
P = the future value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the effective interest rate (also known as the discount rate)
i=nominal Interest rate
n = the number of periods in which payments will be made
FV of annual contribution	=42000*((((1 + 10%) ^10) - 1) / 10%)
FV of annual contribution	669,372
Cabin cost	390,000
Balance funds	279,372
FV of this fund at retirement age	=279372*(1+10%)^20
	1,879,475
Funds required at retirement age	4,172,057
Balance funds required	=4172057-1879475
	2,292,582
So this future value can be accumulated by contribution from year 11 till year 30
Funds required	=Annual saving *(((1-(1 + 10%) ^-20)) /10%)
2,292,582	=Annual saving *(((1-(1 + 10%) ^-20)) /10%)
2,292,582	=Annual saving * 8.513
Annual saving	=2,292,582/8.513
Annual saving	269303.6532
Monthly saving	=269303/12
Monthly saving	22,442