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?

Because the cash flows occur monthly, we must get the effective monthly rate. One way to do this is to find the APR based on monthly compounding, and then divide by 12. So, the pre-retirement APR is:

EAR = .11 = [1 + (APR/12)] 12- 1;

APR = 12[(1.11) 1/12- 1] = .1048 or 10.48%

And the post-retirement APR is:

EAR = .08 = [1 + (APR/12)] 12 -1

APR = 12[(1.08) 1/12 -1] = .0772 or 7.72%

First, we will calculate how much he needs at retirement. The amount needed at retirement is the PV of the monthly spending plus the PV of the inheritance. The PV of these two cash flows is:

PVA = $24500{1 -[1/(1 + .0772/12) 12(25) ]}/(.0772/12) = $3,252,096.21

PV = $1525,000/[1 + (.0772/12)] 300 = $222,723.58

So, at retirement, he needs:

$3,252,096.21+ $222,723.58= $3474819.79

He will be saving $2,600 per month for the next 10 years until he purchases the cabin. The value of his savings after 10 years will be:

FVA = $2,600[{[1 + (.1048/12)] 12(10) -1}/(.1048/12)] = $547,487.10

After he purchases the cabin, the amount he will have left is:

$547,487.10 -345,000 = $202487.10

He still has 20 years until retirement. When he is ready to retire, this amount will have grown to:

FV = $202487.10[1 + (.1048/12)] 12(20) = $1632023.27

So, when he is ready to retire, based on his current savings, he will be short:

$3474819.79-1632023.27 = $1842796.52

This amount is the FV of the monthly savings he must make between years 10 and 30. So, finding the annuity payment using the FVA equation, we find his monthly savings will need to be:

FVA = $1842796.52 = C [{[ 1 + (.1048/12)] 12(20) -1}/(.1048/12)]

C = $2,279.60

He would have to save each month in years 11 through 30 the amount of $2,279.60