Bilbo Baggins wants to save money to meet three objectives. First, he would like

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 $32,000 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 $420,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $1,350,000 to his nephew Frodo. He can afford to save $4,100 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?

Explanation / Answer

The cash flows for this problem occur monthly, and the interest rate is given EAr. Since the cashflows occur monthly, we must get 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 = 0.10= (1+(APR/12)]^12-1, APR = 12*(1.10)^1/12-1] = 0.0957 = 9.57%

And the post-retirement APR is:

EAR = 0.07=[1+(APR/12)]^12-1, APR = 12*(1.07)1/12-1] = 0.0678 = 6.78%

Fisrt we will calculate how much he needs at retirement. the amount neede at retirement is the PV of the monthly spending plus the PV Of the inheritance. The pV of these two cashflows is:

PVA = $32,000*{1-[1/(1+0.0678/12)^12*(25)]}/(0.0678/12) = $4,616,794.32

PV = 1,350,00/[1+(0.0678/12)]^300 = $7,327,034.05

So, at retiremnet he need:

$4,616,794.32+ $7,327,034.05 = $11,943,828.37

He will be saving $4,100 per montyh for the next 10 years until he purchases the cabin. the value of his savings after 10 years will be:

FVA = $4,100*[{[1+(0.0957/12)]^120-1}/(0.0957/12)] = $819,441.81

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

$819,441.81- $4,20,000 = $399,441.81

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

FV = $399,441.81*[1+(0.0957/12)]^240 = $2,687,244.78

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

$11,943,828.37-$2,687,244.78 = $9,256,583.59

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

FVA = $9,256,583.59 = C*[{[1+(0.0957/12)]^240-1}/(0.0957/12)] = $12,885.82

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