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 $27,000 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 15 years at an estimated cost of $469,000. Third, after he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $750,000 to his nephew Frodo. He can afford to save $1,600 per month for the next 15 years.

Required:

If he can earn a 9 percent EAR before he retires and a 5 percent EAR after he retires, how much will he have to save each month in years 16 through 30?

A.) $11,586.50

B.) $11,082.79

C.) $10,865.48

D.) $10,648.17

E.) $11,713.22

Answer 1

Step 1: Calulate APR from EAR, since the cash flows occur monthly and EPR is an annual iterest rate.

Pre- retirement APR: EAR=0.09 = [1 + (APR/12)^{^12} ] - 1 APR = 8.65%

Post-retirement APR: EAR=0.05 =[1 + (APR/12)^{^12} ] - 1 APR = 4.89%

Step 2: Calculate the amount required at retirment.

Amount required at retirement = PV of monthly spending of $27,000 for 20 years + PV of the inhertitence

PV of monthly spendings= 27000{1- [1/(1+0.0489/12)^{^12*20}]} / (0.0489/12) = $4,129,107.97

PV of inhertitence = 750000/ (1+0.0489/12)^{^12*20} = $282,607.92

Amount required at retirement = $4,129,107.97+$282,607.92 = $4,411,715.89

Step 3: Calulate the FV of savings of $1600 per month for next 15 years till purchase of cabin:

FV of savings = 1600[ {[1 + (0.0865/12)]^{^15*12} -1} / (0.0865/12) ] = $586,685.41

Step 4: Calculate the amount of money these savings will grow to in the next 15 years upto his retirement:

After the purchase of cabin, the amount he will be left with is =$586,685.41- $469,000 = $117,685.41

He still has 15 more years until retirement. This amount will grow to:

FV = $117,685.41 [(1+ 0.0865/12)^{^15*12}] = $428,774.42

Step 5: So when at his retirement, based on the current savings amount, he will be short of amount

= $4,411,715.89 - $428,774.42 = $3,892,971.47

Step 6: This is the FV of the monthly savings he must make from years 16 to 30. So, finding the annuity payment using the FVA equation, we find this montly amount is:

FVA = $3,892,971.47 = monthly savings * [{[1 + (0.0865/12)]^{^15*12} -1} / (0.0865/12)

monthly savings = $10,865.30, which is approx equal to answer C.) $10,865.48

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