Bilbo Baggins wants to save money to meet three objectives. First, he would like
ID: 2734655 • Letter: B
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 $30,000 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 20 years at an estimated cost of $1,458,000. 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 $2,000 per month for the next 20 years.
Required:
If he can earn a 12 percent EAR before he retires and a 9 percent EAR after he retires, how much will he have to save each month in years 21 through 30?
Explanation / Answer
Here the cash flow occur monthly and the interest rate is given is the EAR.since the cash flow 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 retainrment APR is , EAR= 0.12=[(APR/12)]^12-1 APR = 12[(1.12)^1/12-1]=0.1139 or 11.39% And the post retirement APR is , EAR= 0.09= [1+APR/12]^12 -1 APR= 12[(1.09)^1/12-1]= 0.0943 or 9.43% First we will calculate how much needs ate retirement .The amount need at retirement is the PV fo the monthly spending plus the PV of the inheritance .The PV of these two ash flow is . PVA= $30,000{1-[1/(1+0.0943/12)^12(20)-1}/(0.0943/12) =$3,234,273.59 PV= 900,000/[1+(0.0943/12)]^240 =$137,519.62 So at retirement he needs , =3,234,273.59+$137,519.62 =$3,371,793.21 He will be saving $2000 per month for the next 10 years until he purchase the cabin .The value of his saving after 10 year will be FVA= $2000+[{(1+(0.1139/12)]12(20) -1}/(0.1134/12)] =1,811,160.77 FVA= $1,811,160.77-$1,458,000 =$353,160.77 He still have 10 year for retirement ,when he ready to retire thiss amount will have grown to FV= $353,160.77[1+(0.1139/12)]^12(10) =1,091,817.22 So when he is ready to retire ,based on his currnet saving ,he will be short =$3,371,793.21-1,091,817.22 =$2,279,976 This amount is the FV of the monthly saving he must take between years 20 and 30.so finding the annuity payment using the FVA equation we find his monthly saving will need to to be , FVA= $2,279,976= C[{(1+0.1134/12)]^12(10) -1/(0.1134/12)] C= $2,279,976/221.329 C= $2,279,976/221.329 =$10,301.30
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