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 $31,500 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 $415,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $1,375,000 to his nephew Frodo. He can afford to save $4,000 per month for the next 10 years. If he can earn a 10 percent EAR before he retires and a 7 percent EAR after he retires, how much will he have to save each month in years 11 through 30?

Explanation / Answer

Lets work backwards. Bilbo needs a corpus after 30 yrs, which will give him 31500 pm for 25 yrs & then leave a corpus of 1375000

EAR = 7% after he reires

EAR = (1+i/12)^12 -1. Here m= 12 as we need monthly Interest rate

ie (1+i/12)^12 -1 = 7%

ie (1+i/12)^12 = 1+7% = 1.07

ie (1+i/12) = 1.07^(1/12) = 1.0057

ie i/12 = 1.0057 - 1 = 0.0057

so i = 12*0.0057 = 6.84%

No of periods = 25Yr*12month = 300 periods

So After 30 Yrs 1month, we have nper =300, Rate=6.84%/12, FV=1375000, PMT=31500

So Corpus at 30Y1M = PV(Rate,nper,pmt,fv)

= PV(6.84%/12,300,-31500,1375000)

= $4,272,029

SO at Y30, we need to find PV at 1Month

ie Corpus at Y30 = $4272029/(1+6.84%/12) = $4,247,816 ...(A)

He can afford to save $4,000 per month for the next 10 years. If he can earn a 10 percent EAR before he retires and a 7 percent EAR after he retires, how much will he have to save each month in years 11 through 30?

EAR = 10% before he reires

EAR = (1+i/12)^12 -1. Here m= 12 as we need monthly Interest rate

ie (1+i/12)^12 -1 = 10%

ie (1+i/12)^12 = 1+10% = 1.10

ie (1+i/12) = 1.10^(1/12) = 1.0080

ie i/12 = 1.0080 - 1 = 0.0080

so i = 12*0.0080 = 9.60%

Lets see how much he has after 10Yrs.

We have nper = 10*12 = 120 month, PMT=4000, Rate=9.6%/12, PV=0

SO At end of 10Yrs, he has =FV(Rate,nper,pmt,pv)

= FV(9.6%/12,120,-4000,0)

= $800,870

Second, he would like to purchase a cabin in Rivendell in 10 years at an estimated cost of $415,000.

So After buying Cabin for 415,000 at Y10, he has 800870-415000 = $385,870

SO This is his starting Corpus at Y10 = PV = $385,870 ....(B)

Now we have A&B to tell us that PV=$385,870, at Y30 FV= 4247816,

nper = 20*12 = 240 month, Rate = 9.6%/12

So Monthly saving = PMT(Rate,nper,pv,fv)

= PMT(9.6%/12,240,385870,4247816)

= $9,513

So he has to Save $9,513 every month from Y11 to Y30.

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