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Question

<div class="txt-body question-body"><div class="feedback-rating abuse-flag" data-questionid="12226125"></div><div class="ugc-base"><p>Bilbo Baggins wants to save money to meet three objectives.First, he would like to be able to retire 30 years from now withretirement income of $32,000 per month for 25 years, with the firstpayment received 30 years and 1 month from now. Second, he wouldlike to purchase a cabin in Rivendell in 10 years at an estimatedcost of $420,000. Third, after he passes on at the end of the 25years of withdrawals, he would like to leave an inheritance of$1,350,000 to his nephew Frodo. He can afford to save $4,100 permonth for the next 10 years. If he can earn an EAR of 10 percentbefore he retires and an EAR of 7 percent after he retires, howmuch will he have to save each month in years 11 through 30?</p></div></div>

Explanation / Answer

&lt;div class=&quot;txt-body answer-body&quot;&gt; &lt;div class=&quot;answer-given-body ugc-base&quot;&gt; &lt;p&gt;The cash flows for this problem occur monthly, and the interestrate is given EAr. Since the cashflows occur monthly, we must geteffective monthly rate. One way to do this is to find the APR basedon monthly compounding, and then divide by 12. So, thepre-retirement APR is:&lt;/p&gt;&lt;p&gt;EAR = 0.10= (1+(APR/12)]^12-1, APR = 12*(1.10)^1/12-1] = 0.0957= 9.57%&lt;/p&gt;&lt;p&gt;And the post-retirement APR is:&lt;/p&gt;&lt;p&gt;EAR = 0.07=[1+(APR/12)]^12-1, APR = 12*(1.07)1/12-1] = 0.0678 =6.78%&lt;/p&gt;&lt;p&gt;Fisrt we will calculate how much he needs at retirement. theamount neede at retirement is the PV of the monthly spending plusthe PV Of the inheritance. The pV of these two cashflows is:&lt;/p&gt;&lt;p&gt;PVA = $32,000*{1-[1/(1+0.0678/12)^12*(25)]}/(0.0678/12) =$4,616,794.32&lt;/p&gt;&lt;p&gt;PV = 1,350,00/[1+(0.0678/12)]^300 = $7,327,034.05&lt;/p&gt;&lt;p&gt;So, at retiremnet he need:&lt;/p&gt;&lt;p&gt;$4,616,794.32+ $7,327,034.05 = $11,943,828.37&lt;/p&gt;&lt;p&gt;He will be saving $4,100 per montyh for the next 10 years untilhe purchases the cabin. the value of his savings after 10 yearswill be:&lt;/p&gt;&lt;p&gt;FVA = $4,100*[{[1+(0.0957/12)]^120-1}/(0.0957/12)] =$819,441.81&lt;/p&gt;&lt;p&gt;After he purchases the cabin, the amount he will left is:&lt;/p&gt;&lt;p&gt;$819,441.81- $4,20,000 = $399,441.81&lt;/p&gt;&lt;p&gt;He still has 20 years untill retirement. When he is ready toretire, this amount will have grown to:&lt;/p&gt;&lt;p&gt;FV = $399,441.81*[1+(0.0957/12)]^240 = $2,687,244.78&lt;/p&gt;&lt;p&gt;SO, when he is ready to retire, based on his current savings, ewill be short:&lt;/p&gt;&lt;p&gt;$11,943,828.37-$2,687,244.78 = $9,256,583.59&lt;/p&gt;&lt;p&gt;This amount is the Fv of the monhtly savings he must makebetween 10 and 30 . So, finding annuity payment using the FVAequation, we find his monthly savings will need to be:&lt;/p&gt;&lt;p&gt;FVA = $9,256,583.59 = C*[{[1+(0.0957/12)]^240-1}/(0.0957/12)] =$12,885.82&lt;/p&gt; &lt;/div&gt; &lt;/div&gt;

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