Lesson Calculating a retirement plan
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<H2>Calculating a retirement plan<H2> <H3>Problem 1</H3>You want to be able to withdraw $45,000 from your account at the beginning of each year for 15 years after you retire. You expect to retire in 25 years. If your account earns 4% interest, how much will you need to deposit at the end of each year until retirement to achieve your retirement goals? <B>Solution</B> Let's solve this problem in two steps. <U>Step 1</U>. <pre> First, let's determine how much money X should be accumulated on the account during 25 years when you make your annual deposits, in order for to have enough to withdraw $45,000 each year for 15 years of your retirement. Withdrawing $45,000 each year, your account (the remaining money) still earns 4% per annum compounded at the end of each year. So, during this period of 15 years, the annuity works as a sinking fund. If you withdraw at the beginning of each year for living, the formula for the starting value of sinking fund is X = {{{W*(1+r)*(((1-(1+0.04)^(-n)))/r)}}}, (1) where W is the annual withdraw amount; r is the annual rate of compounding as the decimal, and n is the number of withdrawals. In this problem, W = $45,000, r = 0.04; n = 15. Substitute these values into the formula (1) and calculate X X = {{{45000*(1+0.04)*(((1-(1+0.04)^(-15)))/0.04)}}} = $520340.53. (2) So, after 25 years of accumulating money, the fund should have $520340.53. Thus, the first step of calculations is complete. </pre> <U>Step 2</U>. <pre> Now we are in position to determine how much should be deposited each year during 25 years when you make annual deposits. I will assume that the annual deposits are made at the end of each of 25 years. Then the fund works as a standard Ordinary Annuity saving plan. So, we use this formula for the future value FV = {{{D*(((1+r)^m-1)/r)}}}, (3) where D is the annual deposit amount; r is the annual rate of compounding as the decimal, and m is the number of deposits. In this problem, FV = $500327.43 as we determined above; D is the unknown value of the deposit to find from the equation; r = 0.04; m = 25. Substitute these values into the formula (3) 520340.53 = {{{D*(((1.04^25-1))/0.04)}}}. (4) The multiplier {{{((1.04^25-1))/0.04}}} is 41.64591, It implies from equation (4) that D = {{{520340.53/41.64591}}} = $12494.40. It is your <U>ANSWER</U>: During 25 years of the accumulating period you should deposit $12494.40 at the end of each year to your account. Then you will be able to withdraw $45000 at the beginning of every year during 15 years. For clarity, notice that in 25 years, you deposit in the bank 25*12494.40 = 312360 dollars. During following 15 years, you withdraw from the bank 15*45000 = 675000 dollars. 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