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You are saving for retirement. You plan to retire when you reach 65 yearsold. To live comfortably during retirement, you plan to withdraw moneyfrom your savings account $48,000 each year. Suppose today is your 25thbirthday, and you decide, starting today and continuing on every yearup to your 64th birthday, that you will put the same amount into a savingsaccount. If the interest rate is 2.4% per annum, compounded annually,how much must you set aside each year to make sure that you will haveenough money in the account on your 65th birthday in order to be ableto withdraw $48,000 per year when you retire and you expect to live untilyour are 90 years old (i.e., the first withdrawal is on your 65th birthday andthe last withdrawal occurs on your 89th birthday)?

Question

You are saving for retirement. You plan to retire when you reach 65 yearsold. To live comfortably during retirement, you plan to withdraw moneyfrom your savings account 48,000eachyear.Supposetodayisyour25thbirthday,andyoudecide,startingtodayandcontinuingoneveryyearuptoyour64thbirthday,thatyouwillputthesameamountintoasavingsaccount.Iftheinterestrateis2.448,000 each year. Suppose today is your 25thbirthday, and you decide, starting today and continuing on every yearup to your 64th birthday, that you will put the same amount into a savingsaccount. If the interest rate is 2.4% per annum, compounded annually,how much must you set aside each year to make sure that you will haveenough money in the account on your 65th birthday in order to be ableto withdraw 48,000 per year when you retire and you expect to live untilyour are 90 years old (i.e., the first withdrawal is on your 65th birthday andthe last withdrawal occurs on your 89th birthday)?

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Solution

To solve this problem, we need to calculate two things: the total amount needed in the account at the time of retirement and the annual savings required to reach that amount.

  1. Calculate the total amount needed at retirement:

    This is a present value calculation because we are finding out how much we need in the account today (at age 65) to be able to withdraw $48,000 per year from age 65 to 89.

    The formula for present value is: PV = PMT * [(1 - (1 + r)^-n) / r]

    Where:

    • PMT is the annual withdrawal amount ($48,000)
    • r is the interest rate (2.4% or 0.024)
    • n is the number of years of withdrawal (90 - 65 + 1 = 25)

    Substituting the values into the formula, we get:

    PV = 48,000∗[(1−(1+0.024)−25)/0.024]PV=48,000 * [(1 - (1 + 0.024)^-25) / 0.024] PV = 48,000 * 16.834 PV = $808,032

  2. Calculate the annual savings required:

    This is a future value calculation because we are finding out how much we need to save each year to reach the future value (the total amount needed at retirement).

    The formula for future value is: FV = PMT * [((1 + r)^n - 1) / r]

    Rearranging the formula to solve for PMT, we get: PMT = FV / [((1 + r)^n - 1) / r]

    Where:

    • FV is the future value (the total amount needed at retirement, $808,032)
    • r is the interest rate (2.4% or 0.024)
    • n is the number of years of saving (65 - 25 = 40)

    Substituting the values into the formula, we get:

    PMT = 808,032/[((1+0.024)40−1)/0.024]PMT=808,032 / [((1 + 0.024)^40 - 1) / 0.024] PMT = 808,032 / 45.761 PMT = $17,664

So, you must set aside approximately 17,664eachyeartomakesurethatyouwillhaveenoughmoneyintheaccountonyour65thbirthdayinordertobeabletowithdraw17,664 each year to make sure that you will have enough money in the account on your 65th birthday in order to be able to withdraw 48,000 per year when you retire and live until you are 90 years old.

This problem has been solved

Similar Questions

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