Charlie is retiring this year and has a $400,000 retirement fund to draw from th
ID: 2774169 • Letter: C
Question
Charlie is retiring this year and has a $400,000 retirement fund to draw from that has an NAR of 2.25% compounded monthly. a If Charlie plans to withdraw $2,000 at the end of each month, how many years would it last? b How many years will the fund last if Charlie withdraws $100,000 up front to pay for a retirement condominium, wants $30,000 in the account at the end, and withdraws $1,500 at the beginning of each month. Charlie is retiring this year and has a $400,000 retirement fund to draw from that has an NAR of 2.25% compounded monthly. a If Charlie plans to withdraw $2,000 at the end of each month, how many years would it last? b How many years will the fund last if Charlie withdraws $100,000 up front to pay for a retirement condominium, wants $30,000 in the account at the end, and withdraws $1,500 at the beginning of each month.Explanation / Answer
Answer (1)
The amount would last for 251 months.
working
Amount available in retirement fund P = $400,000
Nominal Annual Rate = 2.25%
Monthly interest rate r = 2.25%/12 = 0.001875
Monthly withdrawal A = $ 2,000
Let n be the total number of months to make the retirement fund value to be zero, then n can be found out from the below equaltion
Total Amount in retirement fund P = Monthly Withdrawal A * [(1+r)^n -1) /r]
Substituting the above values
400000 = 2000 * [1-(1+0.001875)^-n/0.001875]
400000 * 0.001875 = 2000 – 2000 * 1.001875^-n
750 = 2000 – 2000 * 1.001875^-n
750-2000 = - 2000 * 1.001875^-n
-1250 = -2000 * 1.001875^-n
1.001875^-n = 0.625
Taking logarithms on both sides
log 1.001875^-n = log 0.625
-n * log 1.001875 = log 0.625
- n = log 0.625 / log 1.001875
- n = - 0.2041199827/0.00081353969
n = 250.903533 or 251 months (rounded off)
Thus the amount would last for 251 months.
Answer (2)
If the requirements of Charlie are met an amount $1,500 can be withdrawn at the beginning of the month for a period of 230 months
The retired person needs $ 100,000 upfront for purchase of a condominium, $ 30000 to be available finally and be able to withdraw $ 1500 per month at the beginning of the month
Let n be the number of months the fund can be withdrawn. Then the above details result in an equation
400000 = 100000 + 1500 +1500 * [1-(1.001875)^-(n-1)]/0.001875 + 30000 * (1.001875)^-n
298500 = (1500/0.001875) * (1-1.001875^-(n-1)) + 30000* (1.001875^-n)
298500 = 800000 * 1 – 800000 * 1.001875^-(n-1) + 30000* 1.001875^-n
298500 – 800000 = 800000*1.001875^-(n-1) + 30000 * 1.001875^-n
501500 = 800000*1.001875^-(n-1) + 30000 * 1.001875^-n
501500 - 800000*1.001875^-(n-1) - 30000 * 1.001875^n = 0
501500 – 800000 * 1.001875 (1-n) – 30000 * 1.001875^-n = 0
Applying the exponential rule a^(b+c) = a^b * a^c
501500 – 800000 . 1.001875 * 1.001875^-n – 30000 * 1.001875^-n = 0
501500 – 801500*1.001875^-n – 30000 * 1.001875^-n = 0
501500 – 1.001875^-n * (801500 – 30000) = 0
-1.001875^-n * 771500 = - 501500
1.001875^-n = 501500/771500 --> 1.001875^-n = 0.6500324
Taking logarithms on both sides
-n log 1.001875 = log 0.6500324 --> - n = log 0.6500324 / log 1.001875
- n = -0.18706499/0.00081353969
n = 229.939 or 230 months (rounded off)