Question

You just turned 28. Happy birthday! Now that you’re old, you want to be responsible and start a retirement fund. You plan to invest $465 every month for the next 37 years. You expect to earn a 7.7% annual return until you retire. Once you retire, you will shift your savings into safer investments that you expect will earn 6% per year. You plan to make monthly withdrawals for 23 years of retirement (and don't want to have any money left at the end). Under this plan, how much money will you withdraw from your retirement account each month? (NOTE: Round your answer to the nearest cent)

Answer 1

Here investment made monthly for a fixed amount will be in the form of an annuity

The formula for future value of annuity is ,

Future value of annuity is P*((1+r)ⁿ - 1)/r

Where P = Equalised periodic payment

r = rate of interest

n = Number of periods.

P = $465

Total Years is 37 years , Hence total months is 37 * 12 = 444 months

r = Rate of interest 7.7% per annum

Hence per month = 0.641667 %

Hence the future value = 465 * ((1.006417)⁴⁴⁴-1)/0.006417

= 465 * 16.11403/0.006417

= 465 * 2511.277

= 1167744

Hence the future value will be $1167744.

This amount will generate monthly fixed cash flows for a period of 23 years

Total number of months = 23 * 12 = 276 Months

Here all these retirement cash flows will be at a present value of 1167744. Because this amount invested is generating monthly returns for 23 years

The formula for present value of ordinary annuity when he withdraws at the end of each month.

Present value of annuity - P*((1-(1+r)^-n)/r

Where P = Equalised periodic payment

r = rate of interest

n = Number of periods'

Given annual interest rate is 6% per year . Hence the monthly interest rate is 0.5% nothing but 0.005%

Total Years is 23 years = 23*12 = 276 Months

Let us substitute this is the present value of annuity formula

1167744 = P * ((1 - (1.005)^-276)/0.005

1167744 = P * ((1 - 0.252445)/0.005

1167744 = P * (0.747555)/0.005

1167744 = P* 149.511

P = 7810.422

Hence The monthly withdrawal amount will be $7810.422. provided if he withdraws at the end of each month

For Annuity Due - P + P(1-(1+r)^-(n-1))/r

P + P ( 1- (1.005)^-275)/0.005

P + P (1- 0.253707)/0.005

P + P (0.746293)/0.005

P + P * 149.2585 = 1167744

P * (150.2585) = 1167744

P = 7771.565

Hence The monthly withdrawal amount will be $7771.565. provided if he withdraws at the begining of each month