Question

You are planning to save for retirement over the next 25 years. To do this, you will invest $3,000 a quarter in a stock account and $1,000 a quarter in a bond account. These investments will be made at the beginning of each quarter. The return of the stock account is expected to be 8%, and the bond account will pay 4%. When you retire, you will combine your money into an account with a 6% return. How much can you withdraw each month (starting one month from the retirement date) from your account assuming a 20-year withdrawal period?

*please show work

Answer 1

Assuming rate compounded quarterly before retirement and rate compounded monthly after retirement for simplicity

Step 1:
As the savings are beginning of period it is an annuity due
Future value of annuity due=AMount/(periodic rate)*((1+periodic rate)^n-1)*(1+periodic rate)

Step a) Find the future value of stock account
=3000/(8%/4)*((1+8%/4)^(4*25)-1)*(1+8%/4)
=955430.8561

Step b) Find the future value of bond account
=1000/(4%/4)*((1+4%/4)^(4*25)-1)*(1+4%/4)
=172186.1968

Step c) Find total future value
=955430.8561+172186.1968
=1127617.053

Step 2: Find monthly withdrawals
As the withdrawals are end of the period it is an ordinary annuity
Periodic payments in ordinary annuity=Value*periodic rate/(1-1/(1+periodic rate)^n)
=1127617.053*(6%/12)/(1-1/(1+6%/12)^(12*20))
=8078.598788