You wish to retire at 60 and, at the end of each month thereafter for 25 years, to receive $6,000. Assume that you begin making monthly payments into an account at age 23 and continue these payments until age 60. If the annual interest rate is constant at 12 percent, how much must be deposited monthly between ages 23 and 60 for you to receive your $6,000/month for 25 years? Assume monthly compounding throughout.

I am having difficulty with this problem because once you reach the age of 60, the sum of money you have saved between 23 and 60 (whatever amount that is) keeps accruing interest (at 1 % percent a month) while you you withdraw $6000 a month. I don't know how to find the sum of money you need to save between 23 and 60.

1. calculate the Present worth of those monthly payments of 6K for 25 years. That will be the nest egg you need at age 60.

2. calculate the future value of the monthly payments beginning at age 23 and ending at age 60 (37 years?).

Now your question. If you have a nest egg at 60, its future value is automatically calculated as you had figured in 1) above.

To find the monthly deposit needed between ages 23 and 60, consider the following steps:

Step 1: Calculate the total number of months between ages 23 and 60.
Since there are 12 months in a year and you want to retire at 60, the total number of months is (60 - 23) × 12.

Step 2: Determine the future value (FV) of the monthly deposits using the formula for future value of an ordinary annuity with monthly compounding:
FV = PMT × [(1 + r)^(n) - 1] / r
Where:
- PMT is the monthly deposit
- r is the monthly interest rate (12% / 12)
- n is the total number of months

Step 3: Calculate the future value (FV) of the monthly withdrawals using the same future value formula, but with the monthly withdrawals as the monthly deposits, and n being the number of months in 25 years (25 × 12).

Step 4: Set up an equation to find the monthly deposit (PMT) that will result in an equal future value for both the deposits and withdrawals:
FV_deposits = FV_withdrawals

Step 5: Solve the equation for PMT, which will give you the monthly deposit needed.

Let's go through the calculations:

Step 1:
Total number of months = (60 - 23) × 12 = 37 × 12 = 444 months.

Step 2:
Using the given interest rate of 12% and monthly compounding:
r = 12% / 12 = 1% = 0.01
n = 444 months

FV_deposits = PMT × [(1 + r)^(n) - 1] / r

Step 3:
Since the withdrawals start at age 60 and continue for 25 years (25 × 12 months), the future value of the withdrawals can be calculated using the same formula:
FV_withdrawals = $6,000 × [(1 + r)^(n) - 1] / r
Where $6,000 is the monthly withdrawal amount.

Step 4:
Set the FV_deposits equal to FV_withdrawals:
PMT × [(1 + r)^(n) - 1] / r = $6,000 × [(1 + r)^(n) - 1] / r

Step 5:
Solve for PMT:
PMT = [$6,000 × r] / [(1 + r)^(n) - 1]

Inserting the values:
PMT = [$6,000 × 0.01] / [(1 + 0.01)^(444) - 1] ≈ $235.67

Therefore, you would need to deposit approximately $235.67 per month between the ages of 23 and 60 in order to receive $6,000 per month for 25 years after you retire.