You are trying to calculate how much money you should have at retirement. On your 55th birthday you will retire and immediately make your first withdrawal of ​$4,615.38. You plan to make 26 such withdrawals each year. You plan to continue withdrawing at that level and frequency until you are 90 years old. ​(Assume beginning of period withdrawals with the first withdrawal on your retirement date and no withdrawal on your final​ birthday.) During retirement, your savings will earn only 3.5​% per annum. How much do you have to have saved at retirement to fund these planned​ withdrawals? 

This is a retirement planning problem, where you are seeking to calculate the present value of an annuity based on periodic withdrawals.

### Given information:
- You retire at age 55 and make your first withdrawal immediately.
- You make 26 withdrawals each year (bi-weekly).
- The annual withdrawal amount is \$4,615.38 per period.
- You plan to make these withdrawals until age 90, so the period is 35 years.
- The interest rate during retirement is 3.5% per annum.
- The withdrawals are made at the beginning of each period (beginning-of-period withdrawals).

### Step-by-Step Approach:

1. **Calculate the total number of withdrawals**:
   \[
   \text{Total number of withdrawals} = 26 \times 35 = 910
   \]

2. **Convert the annual interest rate to a rate per withdrawal**:
   The annual interest rate is 3.5%, but since you are withdrawing 26 times per year, we need the interest rate per period (withdrawal).
   \[
   r_{\text{periodic}} = \frac{3.5\%}{26} = \frac{0.035}{26} \approx 0.001346
   \]

3. **Present Value of an Annuity-Due Formula**:
   Since you are making withdrawals at the beginning of each period, we will use the formula for the present value of an annuity-due:
   \[
   PV = W \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
   \]
   Where:
   - $ W = 4,615.38 $ (the amount of each withdrawal),
   - $ r = 0.001346 $ (the interest rate per period),
   - $ n = 910 $ (the total number of withdrawals).

4. **Substitute the values into the formula**:

Let's compute the present value (how much you need to save at retirement). I'll calculate this next.You would need approximately **$2,423,828.18** saved at retirement to fund the planned withdrawals of $4,615.38, made bi-weekly, over 35 years with a 3.5% annual return.

Would you like more details on the calculations or have any other questions?

Here are 5 related questions to consider:
1. How would the amount change if the interest rate during retirement were higher or lower?
2. What would be the effect if you increased or decreased the withdrawal amount?
3. How does the timing of withdrawals (beginning vs. end of period) affect the total amount needed at retirement?
4. What if you plan to live longer or shorter, say until 95 or 85 years old?
5. How does increasing the number of withdrawals per year impact the total savings needed?

**Tip:** When planning for retirement, always consider inflation and how it might affect your withdrawal needs over time.

You are trying to calculate how much money you should have at retirement. On your 55th birthday you will retire and immediately make your first withdrawal of ​$4,615.38. You plan to make 26 such withdrawals each year. You plan to continue withdrawing at that level and frequency until you are 90 years old. ​(Assume beginning of period withdrawals with the first withdrawal on your retirement date and no withdrawal on your final​ birthday.) During retirement, your savings will earn only 3.5​% per annum. How much do you have to have saved at retirement to fund these planned​ withdrawals?

Learn how to calculate the amount of savings required at retirement to fund bi-weekly withdrawals of $4,615.38 over 35 years, with an annual interest rate of 3.5%. This problem involves concepts like the present value of an annuity-due, time value of money, and periodic interest rate conversion. Suitable for financial education or retirement planning.

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