This is a common issue that people face when they retire. A good friend of your family is about to become another year older, and she has decided that she wants to start putting money away for her eventual retirement. She has the following number of years till retirement and she has the following expenditure objectives for retirement: Years until retirement 20 Amount to withdraw each year $50,000 Years to withdraw in retirement 15 Interest rate 7.5% Because your friend is planning ahead, the first withdrawal will not take place until one year after she retires. She wants to make equal annual deposits into her account for her retirement fund. If she starts making these deposits in one year and makes her last deposit in the day she retires, what amount must she deposit annually to be able to make the desired withdrawals at retirement?


To solve this problem, we need to determine the amount your friend needs to deposit annually to meet her retirement goals. This is a typical retirement savings problem involving future value and present value concepts.

### Problem Breakdown

1. **Years until retirement:** 20 years
2. **Annual withdrawal amount during retirement:** $50,000
3. **Years of withdrawals:** 15 years
4. **Interest rate:** 7.5% per year

### Step-by-Step Solution

1. **Calculate the Present Value (PV) of the Withdrawals:**

   The goal is to find the amount needed at the start of retirement (20 years from now) to be able to withdraw $50,000 annually for 15 years. The withdrawals start one year after retirement, and the interest rate remains at 7.5%.

   The formula for the present value of an annuity is:

   \[
   PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}
   \]

   Where:
   - $ PMT = 50,000 $ (annual withdrawal)
   - $ r = 0.075 $ (interest rate)
   - $ n = 15 $ (number of withdrawals)

   Plugging in the values:

   \[
   PV = 50,000 \times \frac{1 - (1 + 0.075)^{-15}}{0.075}
   \]

2. **Calculate the Annual Deposit:**

   Now, calculate the annual deposit required for 20 years to reach the amount calculated above. The formula for the future value of an annuity is:

   \[
   FV = PMT \times \frac{(1 + r)^n - 1}{r}
   \]

   To find the required deposit ($ PMT_{deposit} $):

   \[
   PMT_{deposit} = \frac{FV}{\frac{(1 + r)^n - 1}{r}}
   \]

   Where:
   - $ FV $ is the future value of the retirement account calculated in the first step.
   - $ r = 0.075 $ (interest rate)
   - $ n = 20 $ (number of deposits)

Let's calculate the exact values for these steps.1. **Present Value of Withdrawals:** To fund her retirement, your friend needs approximately **$441,356** at the beginning of her retirement.

2. **Annual Deposit Required:** She needs to make annual deposits of approximately **$10,191.88** over the next 20 years to achieve this amount.

Would you like more details on this calculation, or do you have any questions?

### Related Questions:
1. How does changing the interest rate affect the required annual deposit?
2. What if the number of withdrawal years changes to 20? How would that affect the deposits?
3. How would starting the withdrawals immediately after retirement alter the calculations?
4. What is the impact of making semi-annual deposits instead of annual deposits?
5. How would inflation affect the amount she needs to withdraw annually?

**Tip:** Regularly reviewing and adjusting your savings plan based on changes in interest rates or financial goals can ensure you stay on track for your retirement needs.

This is a common issue that people face when they retire. A good friend of your family is about to become another year older, and she has decided that she wants to start putting money away for her eventual retirement. She has the following number of years till retirement and she has the following expenditure objectives for retirement: Years until retirement 20 Amount to withdraw each year $50,000 Years to withdraw in retirement 15 Interest rate 7.5% Because your friend is planning ahead, the first withdrawal will not take place until one year after she retires. She wants to make equal annual deposits into her account for her retirement fund. If she starts making these deposits in one year and makes her last deposit on the day she retires, what amount must she deposit annually to be able to make the desired withdrawals at retirement?

Learn how to calculate the annual deposits required to achieve your retirement savings goals. This problem involves concepts like future value, present value, and annuities, applying a 7.5% interest rate over 20 years to ensure sufficient withdrawals of $50,000 annually during 15 years of retirement. Suitable for Grades 11-12 or college-level students.

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