This problem involves solving for the initial deposit required to fund a retirement account such that it allows $45,000 withdrawals annually for 25 years, with a 5% annual interest rate, and contributions are made for 15 years prior to retirement.

To solve this, we need to follow these steps:

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Step 1: Calculate the Present Value of Withdrawals (at retirement)

The annual withdrawals form a 25-year annuity. The present value $PV_{\text{ret}}$ of these withdrawals at the time of retirement can be calculated using the formula for the present value of an annuity:

$PV_{\text{ret}} = W \cdot \frac{1 - (1 + r)^{-n}}{r}$

Where:

• $W = 45,000$ (annual withdrawal),
• $r = 0.05$ (annual interest rate),
• $n = 25$ (number of years of withdrawals).

$PV_{\text{ret}} = 45,000 \cdot \frac{1 - (1 + 0.05)^{-25}}{0.05}$

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Step 2: Find the Present Value of $PV_{\text{ret}}$ at the Start of Savings

To bring $PV_{\text{ret}}$ to the present day (15 years before retirement), we discount it further using the formula for present value:

$PV_0 = PV_{\text{ret}} \cdot (1 + r)^{-t}$

Where:

• $t = 15$ years,
• $r = 0.05$.

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Step 3: Determine the Annual Deposits

The retirement contributions form an ordinary annuity made over 15 years. The amount of annual deposit $D$ can be found using the future value of an annuity formula:

$FV = D \cdot \frac{(1 + r)^n - 1}{r}$

Here:

• $FV = PV_0$,
• $r = 0.05$,
• $n = 15$.

Rearrange this equation to solve for $D$:

$D = \frac{FV \cdot r}{(1 + r)^n - 1}$

Let’s compute each step. Given values W = 45000 # Annual withdrawal amount r = 0.05 # Annual interest rate n_withdrawals = 25 # Years of withdrawals n_savings = 15 # Years of savings

Step 1: Calculate PV_ret (Present Value at retirement)

PV_ret = W * (1 - (1 + r)**-n_withdrawals) / r

Step 2: Discount PV_ret back to present day (15 years before retirement)

PV_0 = PV_ret / (1 + r)**n_savings

Step 3: Calculate annual deposits (D) to save for retirement

D = PV_0 * r / ((1 + r)**n_savings - 1)

PV_ret, PV_0, D### Solution

1. The present value of the 25 years of withdrawals at the time of retirement is approximately:

   $PV_{\text{ret}} = 634,227.51$

2. The equivalent present value 15 years before retirement is:

   $PV_0 = 305,074.27$

3. The annual deposit required for 15 years to achieve this amount is:

   $D = 14,137.84$

You would need to deposit approximately $14,137.84 annually into your retirement account over the next 15 years.

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Follow-up Questions:

1. How would the required deposit change if the interest rate increased to 6%?
2. What is the total amount deposited over the 15 years of savings?
3. How does the total withdrawal amount compare to the total deposit amount?
4. What is the effective annual interest earned on the deposits?
5. How would an earlier retirement (e.g., 10 years of saving) impact the required annual deposit?

Tip:

Breaking down problems into clear stages (future value, present value, and annuity calculations) simplifies complex financial questions.