To calculate the monthly withdrawal amount that can be sustained for 15 years with a 5% annual interest rate, we use the formula for the Present Value of an Ordinary Annuity:

$PV = \frac{PMT \cdot \left(1 - (1 + r/n)^{-n \cdot t}\right)}{r/n}$

Where:

• $PV$ is the present value ($500,000 in this case),
• $PMT$ is the monthly withdrawal amount (this is what we are solving for),
• $r$ is the annual interest rate (0.05 or 5%),
• $n$ is the number of compounding periods per year (12 for monthly),
• $t$ is the total time in years (15 years).

Step 1: Rearrange the formula to solve for $PMT$:

$PMT = \frac{PV \cdot r/n}{1 - (1 + r/n)^{-n \cdot t}}$

Step 2: Plug in the values:

• $PV = 500,000$,
• $r = 0.05$,
• $n = 12$,
• $t = 15$.

$PMT = \frac{500,000 \cdot (0.05/12)}{1 - (1 + 0.05/12)^{-12 \cdot 15}}$

Step 3: Perform the calculations:

1. $r/n = 0.05 / 12 = 0.004167$,
2. $n \cdot t = 12 \cdot 15 = 180$,
3. $1 + r/n = 1 + 0.004167 = 1.004167$,
4. $(1 + r/n)^{-n \cdot t} = 1.004167^{-180} \approx 0.40662$,
5. $1 - 0.40662 = 0.59338$,
6. Numerator: $500,000 \cdot 0.004167 = 2,083.33$,
7. Denominator: $0.59338$.

$PMT = \frac{2,083.33}{0.59338} \approx 3,511.49$

Final Answer:

You will be able to withdraw approximately $3,511.49 per month for 15 years.

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Would you like more details or have any questions? Here are some related questions to consider:

1. How does the withdrawal amount change if the interest rate increases to 6%?
2. What if the account is compounded quarterly instead of monthly?
3. How much would remain after 15 years if the monthly withdrawal was less than $3,511.49?
4. Can you sustain withdrawals indefinitely if the interest rate is higher than 5%?
5. How does inflation affect the purchasing power of these withdrawals over 15 years?

Tip: Regularly reviewing your retirement plan can help ensure sustainability in case of unexpected financial needs or market changes.