You have \( \$ 500,000 \) saved for retirement. Your account earns \( 8 \% \) interest. How much will you be able to pull out each month, if you want to be able to take withdrawals for 25 years? Question Help: \( \square \) Video 1 Video 2 Video 3 Message instructor Submit Question

We start with the formula for the present value of an annuity: \[ PV = PMT \cdot \frac{1 - \left(1 + r\right)^{-n}}{r} \] where - \(PV\) is the present amount (\$500,000), - \(PMT\) is the monthly withdrawal amount, - \(r\) is the monthly interest rate, - \(n\) is the total number of withdrawals (months). Our goal is to solve for \(PMT\). ### Step 1. Identify the values - Annual interest rate is \(8\%\). Therefore, the monthly interest rate is \[ r = \frac{0.08}{12} \approx 0.006666667. \] - The number of years is 25, so the number of months is \[ n = 25 \times 12 = 300. \] ### Step 2. Write the formula for \(PMT\) Starting with \[ PV = PMT \cdot \frac{1 - \left(1 + r\right)^{-n}}{r}, \] we solve for \(PMT\): \[ PMT = PV \cdot \frac{r}{1 - \left(1 + r\right)^{-n}}. \] ### Step 3. Substitute the given values Substitute \(PV = 500\,000\), \(r = \frac{0.08}{12}\), and \(n = 300\): \[ PMT = 500\,000 \cdot \frac{\frac{0.08}{12}}{1 - \left(1 + \frac{0.08}{12}\right)^{-300}}. \] ### Step 4. Simplify and compute numerically 1. **Calculate \(r\):** \[ r = \frac{0.08}{12} \approx 0.006666667. \] 2. **Compute \(\left(1 + r\right)^{-300}\):** First, compute \(\left(1 + r\right) = 1.006666667\). Then, \[ \left(1.006666667\right)^{300} \approx 7.293, \] so \[ \left(1.006666667\right)^{-300} \approx \frac{1}{7.293} \approx 0.1371. \] 3. **Calculate the denominator:** \[ 1 - \left(1 + r\right)^{-300} \approx 1 - 0.1371 = 0.8629. \] 4. **Calculate the numerator:** \[ 500\,000 \times 0.006666667 \approx 3\,333.33. \] 5. **Compute \(PMT\):** \[ PMT \approx \frac{3\,333.33}{0.8629} \approx 3\,866.6. \] ### Final Answer The amount you can withdraw each month is approximately \[ \boxed{\$3\,866.60}. \]