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

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UpStudy AI · Accepted Answer

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

$$
\boxed{\$3\,866.60}.
$$
You can withdraw approximately \$3,866.60 each month.

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

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