ou have $$ \$ 400,000 $$ saved for retirement. Your account earns $$ 9 \% $$ interest. How much will you be able to ut each month, if you want to be able to take withdrawals for 20 years?

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We start with the present value of an annuity formula that relates the withdrawal amount $$PMT$$ to the present value $$PV$$:

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

Here:
- $$PV = 400,\!000$$
- The annual interest rate is $$9\%$$, so the monthly interest rate is

$$
r = \frac{0.09}{12} = 0.0075.
$$

- The number of months over 20 years is

$$
n = 20 \times 12 = 240.
$$

To solve for $$PMT$$, we rearrange the formula:

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

Substitute the known values:

$$
PMT = 400,\!000 \times \frac{0.0075}{1 - (1.0075)^{-240}}.
$$

**Step 1. Compute $$(1.0075)^{-240}$$:**

First, compute $$(1.0075)^{240}$$. Often it is easier to use logarithms:

$$
(1.0075)^{240} = e^{240 \ln(1.0075)}.
$$

An approximate value for $$\ln(1.0075)$$ is about $$0.007471$$. Then:

$$
240 \ln(1.0075) \approx 240 \times 0.007471 \approx 1.79304,
$$

so

$$
(1.0075)^{240} \approx e^{1.79304} \approx 6.0128.
$$

Thus,

$$
(1.0075)^{-240} \approx \frac{1}{6.0128} \approx 0.1662.
$$

**Step 2. Compute the denominator:**

$$
1 - (1.0075)^{-240} \approx 1 - 0.1662 = 0.8338.
$$

**Step 3. Compute $$PMT$$:**

$$
PMT \approx 400,\!000 \times \frac{0.0075}{0.8338} \approx 400,\!000 \times 0.0090 \approx 3,\!600.
$$

Thus, you will be able to withdraw approximately $$\$3,\!600$$ per month for 20 years.
You can withdraw approximately \$3,600 each month for 20 years.

ou have saved for retirement. Your account earns interest. How much will you be able to
ut each month, if you want to be able to take withdrawals for 20 years?

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ou have saved for retirement. Your account earns interest. How much will you be able to ut each month, if you want to be able to take withdrawals for 20 years?