You have $$ \$ 300,000 $$ saved for retirement. Your account earns $$ 10 \% $$ interest. How much will you be able to pull out each month, if you want to be able to take withdrawals for 15 years? Check Answer

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UpStudy AI · Accepted Answer

We start with the formula for the present value of an annuity:

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

Here,  
- $$PV = \$300{,}000$$ is the amount saved for retirement.  
- The annual interest rate is $$10\%$$, so the monthly interest rate is

$$
r = \frac{0.10}{12} \approx 0.0083333.
$$

- The withdrawals are for $$15$$ years, so the number of monthly withdrawals is

$$
n = 15 \times 12 = 180.
$$

Solving for the monthly payment $$PMT$$, we rearrange the formula:

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

Substitute the known values:

$$
PMT = \frac{300{,}000 \times 0.0083333}{1 - (1+0.0083333)^{-180}}.
$$

Now, calculate the components step by step.

1. Calculate the numerator:

$$
300{,}000 \times 0.0083333 \approx 2{,}500.
$$

2. Calculate the denominator:
   - First, compute $$1 + r$$:

$$
1 + 0.0083333 = 1.0083333.
$$

- Raise it to the power $$-180$$:

$$
(1.0083333)^{-180}.
$$

To approximate this, note that

$$
(1.0083333)^{180} \approx e^{180 \ln(1.0083333)}.
$$

Since

$$
\ln(1.0083333) \approx 0.008298,
$$

we have:

$$
180 \times 0.008298 \approx 1.49364,
$$

and therefore:

$$
(1.0083333)^{180} \approx e^{1.49364} \approx 4.453.
$$

Then,

$$
(1.0083333)^{-180} \approx \frac{1}{4.453} \approx 0.2246.
$$

- Now, compute the denominator of the PMT formula:

$$
1 - (1.0083333)^{-180} \approx 1 - 0.2246 = 0.7754.
$$

3. Now, compute the payment:

$$
PMT \approx \frac{2{,}500}{0.7754} \approx 3{,}223.
$$

Thus, you will be able to withdraw approximately $$\$3{,}223$$ each month.
You can withdraw approximately \$3,223 each month.

You have saved for retirement. Your account earns interest. How much will you
be able to pull out each month, if you want to be able to take withdrawals for 15 years?
Check Answer

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You have saved for retirement. Your account earns interest. How much will you be able to pull out each month, if you want to be able to take withdrawals for 15 years? Check Answer