Suppose you want to have $$ \$ 600,000 $$ for retirement in 35 years. Your account earns $$ 5 \% $$ interest. How much would you need to deposit in the account each month?

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UpStudy AI · Accepted Answer

We use the future value of an annuity formula given by

$$
FV = PMT \cdot \frac{(1+r)^n - 1}{r},
$$

where

- $$FV$$ is the future value (in this case, $$\$600,\!000$$),
- $$PMT$$ is the monthly deposit,
- $$r$$ is the monthly interest rate,
- $$n$$ is the total number of deposits.

**Step 1. Determine the parameters**

- The annual interest rate is $$5\%$$, so the monthly interest rate is

$$
  r = \frac{0.05}{12} \approx 0.00416667.
  $$

- The total number of months in 35 years is

$$
  n = 35 \times 12 = 420.
  $$

**Step 2. Solve for $$PMT$$**

Rearrange the future value formula to solve for $$PMT$$:

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

Substitute the known values:

$$
PMT = \frac{600,\!000 \cdot 0.00416667}{(1+0.00416667)^{420} - 1}.
$$

**Step 3. Compute $$(1+r)^n$$**

Calculate

$$
(1+0.00416667)^{420}.
$$

Taking natural logarithms,

$$
\ln(1.00416667) \approx 0.004158,
$$

so

$$
(1.00416667)^{420} = e^{420 \times 0.004158} \approx e^{1.74636} \approx 5.7375.
$$

Then,

$$
(1+r)^n - 1 \approx 5.7375 - 1 = 4.7375.
$$

**Step 4. Substitute and calculate $$PMT$$**

Now, the numerator is

$$
600,\!000 \times 0.00416667 = 2500.
$$

Thus

$$
PMT = \frac{2500}{4.7375} \approx 527.99.
$$

**Final Answer:**

You would need to deposit approximately $$\$528$$ each month.
You need to deposit approximately \$528 each month.

Suppose you want to have \( \$ 600,000 \) for retirement in 35 years. Your account earns \( 5 \% \) interest. How much would you need to deposit in the account each month?

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