You would like to have $800,000 when you retire in 40 years. How much should you invest each quarter if you can earn a rate of 7.3% compounded quarterly?

a) How much should you deposit each quarter?

$

b) How much total money will you put into the account?

$

c) How much total interest will you earn?

$

We are dealing with an investment where regular payments are made into an account earning compound interest. The goal is to calculate how much should be invested each quarter to reach a future value of $800,000 in 40 years, with quarterly compounding at an annual interest rate of 7.3%.

We can use the future value of an ordinary annuity formula:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $FV$ is the future value ($800,000).
- $P$ is the quarterly payment (what we are solving for).
- $r$ is the quarterly interest rate.
- $n$ is the total number of payments.

### Step-by-step breakdown:

1. **Interest rate per period** ($r$):
   Since the interest is compounded quarterly, the interest rate per period is:
   \[
   r = \frac{7.3\%}{4} = \frac{0.073}{4} = 0.01825
   \]

2. **Total number of periods** ($n$):
   Since there are 4 quarters per year and the investment lasts for 40 years:
   \[
   n = 40 \times 4 = 160
   \]

3. **Rearranging the formula to solve for $P$**:
   \[
   P = \frac{FV}{\frac{(1 + r)^n - 1}{r}}
   \]

4. **Substitute values into the formula**:
   \[
   P = \frac{800,000}{\frac{(1 + 0.01825)^{160} - 1}{0.01825}}
   \]

### Calculation:

I'll compute this now.a) The amount you should deposit each quarter is approximately **$855.83**.

Next, let's calculate the total amount of money you will put into the account over 40 years.

This can be done by multiplying the quarterly payment by the total number of periods ($n = 160$):

\[
\text{Total deposits} = P \times n
\]b) The total amount of money you will put into the account is approximately **$136,932.07**.

Now, let's calculate the total interest earned by subtracting the total deposits from the future value:

\[
\text{Total interest} = FV - \text{Total deposits}
\]c) The total interest you will earn is approximately **$663,067.93**.

Would you like more details or have any other questions?

Here are 5 related questions you might find useful:
1. What if the interest rate was compounded monthly instead of quarterly?
2. How does the required deposit change if you want to retire in 30 years instead of 40?
3. What would happen if the interest rate were lower, say 5%?
4. How much would you need to invest in a lump sum to reach the same $800,000 goal?
5. How would inflation affect the real value of $800,000 over 40 years?

**Tip:** The longer the investment period and the higher the compounding frequency, the more you benefit from compound interest growth.

You would like to have $800,000 when you retire in 40 years. How much should you invest each quarter if you can earn a rate of 7.3% compounded quarterly?

Learn how to calculate the quarterly investment needed to accumulate $800,000 in 40 years with a 7.3% interest rate compounded quarterly. This detailed solution walks through the use of the future value of annuities formula, interest calculations, and total deposits and interest earned.

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