Use the formula for present value of money to calculate the amount you need to invest now in one lump sum in order to have $1 million after 40 years with an APR of 6% compounded quarterly round your answer to the nearest cent if necessary

The formula for the present value of money (i.e., the amount you need to invest now) when compounded interest is applied is given by:

\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\]

Where:
- $ PV $ is the present value (the amount you need to invest now),
- $ FV $ is the future value (the amount you want in the future, which is $1,000,000 in this case),
- $ r $ is the annual interest rate (expressed as a decimal, so 6% becomes 0.06),
- $ n $ is the number of compounding periods per year (since it’s compounded quarterly, $ n = 4 $),
- $ t $ is the number of years (in this case, $ t = 40 $).

### Substituting the values into the formula:

\[
PV = \frac{1,000,000}{(1 + \frac{0.06}{4})^{4 \times 40}}
\]

Now, let's calculate this step by step.

### Step-by-step calculation:

1. The quarterly interest rate is $ \frac{0.06}{4} = 0.015 $.
2. The total number of compounding periods is $ 4 \times 40 = 160 $.
3. The growth factor is $ (1 + 0.015)^{160} = (1.015)^{160} $.

Let me calculate that for you.The amount you need to invest now in one lump sum to have $1,000,000 after 40 years with an APR of 6%, compounded quarterly, is approximately **$92,349.22**.

Do you have any further questions or would you like a detailed breakdown of the calculation?

Here are some related questions to explore:

1. How does changing the compounding frequency (monthly or annually) affect the required investment?
2. What if the APR is 5% instead of 6%? How much would you need to invest?
3. How much more would you need to invest if the investment horizon is only 30 years?
4. How does continuous compounding change the present value formula and result?
5. What is the future value if you invest $92,349.22 now with 6% APR compounded quarterly?

**Tip**: The more frequently interest is compounded, the smaller the present value required for the same future amount.

Learn how to calculate the present value of an investment to reach $1 million in 40 years with a 6% APR compounded quarterly. This problem uses the time value of money formula and applies compound interest principles. Suitable for advanced high school and college students.

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